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-算法(Algorithm):是指解题方案的准确而完整的描述,是一系列解决问题的清晰指令,算法代表着用系统的方法描述解决问题的策略机制。也就是说,能够对一定规范的输入,在有限时间内获得所要求的输出。 - -数据结构(Data Structures):是计算机存储和组织数据的一种方式,可以用来高效地处理数据。 - -举个例子:二分查找就是一个非常经典的算法,而二分查找经常需要作用在一个有序数组上。这里二分就是一种折半的算法思想, -而数组是我们最常用的一种数据结构,支持根据下标快速访问。很多算法需要特定的数据结构来实现,所以经常把它们放到一块讲。 - -实际上,在真正的项目开发中,大部分时间都是 从数据库取数据 -> 数据操作和结构化 -> 返回给前端,在数据操作过程中需要合理地抽象, -组织、处理数据,如果选用了错误的数据结构,就会造成代码运行低效。这也是我们需要学习算法和数据结构的原因。 - -# 笨方法学算法 -这里我们用一种很原始的『笨』方法来学习算法:纸笔模拟。 - -- 阅读资料了解算法思想 -- 纸笔模拟尝试理解 -- 用自己熟悉的编程语言来实现 -- 单测 - -# 小问题 - -- 你还知道哪些经典的算法和数据结构? -- 学习算法你觉得需要哪些预备知识? -- 我们的业务代码开发中会涉及到算法吗? -- 你了解 redis 吗,你知道它有哪几个常用的数据结构吗?你知道它的底层实现方式吗? diff --git "a/docs/01_\346\212\275\350\261\241\346\225\260\346\215\256\347\261\273\345\236\213\345\222\214\351\235\242\345\220\221\345\257\271\350\261\241\347\274\226\347\250\213/ADT_OOP.md" "b/docs/01_\346\212\275\350\261\241\346\225\260\346\215\256\347\261\273\345\236\213\345\222\214\351\235\242\345\220\221\345\257\271\350\261\241\347\274\226\347\250\213/ADT_OOP.md" deleted file mode 100644 index 31748b5..0000000 --- "a/docs/01_\346\212\275\350\261\241\346\225\260\346\215\256\347\261\273\345\236\213\345\222\214\351\235\242\345\220\221\345\257\271\350\261\241\347\274\226\347\250\213/ADT_OOP.md" +++ /dev/null @@ -1,52 +0,0 @@ -# Python 一切皆对象 - -举个例子,在 python 中我们经常使用的 list - -```py -l = list() # 实例化一个 list 对象 l -l.append(1) # 调用 l 的 append 方法 -l.append(2) -l.remove(1) -print(len(l)) # 调用对象的 `__len__` 方法 -``` - -在后面实现新的数据类型时,我们将使用 python 的 class 实现,它包含属性和方法。 -属性一般是使用某种特定的数据类型,而方法一般是对属性的操作。 -这里你只需了解这么多就行了, 我们不会使用继承等特性。 - - -# 什么是抽象数据类型 ADT - -实际上 python 内置的 list 就可以看成一种抽象数据类型。 - -ADT: Abstract Data Type,抽象数据类型,我们在组合已有的数据结构来实现一种新的数据类型, ADT 定义了类型的数据和操作。 - -我们以抽象一个背包(Bag) 数据类型来说明,背包是一种容器类型,我们可以给它添加东西,也可以移除东西,并且我们想知道背包里 -有多少东西。于是我们可以定义一个新的数据类型叫做 Bag. - -```py -class Bag: - """ 背包类型 """ - pass -``` - - -# 实现一个 Bag ADT -视频中我们将使用 python 的 class 来实现一个新的容器类型叫做 Bag。 - - -# 实现 ADT 我们应该注意什么? -- 如何选用恰当的数据结构作为存储? -- 选取的数据结构能否满足 ADT 的功能需求 -- 实现效率如何? - - -# 小问题: -- 你了解 python 的魔术方法吗? 比如 `__len__` ,调用 len(l) 的时候发生了什么? -- 你了解单测吗?我们以后将使用 pytest 运行单元测试,保证我们实现的数据结构和算法是正确的。你可以网上搜索下它的简单用法 - -# 延伸阅读: - -[数据结构与算法--ADT](http://www.atjiang.com/data-structures-using-python-ADT/) - -[http://www.nhu.edu.tw/~chun/CS-ch12-Abstract%20Data%20Types.pdf](http://www.nhu.edu.tw/~chun/CS-ch12-Abstract%20Data%20Types.pdf) diff --git "a/docs/01_\346\212\275\350\261\241\346\225\260\346\215\256\347\261\273\345\236\213\345\222\214\351\235\242\345\220\221\345\257\271\350\261\241\347\274\226\347\250\213/bag_adt.py" "b/docs/01_\346\212\275\350\261\241\346\225\260\346\215\256\347\261\273\345\236\213\345\222\214\351\235\242\345\220\221\345\257\271\350\261\241\347\274\226\347\250\213/bag_adt.py" deleted file mode 100644 index 4e5efc8..0000000 --- "a/docs/01_\346\212\275\350\261\241\346\225\260\346\215\256\347\261\273\345\236\213\345\222\214\351\235\242\345\220\221\345\257\271\350\261\241\347\274\226\347\250\213/bag_adt.py" +++ /dev/null @@ -1,43 +0,0 @@ -# coding: utf8 - - -class Bag(object): - - def __init__(self, maxsize=10): - self.maxsize = maxsize - self._items = list() - - def add(self, item): - if len(self) >= self.maxsize: - raise Exception('Full') - self._items.append(item) - - def remove(self, item): - self._items.remove(item) - - def __len__(self): - return len(self._items) - - def __iter__(self): - for item in self._items: - yield item - - -def test_bag(): - bag = Bag() - - bag.add(1) - bag.add(2) - bag.add(3) - - assert len(bag) == 3 - - bag.remove(3) - assert len(bag) == 2 - - for i in bag: - print(i) - - -if __name__ == '__main__': - test_bag() diff --git "a/docs/02_\346\225\260\347\273\204\345\222\214\345\210\227\350\241\250/array_and_list.md" "b/docs/02_\346\225\260\347\273\204\345\222\214\345\210\227\350\241\250/array_and_list.md" deleted file mode 100644 index 21b5d90..0000000 --- "a/docs/02_\346\225\260\347\273\204\345\222\214\345\210\227\350\241\250/array_and_list.md" +++ /dev/null @@ -1,54 +0,0 @@ -# 线性结构 -本节我们从最简单和常用的线性结构开始,并结合 Python 语言本身内置的数据结构和其底层实现方式来讲解。 -虽然本质上数据结构的思想是语言无关的,但是了解 Python 的实现方式有助于你避免一些坑。 - -我们会在代码中注释出操作的时间复杂度。 - - -# 数组 array - -数组是最常用到的一种线性结构,其实 python 内置了一个 array 模块,但是大部人甚至从来没用过它。 -Python 的 array 是内存连续、存储的都是同一数据类型的结构,而且只能存数值和字符。 - -我建议你课下看下 array 的文档:https://docs.python.org/2/library/array.html - -你可能很少会使用到它(我推荐你用 numpy.array),我将在视频里简单介绍下它的使用和工作方式,最常用的还是接下来要说的 list, -本章最后我们会用 list 来实现一个固定长度、并且支持所有 Python 数据类型的数组 Array. - - -# 列表 list -如果你学过 C++,list 其实和 C++ STL(标准模板库)中的 vector 很类似,它可能是你的 Python 学习中使用最频繁的数据结构之一。 -这里我们不再去自己实现 list,因为这是个 Python 提供的非常基础的数据类型,我会在视频中讲解它的工作方式和内存分配策略, -避免使用过程中碰到一些坑。当然如果你有毅力或者兴趣的了解底层是如何实现的,可以看看 cpython 解释器的具体实现。 - - -操作 | 平均时间复杂度 | ---------------------------------------|----------------| -list[index] | O(1) | -list.append | O(1) | -list.insert | O(n) | -list.pop(index), default last element | O(1) | -list.remove | O(n) | - -![](./list.png) - -# 用 list 实现 Array ADT -讲完了 list 让我们来实现一个定长的数组 Array ADT,在其他一些语言中,内置的数组结构就是定长的。 -这里我们会使用 list 作为 Array 的一个成员(代理)。具体请参考视频讲解和代码示例,后边我们会使用到这个 Array 类。 - - -# 小问题 -- 你知道线性结构的查找,删除,访问一个元素的平均时间复杂度吗?(后边我们会介绍这个概念,现在你可以简单地理解为一个操作需要的平均步骤) -- list 内存重新分配的时候为什么要有冗余?不会浪费空间吗? -- 当你频繁的pop list 的第一个元素的时候,会发生什么?如果需要频繁在两头增添元素,你知道更高效的数据结构吗?后边我们会讲到 - - -# 延伸阅读 - -[Python list implementation](https://www.laurentluce.com/posts/python-list-implementation/) - -[https://github.com/python/cpython/blob/master/Objects/listobject.c](https://github.com/python/cpython/blob/master/Objects/listobject.c) - - -# 勘误 -视频里的 Array.clear 方法有误。应该是 `for i in range(len(self._items))`,已经在后续所有使用到 Array 的代码里修正 diff --git "a/docs/02_\346\225\260\347\273\204\345\222\214\345\210\227\350\241\250/array_and_list.py" "b/docs/02_\346\225\260\347\273\204\345\222\214\345\210\227\350\241\250/array_and_list.py" deleted file mode 100644 index 11072a4..0000000 --- "a/docs/02_\346\225\260\347\273\204\345\222\214\345\210\227\350\241\250/array_and_list.py" +++ /dev/null @@ -1,45 +0,0 @@ -# -*- coding: utf-8 -*- - -# https://docs.python.org/2/library/array.html -from array import array # python 提供的比较原始的 array 类 - - -arr = array('u', 'asdf') - -print(arr[0], arr[1], arr[2], arr[3]) - - -# 实现定长的 Array ADT,省略了边界检查等 - -class Array(object): - - def __init__(self, size=32): - self._size = size - self._items = [None] * size - - def __getitem__(self, index): - return self._items[index] - - def __setitem__(self, index, value): - self._items[index] = value - - def __len__(self): - return self._size - - def clear(self, value=None): - for i in range(len(self._items)): - self._items[i] = value - - def __iter__(self): - for item in self._items: - yield item - - -def test_array(): - size = 10 - a = Array(size) - a[0] = 1 - assert a[0] == 1 - assert len(a) == 10 - -# py.test array_and_list.py diff --git "a/docs/02_\346\225\260\347\273\204\345\222\214\345\210\227\350\241\250/list.png" "b/docs/02_\346\225\260\347\273\204\345\222\214\345\210\227\350\241\250/list.png" deleted file mode 100644 index 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save memory - - def __init__(self, value=None, prev=None, next=None): - self.value, self.prev, self.next = value, prev, next - - -class CircularDoubleLinkedList(object): - """循环双端链表 ADT - 多了个循环其实就是把 root 的 prev 指向 tail 节点,串起来 - """ - - def __init__(self, maxsize=None): - self.maxsize = maxsize - node = Node() - node.next, node.prev = node, node - self.root = node - self.length = 0 - - def __len__(self): - return self.length - - def headnode(self): - return self.root.next - - def tailnode(self): - return self.root.prev - - def append(self, value): # O(1), 你发现一般不用 for 循环的就是 O(1),有限个步骤 - if self.maxsize is not None and len(self) >= self.maxsize: - raise Exception('LinkedList is Full') - node = Node(value=value) - tailnode = self.tailnode() or self.root - - tailnode.next = node - node.prev = tailnode - node.next = self.root - self.root.prev = node - self.length += 1 - - def appendleft(self, value): - if self.maxsize is not None and len(self) >= self.maxsize: - raise Exception('LinkedList is Full') - node = Node(value=value) - if self.root.next is self.root: # empty - node.next = self.root - node.prev = self.root - self.root.next = node - self.root.prev = node - else: - node.prev = self.root - headnode = self.root.next - node.next = headnode - headnode.prev = node - self.root.next = node - self.length += 1 - - def remove(self, node): # O(1),传入node 而不是 value 我们就能实现 O(1) 删除 - """remove - :param node # 在 lru_cache 里实际上根据key 保存了整个node: - """ - if node is self.root: - return - else: # - node.prev.next = node.next - node.next.prev = node.prev - self.length -= 1 - return node - - def iter_node(self): - if self.root.next is self.root: - return - curnode = self.root.next - while curnode.next is not self.root: - yield curnode - curnode = curnode.next - yield curnode - - def __iter__(self): - for node in self.iter_node(): - yield node.value - - def iter_node_reverse(self): - """相比单链表独有的反序遍历""" - if self.root.prev is self.root: - return - curnode = self.root.prev - while curnode.prev is not self.root: - yield curnode - curnode = curnode.prev - yield curnode - - -def test_double_link_list(): - dll = CircularDoubleLinkedList() - assert len(dll) == 0 - - dll.append(0) - dll.append(1) - dll.append(2) - - assert list(dll) == [0, 1, 2] - - assert [node.value for node in dll.iter_node()] == [0, 1, 2] - assert [node.value for node in dll.iter_node_reverse()] == [2, 1, 0] - - headnode = dll.headnode() - assert headnode.value == 0 - dll.remove(headnode) - assert len(dll) == 2 - assert [node.value for node in dll.iter_node()] == [1, 2] - - dll.appendleft(0) - assert [node.value for node in dll.iter_node()] == [0, 1, 2] - - -if __name__ == '__main__': - test_double_link_list() diff --git "a/docs/03_\351\223\276\350\241\250/linked_list.md" "b/docs/03_\351\223\276\350\241\250/linked_list.md" deleted file mode 100644 index 9b3f81f..0000000 --- "a/docs/03_\351\223\276\350\241\250/linked_list.md" +++ /dev/null @@ -1,107 +0,0 @@ -# 链式结构 - -上一节讲到了支持随机访问的线性结构,这次我们开始讲链式结构, 视频里我会说下这两种结构的区别,然后讲解最常见的单链表和双链表。 -之前在专栏文章[那些年,我们一起跪过的算法题[视频]](https://zhuanlan.zhihu.com/p/35175401)里实现过一个 lru_cache, -使用到的就是循环双端链表,如果感觉这篇文章有点难理解,我们这里将会循序渐进地来实现。 -后边讲到哈希表的冲突解决方式的时候,我们会再次提到链表。 - -上一节我们分析了 list 的各种操作是如何实现的,如果你还有印象的话,list -在头部进行插入是个相当耗时的操作(需要把后边的元素一个一个挪个位置)。假如你需要频繁在数组两头增删,list 就不太合适。 -今天我们介绍的链式结构将摆脱这个缺陷,当然了链式结构本身也有缺陷,比如你不能像数组一样随机根据下标访问,你想查找一个元素只能老老实实从头遍历。 -所以嘛,学习和了解数据结构的原理和实现你才能准确地选择到底什么时候该用什么数据结构,而不是瞎选导致代码性能很差。 - - -# 单链表 -和线性结构不同,链式结构内存不连续的,而是一个个串起来的,这个时候就需要每个链接表的节点保存一个指向下一个节点的指针。 -这里可不要混淆了列表和链表(它们的中文发音类似,但是列表 list 底层其实还是线性结构,链表才是真的通过指针关联的链式结构)。 -看到指针你也不用怕,这里我们用的 python,你只需要一个简单赋值操作就能实现,不用担心 c 语言里复杂的指针。 - -先来定义一个链接表的节点,刚才说到有一个指针保存下一个节点的位置,我们叫它 next, 当然还需要一个 value 属性保存值 - -```py -class Node(object): - def __init__(self, value, next=None): - self.value = value - self.next = next -``` -然后就是我们的单链表 LinkedList ADT: - -```py -class LinkedList(object): - """ 链接表 ADT - [root] -> [node0] -> [node1] -> [node2] - """ -``` -实现我们会在视频中用画图来模拟并且手动代码实现,代码里我们会标识每个步骤的时间复杂度。这里请高度集中精力, -虽然链表的思想很简单,但是想要正确写对链表的操作代码可不容易,稍不留神就可能丢失一些步骤。 -这里我们还是会用简单的单测来验证代码是否按照预期工作。 - -来看下时间复杂度: - -链表操作 | 平均时间复杂度 | -------------------------------|----------------| -linked_list.append(value) | O(1) | -linked_list.appendleft(value) | O(1) | -linked_list.find(value) | O(n) | -linked_list.remove(value) | O(n) | - - -# 双链表 -上边我们亲自实现了一个单链表,但是能看到很明显的问题,单链表虽然 append 是 O(1),但是它的 find 和 remove 都是 O(n)的, -因为删除你也需要先查找,而单链表查找只有一个方式就是从头找到尾,中间找到才退出。 -这里我之前提到过如果要实现一个 lru 缓存(访问时间最久的踢出),我们需要在一个链表里能高效的删除元素, -并把它追加到访问表的最后一个位置,这个时候单链表就满足不了了, -因为缓存在 dict 里查找的时间是 O(1),你更新访问顺序就 O(n)了,缓存就没了优势。 - -这里就要使用到双链表了,相比单链表来说,每个节点既保存了指向下一个节点的指针,同时还保存了上一个节点的指针。 - -```py -class Node(object): - # 如果节点很多,我们可以用 __slots__ 来节省内存,把属性保存在一个 tuple 而不是 dict 里 - # 感兴趣可以自行搜索 python __slots__ - __slots__ = ('value', 'prev', 'next') - - def __init__(self, value=None, prev=None, next=None): - self.value, self.prev, self.next = value, prev, next -``` - -对, 就多了 prev,有啥优势嘛? - -- 看似我们反过来遍历双链表了。反过来从哪里开始呢?我们只要让 root 的 prev 指向 tail 节点,不就串起来了吗? -- 直接删除节点,当然如果给的是一个值,我们还是需要查找这个值在哪个节点? - 但是如果给了一个节点,我们把它拿掉,直接让它的前后节点互相指过去不就行了?哇欧,删除就是 O(1) 了,两步操作就行啦 - -好,废话不多说,我们在视频里介绍怎么实现一个双链表 ADT。你可以直接在本项目的 `docs/03_链表/double_link_list.py` 找到代码。 -最后让我们看下它的时间复杂度:(这里 CircularDoubleLinkedList 取大写字母缩写为 cdll) - -循环双端链表操作 | 平均时间复杂度 | ----------------------------------------|----------------| -cdll.append(value) | O(1) | -cdll.appendleft(value) | O(1) | -cdll.remove(node),注意这里参数是 node | O(1) | -cdll.headnode() | O(1) | -cdll.tailnode() | O(1) | - - -# 小问题: -- 这里单链表我没有实现 insert 方法,你能自己尝试实现吗? insert(value, new_value),我想在某个值之前插入一个值。你同样需要先查找,所以这个步骤也不够高效。 -- 你能尝试自己实现个 lru cache 吗?需要使用到我们这里提到的循环双端链表 -- 借助内置的 collections.OrderedDict,它有两个方法 popitem 和 move_to_end,我们可以迅速实现一个 LRU cache。请你尝试用 OrderedDict 来实现。 -- python 内置库的哪些数据结构使用到了本章讲的链式结构? - - -# 相关阅读 - -[那些年,我们一起跪过的算法题- Lru cache[视频]](https://zhuanlan.zhihu.com/p/35175401) - -# 勘误: - -视频中 LinkedList.remove 方法讲解有遗漏, linked_list.py 文件已经修正,请读者注意。具体请参考 [fix linked_list & add gitigonre](https://github.com/PegasusWang/python_data_structures_and_algorithms/pull/3)。视频最后增加了一段勘误说明。 - -# Leetcode - -反转链表 [reverse-linked-list](https://leetcode.com/problems/reverse-linked-list/) - -这里有一道关于 LRU 的练习题你可以尝试下。 -[LRU Cache](https://leetcode.com/problems/lru-cache/description/) - -合并两个有序链表 [merge-two-sorted-lists](/https://leetcode.com/problems/merge-two-sorted-lists/submissions/) diff --git "a/docs/03_\351\223\276\350\241\250/linked_list.py" "b/docs/03_\351\223\276\350\241\250/linked_list.py" deleted file mode 100644 index b64eb1c..0000000 --- "a/docs/03_\351\223\276\350\241\250/linked_list.py" +++ /dev/null @@ -1,219 +0,0 @@ -# -*- coding: utf-8 -*- - - -class Node(object): - def __init__(self, value=None, next=None): # 这里我们 root 节点默认都是 None,所以都给了默认值 - self.value = value - self.next = next - - def __str__(self): - """方便你打出来调试,复杂的代码可能需要断点调试""" - return ''.format(self.value, self.next) - - __repr__ = __str__ - - -class LinkedList(object): - """ 链接表 ADT - [root] -> [node0] -> [node1] -> [node2] - """ - - def __init__(self, maxsize=None): - """ - :param maxsize: int or None, 如果是 None,无限扩充 - """ - self.maxsize = maxsize - self.root = Node() # 默认 root 节点指向 None - self.tailnode = None - self.length = 0 - - def __len__(self): - return self.length - - def append(self, value): # O(1) - if self.maxsize is not None and len(self) >= self.maxsize: - raise Exception('LinkedList is Full') - node = Node(value) # 构造节点 - tailnode = self.tailnode - if tailnode is None: # 还没有 append 过,length = 0, 追加到 root 后 - self.root.next = node - else: # 否则追加到最后一个节点的后边,并更新最后一个节点是 append 的节点 - tailnode.next = node - self.tailnode = node - self.length += 1 - - def appendleft(self, value): - if self.maxsize is not None and len(self) >= self.maxsize: - raise Exception('LinkedList is Full') - node = Node(value) - if self.tailnode is None: # 如果原链表为空,插入第一个元素需要设置 tailnode - self.tailnode = node - - headnode = self.root.next - self.root.next = node - node.next = headnode - self.length += 1 - - def __iter__(self): - for node in self.iter_node(): - yield node.value - - def iter_node(self): - """遍历 从 head 节点到 tail 节点""" - curnode = self.root.next - while curnode is not self.tailnode: # 从第一个节点开始遍历 - yield curnode - curnode = curnode.next # 移动到下一个节点 - if curnode is not None: - yield curnode - - def remove(self, value): # O(n) - """ 删除包含值的一个节点,将其前一个节点的 next 指向被查询节点的下一个即可 - - :param value: - """ - prevnode = self.root # - for curnode in self.iter_node(): - if curnode.value == value: - prevnode.next = curnode.next - if curnode is self.tailnode: # NOTE: 注意更新 tailnode - if prevnode is self.root: - self.tailnode = None - else: - self.tailnode = prevnode - del curnode - self.length -= 1 - return 1 # 表明删除成功 - else: - prevnode = curnode - return -1 # 表明删除失败 - - def find(self, value): # O(n) - """ 查找一个节点,返回序号,从 0 开始 - - :param value: - """ - index = 0 - for node in self.iter_node(): # 我们定义了 __iter__,这里就可以用 for 遍历它了 - if node.value == value: - return index - index += 1 - return -1 # 没找到 - - def popleft(self): # O(1) - """ 删除第一个链表节点 - """ - if self.root.next is None: - raise Exception('pop from empty LinkedList') - headnode = self.root.next - self.root.next = headnode.next - self.length -= 1 - value = headnode.value - - if self.tailnode is headnode: # 勘误:增加单节点删除 tailnode 处理 - self.tailnode = None - del headnode - return value - - def clear(self): - for node in self.iter_node(): - del node - self.root.next = None - self.length = 0 - self.tailnode = None - - def reverse(self): - """反转链表""" - curnode = self.root.next - self.tailnode = curnode # 记得更新 tailnode,多了这个属性处理起来经常忘记 - prevnode = None - - while curnode: - nextnode = curnode.next - curnode.next = prevnode - - if nextnode is None: - self.root.next = curnode - - prevnode = curnode - curnode = nextnode - - -def test_linked_list(): - ll = LinkedList() - - ll.append(0) - ll.append(1) - ll.append(2) - ll.append(3) - - assert len(ll) == 4 - assert ll.find(2) == 2 - assert ll.find(-1) == -1 - - assert ll.remove(0) == 1 - assert ll.remove(10) == -1 - assert ll.remove(2) == 1 - assert len(ll) == 2 - assert list(ll) == [1, 3] - assert ll.find(0) == -1 - - ll.appendleft(0) - assert list(ll) == [0, 1, 3] - assert len(ll) == 3 - - headvalue = ll.popleft() - assert headvalue == 0 - assert len(ll) == 2 - assert list(ll) == [1, 3] - - assert ll.popleft() == 1 - assert list(ll) == [3] - ll.popleft() - assert len(ll) == 0 - assert ll.tailnode is None - - ll.clear() - assert len(ll) == 0 - assert list(ll) == [] - - -def test_linked_list_remove(): - ll = LinkedList() - ll.append(3) - ll.append(4) - ll.append(5) - ll.append(6) - ll.append(7) - ll.remove(7) - print(list(ll)) - -def test_single_node(): - # https://github.com/PegasusWang/python_data_structures_and_algorithms/pull/21 - ll = LinkedList() - ll.append(0) - ll.remove(0) - ll.appendleft(1) - assert list(ll) == [1] - -def test_linked_list_reverse(): - ll = LinkedList() - n = 10 - for i in range(n): - ll.append(i) - ll.reverse() - assert list(ll) == list(reversed(range(n))) - - -def test_linked_list_append(): - ll = LinkedList() - ll.appendleft(1) - ll.append(2) - assert list(ll) == [1, 2] - - -if __name__ == '__main__': - test_single_node() - test_linked_list() - test_linked_list_append() - test_linked_list_reverse() diff --git "a/docs/03_\351\223\276\350\241\250/lru_cache.py" "b/docs/03_\351\223\276\350\241\250/lru_cache.py" deleted file mode 100644 index 426a309..0000000 --- "a/docs/03_\351\223\276\350\241\250/lru_cache.py" +++ /dev/null @@ -1,134 +0,0 @@ -""" -python3 only -LRU cache -""" -from collections import OrderedDict -from functools import wraps - - -def fib(n): - if n <= 1: # 0 or 1 - return n - return f(n - 1) + f(n - 2) # 由于涉及到重复计算,这个递归函数在 n 大了以后会非常慢。 O(2^n) - - -""" -下边就来写一个缓存装饰器来优化它。传统方法是用个数组记录之前计算过的值,但是这种方式不够 Pythonic -""" - - -def cache(func): - """先引入一个简单的装饰器缓存,其实原理很简单,就是内部用一个字典缓存已经计算过的结果""" - store = {} - - @wraps(func) - def _(n): # 这里函数没啥意义就随便用下划线命名了 - if n in store: - return store[n] - else: - res = func(n) - store[n] = res - return res - return _ - - -@cache -def f(n): - if n <= 1: # 0 or 1 - return n - return f(n - 1) + f(n - 2) - - -""" -问题来了,假如空间有限怎么办,我们不可能一直向缓存塞东西,当缓存达到一定个数之后,我们需要一种策略踢出一些元素, -用来给新的元素腾出空间。 -一般缓存失效策略有 -- LRU(Least-Recently-Used): 替换掉最近请求最少的对象,实际中使用最广。cpu缓存淘汰和虚拟内存效果好,web应用欠佳 -- LFU(Least-Frequently-Used): 缓存污染问题(一个先前流行的缓存对象会在缓存中驻留很长时间) -- First in First out(FIFO) -- Random Cache: 随机选一个删除 - -LRU 是常用的一个,比如 redis 就实现了这个策略,这里我们来模拟实现一个。 -要想实现一个 LRU,我们需要一种方式能够记录访问的顺序,并且每次访问之后我们要把最新使用到的元素放到最后(表示最新访问)。 -当容量满了以后,我们踢出最早访问的元素。假如用一个链表来表示的话: - -[1] -> [2] -> [3] - -假设最后边是最后访问的,当访问到一个元素以后,我们把它放到最后。当容量满了,我们踢出第一个元素就行了。 -一开始的想法可能是用一个链表来记录访问顺序,但是单链表有个问题就是如果访问了中间一个元素,我们需要拿掉它并且放到链表尾部。 -而单链表无法在O(1)的时间内删除一个节点(必须要先搜索到它),但是双端链表可以,因为一个节点记录了它的前后节点, -只需要把要删除的节点的前后节点链接起来就行了。 -还有个问题是如何把删除后的节点放到链表尾部,如果是循环双端链表就可以啦,我们有个 root 节点链接了首位节点, -只需要让 root 的前一个指向这个被删除节点,然后让之前的最后一个节点也指向它就行了。 - -使用了循环双端链表之后,我们的操作就都是 O(1) 的了。这也就是使用一个 dict 和一个 循环双端链表 实现LRU 的思路。 -不过一般我们使用内置的 OrderedDict(原理和这个类似)就好了,要实现一个循环双端链表是一个不简单的事情,因为链表操作很容易出错。 - -补充:其实 lru 有个缺点就是额外的链表比较占用空间,如果你感兴趣的话可以看看 redis 如何实现的 lru 算法 -""" - - -class LRUCache: - def __init__(self, capacity=128): - self.capacity = capacity - # 借助 OrderedDict 我们可以快速实现一个 LRUCache,OrderedDict 内部其实也是使用循环双端链表实现的 - # OrderedDict 有两个重要的函数用来实现 LRU,一个是 move_to_end,一个是 popitem,请自己看文档 - self.od = OrderedDict() - - def get(self, key, default=None): - val = self.od.get(key, default) # 如果没有返回 default,保持 dict 语义 - self.od.move_to_end(key) # 每次访问就把key 放到最后表示最新访问 - return val - - def add_or_update(self, key, value): - if key in self.od: # update - self.od[key] = value - self.od.move_to_end(key) - else: # insert - self.od[key] = value - if len(self.od) > self.capacity: # full - self.od.popitem(last=False) - - def __call__(self, func): - """ - 一个简单的 LRU 实现。有一些问题需要思考下: - - - 这里为了简化默认参数只有一个数字 n,假如可以传入多个参数,如何确定缓存的key 呢? - - 这里实现没有考虑线程安全的问题,要如何才能实现线程安全的 LRU 呢?当然如果不是多线程环境下使用是不需要考虑的 - - 假如这里没有用内置的 dict,你能使用 redis 来实现这个 LRU 吗,如果使用了 redis,我们可以存储更多数据到服务器。而使用字典实际上是缓存了Python进程里(localCache)。 - - 这里只是实现了 lru 策略,你能同时实现一个超时 timeout 参数吗?比如像是memcache 实现的 lazy expiration 策略 - - LRU有个缺点就是,对于周期性的数据访问会导致命中率迅速下降,有一种优化是 LRU-K,访问了次数达到 k 次才会将数据放入缓存 - """ - def _(n): - if n in self.od: - return self.get(n) - else: - val = func(n) - self.add_or_update(n, val) - return val - return _ - - -@LRUCache(10) -def f_use_lru(n): - if n <= 1: # 0 or 1 - return n - return f_use_lru(n - 1) + f_use_lru(n - 2) - - -def test(): - import time - beg = time.time() - for i in range(34): - print(f(i)) - print(time.time() - beg) - beg = time.time() - for i in range(34): - print(f_use_lru(i)) - print(time.time() - beg) - - -# TODO 要怎么给 lru 写单测? - -if __name__ == '__main__': - test() diff --git "a/docs/04_\351\230\237\345\210\227/array_queue.png" "b/docs/04_\351\230\237\345\210\227/array_queue.png" deleted file mode 100644 index 25368872ba134e559a6110909dd62486a0079b5f..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 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in (HashTable.EMPTY, HashTable.UNUSED): - yield slot.key - -######################################### -# 上边是从 哈希表章 拷贝过来的代码,我们会直接继承 HashTable 实现 dict -######################################### - - -class DictADT(HashTable): - - def _iter_slot(self): - for slot in self._table: - if slot not in (HashTable.EMPTY, HashTable.UNUSED): - yield slot - - def __setitem__(self, key, value): - self.add(key, value) - - def __getitem__(self, key): - if key not in self: - raise KeyError() - else: - return self.get(key) - - def items(self): - for slot in self._iter_slot(): - yield (slot.key, slot.value) - - def keys(self): - for slot in self._iter_slot(): - yield slot.key - - def values(self): - for slot in self._iter_slot(): - yield slot.value - - -def test_dict_adt(): - import random - d = DictADT() - - d['a'] = 1 - assert d['a'] == 1 - d.remove('a') - - l = list(range(30)) - random.shuffle(l) - for i in l: - d.add(i, i) - - for i in range(30): - assert d.get(i) == i - - assert sorted(list(d.keys())) == sorted(l) - - -test_dict_adt() diff --git "a/docs/09_\351\233\206\345\220\210/set.md" "b/docs/09_\351\233\206\345\220\210/set.md" deleted file mode 100644 index 6a45488..0000000 --- "a/docs/09_\351\233\206\345\220\210/set.md" +++ /dev/null @@ -1,52 +0,0 @@ -# 集合 set - -集合是一种不包含重复元素的数据结构,经常用来判断是否重复这种操作,或者集合中是否存在一个元素。 -这一章讲集合,实际上它的底层也是哈希表实现的,所以像实现 DictADT 一样,借助 HashTable 实现它也比较简单。 - - -# 集合操作 -集合可能最常用的就是去重,判断是否存在一个元素等,但是 set 相比 dict 有更丰富的操作,主要是数学概念上的。 -如果你学过《离散数学》中集合相关的概念,基本上是一致的。 python 的 set 提供了如下基本的集合操作, -假设有两个集合 A,B,有以下操作: - -- 交集: A & B,表示同时在 A 和 B 中的元素。 python 中重载 `__and__` 实现 -- 并集: A | B,表示在 A 或者 B 中的元素,两个集合相加。python 中重载 `__or__` 实现 -- 差集: A - B,表示在 A 中但是不在 B 中的元素。 python 中重载 `__sub__` 实现 -- 对称差: A ^ B,返回在 A 或 B 但是不在 A、B 中都出现的元素。其实就是 (A|B) - (A&B), python 中重载 `__xor__` 实现 - -这里使用的 &, |, -, ^ 在 python 内置的 set 实现中都是重载了内置的运算符。这里我们也用这种方式实现, -具体实现我会在视频里演示。python 同样实现了 intersection, union, difference, symmetric_difference 这四个方法, -和使用运算符的功能是一样的。 - -![](./set.png) - -# python frozenset -在 python 里还有一个 frozenset,看它的名字就知道这种也是集合,但是它的内容是无法变动的。一般我们使用 -它的常见就是用一个可迭代对象初始化它,然后只用来判重等操作。 - - -# 实现一个 set ADT -如何实现一个集合的 ADT 呢,其实还是个哈希表,哈希表不是有 key 和 value 嘛,咱把 value 置为 1 不就行了。 - -```py -class SetADT(HashTable): - - def add(self, key): - # 集合其实就是一个 dict,只不过我们把它的 value 设置成 1 - return super(SetADT, self).add(key, True) -``` - -当然其它数学上的操作就麻烦点了,不过也很容易实现。 - - -# 思考题 -- 集合判断一个元素是否存在的时间复杂度是多少? -- 集合的元素 key 需要满足什么概念?可变对象可以吗? -- 请你在 SetADT 基础上实现集合的 remove 操作和 pop 操作 -- 你能尝试实现对称差操作吗?这里我没有实现,留给你作为练习 -- 你知道如何重载 python 的内置运算符吗?这里我们实现 set 的集合操作就是用到了重载,请阅读相关 python 文档。 -- 当元素个数不多的时候,我们可以用 set 来判重,但是如果是大量元素会非常耗费内存。请你了解下 Bloom Filter - - -# 延伸阅读 -阅读 python 文档关于 set 的相关章节,了解 set 还有哪些操作?比如比较运算符的概念,比较两个集合意味着什么。 diff --git "a/docs/09_\351\233\206\345\220\210/set.png" "b/docs/09_\351\233\206\345\220\210/set.png" deleted file mode 100644 index 5124a1ee3e32639d950f93ae84891c8b8ae39281..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 82401 zcmZU(18`ofItCp 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zo4h5KCqT_4^#RWRxTyU-U$pogWgMTZ+_ykg$~-|U_yH0~r;O+RYqVaVhjA#g9`Sn* zI}t30oB+m{m?{y-+{ags9yiFtXqB2%!0g{T(sM9yF*1cDu^pKMXKx ztro0Zs+bnpgRueckOxNfEKKHDWBs{gUkPi+8(6{j@0ka~Jxew)xY1Z5GfnaLk}0g6 zEs}}DpV73;!T^tlc65dC-%A6qcJt`HD-Zj5re6)rO!yN_VdDAsQYws(sV9zyJm^C& z;p}i>1GW+FJ^PDgcz-9qa{_=5MFITa|8GuI%N;^P3$vTfx2}puu+M8L1<6uzgMj}4 DLxfdV diff --git "a/docs/09_\351\233\206\345\220\210/set_adt.py" "b/docs/09_\351\233\206\345\220\210/set_adt.py" deleted file mode 100644 index 3efbe78..0000000 --- "a/docs/09_\351\233\206\345\220\210/set_adt.py" +++ /dev/null @@ -1,191 +0,0 @@ -# -*- coding: utf-8 -*- - -# 从数组和列表章复制的代码 - - -class Array(object): - - def __init__(self, size=32, init=None): - self._size = size - self._items = [init] * size - - def __getitem__(self, index): - return self._items[index] - - def __setitem__(self, index, value): - self._items[index] = value - - def __len__(self): - return self._size - - def clear(self, value=None): - for i in range(len(self._items)): - self._items[i] = value - - def __iter__(self): - for item in self._items: - yield item - - -class Slot(object): - """定义一个 hash 表 数组的槽 - 注意,一个槽有三种状态,看你能否想明白。相比链接法解决冲突,二次探查法删除一个 key 的操作稍微复杂。 - 1.从未使用 HashMap.UNUSED。此槽没有被使用和冲突过,查找时只要找到 UNUSED 就不用再继续探查了 - 2.使用过但是 remove 了,此时是 HashMap.EMPTY,该探查点后边的元素扔可能是有key - 3.槽正在使用 Slot 节点 - """ - - def __init__(self, key, value): - self.key, self.value = key, value - - -class HashTable(object): - - UNUSED = None # 没被使用过 - EMPTY = Slot(None, None) # 使用却被删除过 - - def __init__(self): - self._table = Array(8, init=HashTable.UNUSED) # 保持 2*i 次方 - self.length = 0 - - @property - def _load_factor(self): - # load_factor 超过 0.8 重新分配 - return self.length / float(len(self._table)) - - def __len__(self): - return self.length - - def _hash(self, key): - return abs(hash(key)) % len(self._table) - - def _find_key(self, key): - index = self._hash(key) - _len = len(self._table) - while self._table[index] is not HashTable.UNUSED: - if self._table[index] is HashTable.EMPTY: - index = (index*5 + 1) % _len - continue - elif self._table[index].key == key: - return index - else: - index = (index*5 + 1) % _len - return None - - def _find_slot_for_insert(self, key): - index = self._hash(key) - _len = len(self._table) - while not self._slot_can_insert(index): - index = (index*5 + 1) % _len - return index - - def _slot_can_insert(self, index): - return (self._table[index] is HashTable.EMPTY or self._table[index] is HashTable.UNUSED) - - def __contains__(self, key): # in operator - index = self._find_key(key) - return index is not None - - def add(self, key, value): - if key in self: - index = self._find_key(key) - self._table[index].value = value - return False - else: - index = self._find_slot_for_insert(key) - self._table[index] = Slot(key, value) - self.length += 1 - if self._load_factor >= 0.8: - self._rehash() - return True - - def _rehash(self): - old_table = self._table - newsize = len(self._table) * 2 - self._table = Array(newsize, HashTable.UNUSED) - - self.length = 0 - - for slot in old_table: - if slot is not HashTable.UNUSED and slot is not HashTable.EMPTY: - index = self._find_slot_for_insert(slot.key) - self._table[index] = slot - self.length += 1 - - def get(self, key, default=None): - index = self._find_key(key) - if index is None: - return default - else: - return self._table[index].value - - def remove(self, key): - index = self._find_key(key) - if index is None: - raise KeyError() - value = self._table[index].value - self.length -= 1 - self._table[index] = HashTable.EMPTY - return value - - def __iter__(self): - for slot in self._table: - if slot not in (HashTable.EMPTY, HashTable.UNUSED): - yield slot.key - -######################################### -# 上边是从 哈希表章 拷贝过来的代码,我们会直接继承 HashTable 实现 集合 set -######################################### - - -class SetADT(HashTable): - - def add(self, key): - # 集合其实就是一个 dict,只不过我们把它的 value 设置成 1 - return super(SetADT, self).add(key, True) - - def __and__(self, other_set): - """交集 A&B""" - new_set = SetADT() - for element_a in self: - if element_a in other_set: - new_set.add(element_a) - return new_set - - def __sub__(self, other_set): - """差集 A-B""" - new_set = SetADT() - for element_a in self: - if element_a not in other_set: - new_set.add(element_a) - return new_set - - def __or__(self, other_set): - """并集 A|B""" - new_set = SetADT() - for element_a in self: - new_set.add(element_a) - for element_b in other_set: - new_set.add(element_b) - return new_set - - -def test_set_adt(): - sa = SetADT() - sa.add(1) - sa.add(2) - sa.add(3) - assert 1 in sa # 测试 __contains__ 方法,实现了 add 和 __contains__,集合最基本的功能就实现啦 - - sb = SetADT() - sb.add(3) - sb.add(4) - sb.add(5) - - assert sorted(list(sa & sb)) == [3] - assert sorted(list(sa - sb)) == [1, 2] - assert sorted(list(sa | sb)) == [1, 2, 3, 4, 5] - - -if __name__ == '__main__': - test_set_adt() diff --git "a/docs/10_\351\200\222\345\275\222/fact.png" "b/docs/10_\351\200\222\345\275\222/fact.png" deleted file mode 100644 index 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-递归用一种通俗的话来说就是自己调用自己,但是需要分解它的参数,让它解决一个更小一点的问题,当问题小到一定规模的时候,需要一个递归出口返回。 -这里举一个和其他很多老套的教科书一样喜欢举的例子,阶乘函数,我觉得用来它演示再直观不过。它的定义是这样的: - -![](./fact.png) - -我们很容易根据它的定义写出这样一个递归函数,因为它本身就是递归定义的。 - -```py -def fact(n): - if n == 0: - return 1 - else: - return n * fact(n-1) -``` -看吧,几乎完全是按照定义来写的。我们来看下递归函数的几个特点: - -- 递归必须包含一个基本的出口(base case),否则就会无限递归,最终导致栈溢出。比如这里就是 n == 0 返回 1 -- 递归必须包含一个可以分解的问题(recursive case)。 要想求得 fact(n),就需要用 n * fact(n-1) -- 递归必须必须要向着递归出口靠近(toward the base case)。 这里每次递归调用都会 n-1,向着递归出口 n == 0 靠近 - - -# 调用栈 -看了上一个例子你可能觉得递归好简单,先别着急,我们再举个简单的例子,上边我们并没有讲递归如何工作的。 -假如让你输出从 1 到 10 这十个数字,如果你是个正常人的话,我想你的第一反应都是这么写: - -```py -def print_num(n): - for i in range(1, n + 1): # 注意很多编程语言使用的都是 从 0 开始的左闭右开区间, python 也不例外 - print(i) - - -if __name__ == '__main__': - print_num(10) -``` - -我们尝试写一个递归版本,不就是自己调用自己嘛: - -```py -def print_num_recursive(n): - if n > 0: - print_num_recursive(n-1) - print(n) -``` - -你猜下它的输出?然后我们调换下 print 顺序,你再猜下它的输出 - -```py -def print_num_recursive_revserve(n): - if n > 0: - print(n) - print_num_recursive_revserve(n-1) -``` -你能明白是为什么吗?我建议你运行下这几个小例子,它们很简单但是却能说明问题。 -计算机内部使用调用栈来实现递归,这里的栈一方面指的是内存中的栈区,一方面栈又是之前讲到的后进先出这种数据结构。 -每当进入递归函数的时候,系统都会为当前函数开辟内存保存当前变量值等信息,每个调用栈之间的数据互不影响,新调用的函数 -入栈的时候会放在栈顶。视频里我们会画图来演示这个过程。 - -递归只用大脑不用纸笔模拟的话很容易晕,因为明明是同一个变量名字,但是在不同的调用栈里它是不同的值,所以我建议 -你最好手动画画这个过程。 - -![](./print_rec.png) - -# 用栈模拟递归 -刚才说到了调用栈,我们就用栈来模拟一把。之前栈这一章我们讲了如何自己实现栈,不过这里为了不拷贝太多代码,我们直接用 collections.deque 就可以 -快速实现一个简单的栈。 - -```py -from collections import deque - - -class Stack(object): - def __init__(self): - self._deque = deque() - - def push(self, value): - return self._deque.append(value) - - def pop(self): - return self._deque.pop() - - def is_empty(self): - return len(self._deque) == 0 - - -def print_num_use_stack(n): - s = Stack() - while n > 0: # 不断将参数入栈 - s.push(n) - n -= 1 - - while not s.is_empty(): # 参数弹出 - print(s.pop()) -``` -这里结果也是输出 1 到 10,只不过我们是手动模拟了入栈和出栈的过程,帮助你理解计算机是如何实现递归的,是不是挺简单!现在你能明白为什么上边 print_num_recursive print_num_recursive_revserve 两个函数输出的区别了吗? - - -# 尾递归 -上边的代码示例(麻雀虽小五脏俱全)中实际上包含了两种形式的递归,一种是普通的递归,还有一种叫做尾递归: - -```py -def print_num_recursive(n): - if n > 0: - print_num_recursive(n-1) - print(n) - - -def print_num_recursive_revserve(n): - if n > 0: - print(n) - print_num_recursive_revserve(n-1) # 尾递归 -``` - -概念上它很简单,就是递归调用放在了函数的最后。有什么用呢? -普通的递归, 每一级递归都产生了新的局部变量, 必须创建新的调用栈, 随着递归深度的增加, 创建的栈越来越多, 造成爆栈。虽然尾递归调用也会创建新的栈, -但是我们可以优化使得尾递归的每一级调用共用一个栈!, 如此便可解决爆栈和递归深度限制的问题! -不幸的是 python 默认不支持尾递归优化(见延伸阅读),不过一般尾递归我们可以用一个迭代来优化它。 - - -# 汉诺塔问题 - -有三根杆子A,B,C。A杆上有N个(N>1)穿孔圆盘,盘的尺寸由下到上依次变小。要求按下列规则将所有圆盘移至C杆: -但是有两个条件: - -- 每次只能移动一个圆盘; -- 大盘不能叠在小盘上面。 - -> 最早发明这个问题的人是法国数学家爱德华·卢卡斯。 -> 传说越南河内某间寺院有三根银棒,上串64个金盘。寺院里的僧侣依照一个古老的预言,以上述规则移动这些盘子;预言说当这些盘子移动完毕,世界就会灭亡。 -> 这个传说叫做梵天寺之塔问题(Tower of Brahma puzzle)。但不知道是卢卡斯自创的这个传说,还是他受他人启发。 - - - -![五个盘子的汉诺塔问题](./hanoi_tower.png) - -理解这个问题需要我们一些思维上的转换,因为我们正常的思维可能都是从上边最小的盘子开始移动,但是这里我们从移动最底下的盘子开始思考。 -假设我们已经知道了如何移动上边的四个盘子到 B(pole2),现在把最大的盘子从 A -> C 就很简单了。当把最大的盘子移动到 -C 之后,只需要把 B 上的 4 个盘子从 B -> C 就行。(这里的 pole1, 2, 3 分别就是 A, B, C 杆) - -![](./hanoi_four_disks.png) - -问题是仍要想办法如何移动上边的 4 个盘子,我们可以同样的方式来移动上边的 4 个盘子,这就是一种递归的解法。 -给定 n 个盘子和三个杆分别是 源杆(Source), 目标杆(Destination),和中介杆(Intermediate),我们可以定义如下递归操作: - -- 把上边的 n-1 个盘子从 S 移动到 I,借助 D 杆 -- 把最底下的盘子从 S 移动到 D -- 把 n-1 个盘子从 I 移动到 D,借助 S - -我们把它转换成代码: - -```py -def hanoi_move(n, source, dest, intermediate): - if n >= 1: # 递归出口,只剩一个盘子 - hanoi_move(n-1, source, intermediate, dest) - print("Move %s -> %s" % (source, dest)) - hanoi_move(n-1, intermediate, dest, source) -hanoi_move(3, 'A', 'C', 'B') - -# 输出,建议你手动模拟下。三个盘子 A(Source), B(intermediate), C(Destination) -""" -Move A -> C -Move A -> B -Move C -> B -Move A -> C -Move B -> A -Move B -> C -Move A -> C -""" -``` - -
-![三个盘子的汉诺塔解法](./hanoi.gif) -
- -是不是很神奇,但是老实说这个过程仅凭大脑空想是比较难以想象出来的。人的大脑『栈』深度很有限,因为你甚至都没法同时记住超过 8 个以上的 -无意义数字,所以用大脑模拟不如用纸笔来模拟下。(不排除有些聪明的同学能迅速在脑瓜里完成这个过程) - -# 延伸阅读 -递归是个非常重要的概念,我们后边的数据结构和算法中还会多次碰到它,我建议你多阅读一些资料加深理解: - -- 《算法图解》第三章 递归 -- 《Data Structures and Algorithms in Python》 第 10 章 Recursion -- [《Python开启尾递归优化!》](https://segmentfault.com/a/1190000007641519) -- [尾调用优化](http://www.ruanyifeng.com/blog/2015/04/tail-call.html) -- [汉诺塔](https://zh.wikipedia.org/wiki/%E6%B1%89%E8%AF%BA%E5%A1%94) - -# 思考题 -- 你能举出其他一些使用到递归的例子吗? -- 实现一个 flatten 函数,把嵌套的列表扁平化,你需要用递归函数来实现。比如 [[1,2], [1,2,3] -> [1,2,1,2,3] -- 使用递归和循环各有什么优缺点,你能想到吗?怎么把一个尾递归用迭代替换? -- 递归有时候虽然很优雅直观,但是时间复杂度却不理想,比如斐波那契数列,它的表达式是 F(n) = F(n-1) + F(n-2),你能计算它的时间复杂度吗?请你画个树来表示它的计算过程,为什么这个时间复杂度很不理想?我们怎样去优化它。 -- python 内置的 dict 只能用 dict['key'] 的形式访问比较麻烦,我们想用 dict.key 的形式访问。tornado web 框架中提供了一个 ObjectDict,请你实现一个递归函数接收一个字典,并返回一个可以嵌套访问的 ObjectDict diff --git "a/docs/10_\351\200\222\345\275\222/recursion.py" "b/docs/10_\351\200\222\345\275\222/recursion.py" deleted file mode 100644 index 86d2f90..0000000 --- "a/docs/10_\351\200\222\345\275\222/recursion.py" +++ /dev/null @@ -1,72 +0,0 @@ -# -*- coding: utf-8 -*- - - -def fact(n): - if n == 0: - return 1 - else: - return n * fact(n - 1) - - -def print_num(n): - for i in range(1, n + 1): # 注意很多编程语言使用的都是 从 0 开始的左闭右开区间, python 也不例外 - print(i) - - -def print_num_recursive(n): - if n > 0: - print_num_recursive(n - 1) - print(n) - - -def print_num_recursive_revserve(n): - if n > 0: - print(n) - print_num_recursive_revserve(n - 1) - - -from collections import deque - - -class Stack(object): - def __init__(self): - self._deque = deque() - - def push(self, value): - return self._deque.append(value) - - def pop(self): - return self._deque.pop() - - def is_empty(self): - return len(self._deque) == 0 - - -def print_num_use_stack(n): - s = Stack() - while n > 0: # 不断将参数入栈 - s.push(n) - n -= 1 - - while not s.is_empty(): # 参数弹出 - print(s.pop()) - - -def hanoi_move(n, source, dest, intermediate): - if n >= 1: # 递归出口,只剩一个盘子 - hanoi_move(n - 1, source, intermediate, dest) - print("Move %s -> %s" % (source, dest)) - hanoi_move(n - 1, intermediate, dest, source) - - -def flatten(rec_list): - for i in rec_list: - if isinstance(i, list): - for i in flatten(i): - yield i - else: - yield i - - -def test_flatten(): - assert list(flatten([[[1], 2, 3], [1, 2, 3]])) == [1, 2, 3, 1, 2, 3] diff --git "a/docs/11_\347\272\277\346\200\247\346\237\245\346\211\276\344\270\216\344\272\214\345\210\206\346\237\245\346\211\276/search.md" "b/docs/11_\347\272\277\346\200\247\346\237\245\346\211\276\344\270\216\344\272\214\345\210\206\346\237\245\346\211\276/search.md" deleted file mode 100644 index 8ead740..0000000 --- "a/docs/11_\347\272\277\346\200\247\346\237\245\346\211\276\344\270\216\344\272\214\345\210\206\346\237\245\346\211\276/search.md" +++ /dev/null @@ -1,118 +0,0 @@ -# 查找 - -查找可以说是我们业务代码里用得最多的操作,比如我们经常需要在一个列表里找到我们需要的一个元素,然后返回它的位置。 -其实之前我们介绍的哈希表就是非常高效率的查找数据结构,很明显地它是用空间换时间。这一节介绍两个基本的基于线性结构的查找。 - -# 线性查找 -线性查找就是从头找到尾,直到符合条件了就返回。比如在一个 list 中找到一个等于 5 的元素并返回下标: - -```py -number_list = [0, 1, 2, 3, 4, 5, 6, 7] - - -def linear_search(value, iterable): - for index, val in enumerate(iterable): - if val == value: - return index - return -1 - - -assert linear_search(5, number_list) == 5 - -``` -是不是 so easy。当然我们需要来一点花样,比如传一个谓词进去,你要知道,在 python 里一切皆对象,所以我们可以把函数当成一个参数传给另一个函数。 - -```py -def linear_search_v2(predicate, iterable): - for index, val in enumerate(iterable): - if predicate(val): - return index - return -1 - - -assert linear_search_v2(lambda x: x == 5, number_list) == 5 -``` - -效果是一样的,但是传入一个谓词函数进去更灵活一些,比如我们可以找到第一个大于或者小于 5 的,从而控制函数的行为。 -还能玩出什么花样呢?前面我们刚学习了递归,能不能发挥自虐精神没事找事用递归来实现呢? - -```py -def linear_search_recusive(array, value): - if len(array) == 0: - return -1 - index = len(array)-1 - if array[index] == value: - return index - return linear_search_recusive(array[0:index], value) - - -assert linear_search_recusive(number_list, 5) == 5 -assert linear_search_recusive(number_list, 8) == -1 -assert linear_search_recusive(number_list, 7) == 7 -assert linear_search_recusive(number_list, 0) == 0 -``` -这里的 assert 我多写了几个,包括正常情况、异常情况和边界值等,因为递归比较容易出错。注意这里的两个递归出口。 -当然业务代码里如果碰到这种问题我们肯定是选上边最直白的方式来实现,要不你的同事肯定想打你。 - -# 二分查找 -上一小节说的线性查找针对的是无序序列,假如一个序列已经有序了呢,我们还需要从头找到尾吗?当然不用,折半(二分)是一种经典思想。日常生活中还有哪些经典的二分思想呢? - -- 猜数字游戏 -- 一尺之棰,日取其半,万世不竭 -- 有些民间股神,告诉一堆人某个股票会涨,告诉另一半人会跌。后来真涨了,慢慢又告诉信了他的一半人另一个股票会涨,另一半说会跌。就这样韭菜多了总有一些人信奉他为股神。。。 - -其实之前写过博客[《抱歉,我是开发,你居然让我写单测[视频]》](https://zhuanlan.zhihu.com/p/35352024)讲过二分查找,当时主要是为了引入单元测试这个概念的,因为很多不正规的项目代码很糙,更别说写单测了。这里我就直接贴代码啦 - -```py -def binary_search(sorted_array, val): - if not sorted_array: - return -1 - - beg = 0 - end = len(sorted_array) - 1 - - while beg <= end: - mid = int((beg + end) / 2) # beg + (end-beg)/2, 为了屏蔽 python 2/3 差异我用了强转 - if sorted_array[mid] == val: - return mid - elif sorted_array[mid] > val: - end = mid - 1 - else: - beg = mid + 1 - return -1 - - -def test_binary_search(): - a = list(range(10)) - - # 正常值 - assert binary_search(a, 1) == 1 - assert binary_search(a, -1) == -1 - - # 异常值 - assert binary_search(None, 1) == -1 - - # 边界值 - assert binary_search(a, 0) == 0 -``` - - -# 思考题 -- 给你个挑战,用递归来实现本章的二分查找。你要十分注意边界条件,注意用单测测试呦,在你写代码的时候,可能会碰到边界问题或者无穷递归等。 如果你想不起来,可以看看本章的代码示例 -- 二分查找有一个变形,比如我们想在一个有序数组中插入一个值之后,数组仍保持有序,请你找出这个位置。(bisect 模块) - - -# 延伸阅读 -这里没给链接,请善用 google 等搜索引擎和 Dash(mac) 等文档查询工具,在你学习代码的过程中你会非常频繁地使用它们。 -或者如果你有时间也可以跳转到这些模块的源码,看看它们的实现方式。标准库都是些高手写的,肯定能学到一些姿势。 - -- 阅读 python 文档关于二分的 bisect 模块。 -- 阅读 python 文档 itertools 相关模块和常见的几个函数 takewhile, dropwhile, from_iterable, count, tee 等用法 -- [每个程序员都应该会点形式化证明](https://zhuanlan.zhihu.com/p/35364999?group_id=967109293607129088) - - -# Leetcode - -[找旋转过的排序数组中最小的数 find-minimum-in-rotated-sorted-array](https://leetcode.com/problems/find-minimum-in-rotated-sorted-array/description/) - -[已排序的数组中找到第一和最后一个元素 find-first-and-last-position-of-element-in-sorted-array/](https://leetcode.com/problems/find-first-and-last-position-of-element-in-sorted-array/submissions/) diff --git "a/docs/11_\347\272\277\346\200\247\346\237\245\346\211\276\344\270\216\344\272\214\345\210\206\346\237\245\346\211\276/search.py" "b/docs/11_\347\272\277\346\200\247\346\237\245\346\211\276\344\270\216\344\272\214\345\210\206\346\237\245\346\211\276/search.py" deleted file mode 100644 index 6a9e708..0000000 --- "a/docs/11_\347\272\277\346\200\247\346\237\245\346\211\276\344\270\216\344\272\214\345\210\206\346\237\245\346\211\276/search.py" +++ /dev/null @@ -1,60 +0,0 @@ -# -*- coding: utf-8 -*- - -number_list = [0, 1, 2, 3, 4, 5, 6, 7] - - -def linear_search(value, iterable): - for index, val in enumerate(iterable): - if val == value: - return index - return -1 - - -assert linear_search(5, number_list) == 5 - - -def linear_search_v2(predicate, iterable): - for index, val in enumerate(iterable): - if predicate(val): - return index - return -1 - - -assert linear_search_v2(lambda x: x == 5, number_list) == 5 - - -def linear_search_recusive(array, value): - if len(array) == 0: - return -1 - index = len(array)-1 - if array[index] == value: - return index - return linear_search_recusive(array[0:index], value) - - -assert linear_search_recusive(number_list, 5) == 5 -assert linear_search_recusive(number_list, 8) == -1 -assert linear_search_recusive(number_list, 7) == 7 -assert linear_search_recusive(number_list, 0) == 0 - - -def binary_search_recursive(sorted_array, beg, end, val): - if beg >= end: - return -1 - mid = int((beg + end) / 2) # beg + (end-beg)/2 - if sorted_array[mid] == val: - return mid - elif sorted_array[mid] > val: - return binary_search_recursive(sorted_array, beg, mid, val) # 注意我依然假设 beg, end 区间是左闭右开的 - else: - return binary_search_recursive(sorted_array, mid+1, end, val) - - -def test_binary_search_recursive(): - # 我们测试所有值和边界条件 - a = list(range(10)) - for i in a: - assert binary_search_recursive(a, 0, len(a), i) == i - - assert binary_search_recursive(a, 0, len(a), -1) == -1 - assert binary_search_recursive(a, 0, len(a), 10) == -1 diff --git "a/docs/12_\345\237\272\346\234\254\346\216\222\345\272\217\347\256\227\346\263\225/basic_sort.md" "b/docs/12_\345\237\272\346\234\254\346\216\222\345\272\217\347\256\227\346\263\225/basic_sort.md" deleted file mode 100644 index 9804502..0000000 --- "a/docs/12_\345\237\272\346\234\254\346\216\222\345\272\217\347\256\227\346\263\225/basic_sort.md" +++ /dev/null @@ -1,145 +0,0 @@ -# 基本排序算法 -从本章开始讲常见的基于比较的排序算法,先讲三个简单的但是时间复杂度却不太理想的排序算法,包括冒泡排序、选择排序和插入排序。 - - -# 冒泡排序 -bubble sort 可以说是最简单的一种排序算法了,它的思想如下。对一个数组进行 n-1 轮迭代,每次比较相邻两个元素, -如果相邻的元素前者大于后者,就交换它们。因为直接在元素上操作而不是返回新的数组,所以是一个 inplace 的操作。 -这里冒泡的意思其实就是每一轮冒泡一个最大的元素就会通过不断比较和交换相邻元素使它转移到最右边。 - -你可以想象假如有 10 个小盆友从左到右站成一排,个头不等。老师想让他们按照个头从低到高站好,于是他开始喊口号。 -每喊一次,从第一个小盆友开始,相邻的小朋友如果身高不是正序就会两两调换,就这样第一轮个头最高的排到了最右边。(冒泡到最右边) -第二轮依次这么来,从第一个小朋友开始两两交换,这样次高的小盆友又排到了倒数第二个位置。依次类推。 - - -我们在视频里手动模拟下它的过程。 - - -```py -import random - - -def bubble_sort(seq): # O(n^2), n(n-1)/2 = 1/2(n^2 + n) - n = len(seq) - for i in range(n-1): - print(seq) # 我打印出来让你看清楚每一轮最高、次高、次次高...的小朋友会冒泡到右边 - for j in range(n-1-i): # 这里之所以 n-1 还需要 减去 i 是因为每一轮冒泡最大的元素都会冒泡到最后,无需再比较 - if seq[j] > seq[j+1]: - seq[j], seq[j+1] = seq[j+1], seq[j] - print(seq) - - -def test_bubble_sort(): - seq = list(range(10)) # 注意 python3 返回迭代器,所以我都用 list 强转了,python2 range 返回的就是 list - random.shuffle(seq) # shuffle inplace 操作,打乱数组 - bubble_sort(seq) - assert seq == sorted(seq) # 注意呦,内置的 sorted 就不是 inplace 的,它返回一个新的数组,不影响传入的参数 - -""" 我打印出来让你看到每次从最高到次高的小盆友就这么排好序了,因为是随机数,你第一个没有排序的数组应该和我的不一样 -[3, 4, 5, 0, 9, 1, 7, 8, 6, 2] -[3, 4, 0, 5, 1, 7, 8, 6, 2, 9] -[3, 0, 4, 1, 5, 7, 6, 2, 8, 9] -[0, 3, 1, 4, 5, 6, 2, 7, 8, 9] -[0, 1, 3, 4, 5, 2, 6, 7, 8, 9] -[0, 1, 3, 4, 2, 5, 6, 7, 8, 9] -[0, 1, 3, 2, 4, 5, 6, 7, 8, 9] -[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] -[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] -[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] -""" -``` - - - -# 选择排序 -刚才看到冒泡是每轮迭代中,如果相邻的两个元素前者大于后者了就交换两个相邻元素(假设正序排序)。其实还有一种思路就是, -每次我们找到最小的元素插入迭代的起始位置,这样每个位置从它自己的位置开始它就是最小的了,一圈下来数组就有序了。 -选择可以理解为 一个 0 到 n-1 的迭代,每次向后查找选择一个最小的元素。 - -同样小盆友又来啦,这次我们从第一个开始,从头到尾找一个个头最小的小盆友,然后把它和第一个小盆友交换。 -然后从第二个小盆友开始采取同样的策略,这样一圈下来小盆友就有序了。 - -```py -def select_sort(seq): - n = len(seq) - for i in range(n-1): - min_idx = i # 我们假设当前下标的元素是最小的 - for j in range(i+1, n): # 从 i 的后边开始找到最小的元素,得到它的下标 - if seq[j] < seq[min_idx]: - min_idx = j # 一个 j 循环下来之后就找到了最小的元素它的下标 - if min_idx != i: # swap - seq[i], seq[min_idx] = seq[min_idx], seq[i] - - -def test_select_sort(): - seq = list(range(10)) - random.shuffle(seq) - select_sort(seq) - assert seq == sorted(seq) - -""" -[4, 7, 5, 3, 6, 0, 2, 9, 8, 1] -[0, 7, 5, 3, 6, 4, 2, 9, 8, 1] -[0, 1, 5, 3, 6, 4, 2, 9, 8, 7] -[0, 1, 2, 3, 6, 4, 5, 9, 8, 7] -[0, 1, 2, 3, 6, 4, 5, 9, 8, 7] -[0, 1, 2, 3, 4, 6, 5, 9, 8, 7] -[0, 1, 2, 3, 4, 5, 6, 9, 8, 7] -[0, 1, 2, 3, 4, 5, 6, 9, 8, 7] -[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] -[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] - -""" -``` - - -# 插入排序 -插入排序很多教科书都是用扑克牌的例子讲的,想象你手里有一些扑克牌,它们顺序是散乱的,现在需要你把它们整理成有序的,你会怎么做呢? -首先拿最顶上的一张,然后拿第二张,第二张点数大,你就把第二张放在第一张的下边,否则放在第一张上边。 -当你拿第三张的时候,你同样会找到适合它大小的位置插入进去。 - -换成小朋友一样,第一个小盆友只有一个人我们假设是有序的,然后第二个小盆友会跟第一个比,如果第一个高就交换位置。 -接下来第三个小盆友从第二个位置开始比较,如果没第二个高就交换位置,然后没第一个高也交换位置,保持前边三个小盆友身高有序就好。 -依次类推,等到最后一个小盆友也转移到合适的位置,整个队列就是有序的了。 - -插入排序就是这个道理, 每次挑选下一个元素插入已经排序的数组中,初始时已排序数组只有一个元素。我们就直接上代码吧。 - - -```py -def insertion_sort(seq): - """ 每次挑选下一个元素插入已经排序的数组中,初始时已排序数组只有一个元素""" - n = len(seq) - print(seq) - for i in range(1, n): - value = seq[i] # 保存当前位置的值,因为转移的过程中它的位置可能被覆盖 - # 找到这个值的合适位置,使得前边的数组有序 [0,i] 有序 - pos = i - while pos > 0 and value < seq[pos-1]: - seq[pos] = seq[pos-1] # 如果前边的元素比它大,就让它一直前移 - pos -= 1 - seq[pos] = value # 找到了合适的位置赋值就好 - print(seq) - - -""" 不断把新元素放到已经有序的数组中 -[1, 7, 3, 0, 9, 4, 8, 2, 6, 5] -[1, 7, 3, 0, 9, 4, 8, 2, 6, 5] -[1, 3, 7, 0, 9, 4, 8, 2, 6, 5] -[0, 1, 3, 7, 9, 4, 8, 2, 6, 5] -[0, 1, 3, 7, 9, 4, 8, 2, 6, 5] -[0, 1, 3, 4, 7, 9, 8, 2, 6, 5] -[0, 1, 3, 4, 7, 8, 9, 2, 6, 5] -[0, 1, 2, 3, 4, 7, 8, 9, 6, 5] -[0, 1, 2, 3, 4, 6, 7, 8, 9, 5] -[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] -""" -``` - - -# 思考题 -- 本章介绍的几个排序算法平均时间复杂度是多少? -- 请你补充插入排序的单元测试代码 - - -# 延伸阅读 -- 《Data Structures and Algorithms in Python》第5章 diff --git "a/docs/12_\345\237\272\346\234\254\346\216\222\345\272\217\347\256\227\346\263\225/basic_sort.py" "b/docs/12_\345\237\272\346\234\254\346\216\222\345\272\217\347\256\227\346\263\225/basic_sort.py" deleted file mode 100644 index d25682d..0000000 --- "a/docs/12_\345\237\272\346\234\254\346\216\222\345\272\217\347\256\227\346\263\225/basic_sort.py" +++ /dev/null @@ -1,56 +0,0 @@ -# -*- coding: utf-8 -*- - - -import random - - -def bubble_sort(seq): # O(n^2), n(n-1)/2 = 1/2(n^2 + n) - n = len(seq) - for i in range(n-1): - print(seq) # 我打印出来让你看清楚每一轮最高、次高、次次高...的小朋友会冒泡到右边 - for j in range(n-1-i): # 这里之所以 n-1 还需要 减去 i 是因为每一轮冒泡最大的元素都会冒泡到最后,无需再比较 - if seq[j] > seq[j+1]: - seq[j], seq[j+1] = seq[j+1], seq[j] - print(seq) - - -def test_bubble_sort(): - seq = list(range(10)) # 注意 python3 返回迭代器,所以我都用 list 强转了,python2 range 返回的就是 list - random.shuffle(seq) # shuffle inplace 操作,打乱数组 - sorted_seq = sorted(seq) # 注意呦,内置的 sorted 就不是 inplace 的,它返回一个新的数组,不影响传入的参数 - bubble_sort(seq) - assert seq == sorted_seq - - -def select_sort(seq): - n = len(seq) - for i in range(n-1): - min_idx = i # 我们假设当前下标的元素是最小的 - for j in range(i+1, n): # 从 i 的后边开始找到最小的元素,得到它的下标 - if seq[j] < seq[min_idx]: - min_idx = j # 一个 j 循环下来之后就找到了最小的元素它的下标 - if min_idx != i: # swap - seq[i], seq[min_idx] = seq[min_idx], seq[i] - - -def test_select_sort(): - seq = list(range(10)) - random.shuffle(seq) - sorted_seq = sorted(seq) - select_sort(seq) - assert seq == sorted_seq - - -def insertion_sort(seq): - """ 每次挑选下一个元素插入已经排序的数组中,初始时已排序数组只有一个元素""" - n = len(seq) - print(seq) - for i in range(1, n): - value = seq[i] # 保存当前位置的值,因为转移的过程中它的位置可能被覆盖 - # 找到这个值的合适位置,使得前边的数组有序 [0,i] 有序 - pos = i - while pos > 0 and value < seq[pos-1]: - seq[pos] = seq[pos-1] # 如果前边的元素比它大,就让它一直前移 - pos -= 1 - seq[pos] = value # 找到了合适的位置赋值就好 - print(seq) diff --git "a/docs/13_\351\253\230\347\272\247\346\216\222\345\272\217\347\256\227\346\263\225/advanced_sorting.md" "b/docs/13_\351\253\230\347\272\247\346\216\222\345\272\217\347\256\227\346\263\225/advanced_sorting.md" deleted file mode 100644 index 2f3c835..0000000 --- "a/docs/13_\351\253\230\347\272\247\346\216\222\345\272\217\347\256\227\346\263\225/advanced_sorting.md" +++ /dev/null @@ -1,10 +0,0 @@ -# 高级排序算法 - -本章开始讲几个高级一些的排序算法,因为涉及到分治、递归和一些高级数据结构等,所以比前一章节的基本排序要稍微难理解一些。包括: - -- [分治法与归并排序](./merge_sort.md) -- [快速排序](./quick_sort.md) - -在讲完二叉树之后,我们会看下它的应用: - -- 堆和堆排序 diff --git "a/docs/13_\351\253\230\347\272\247\346\216\222\345\272\217\347\256\227\346\263\225/merge_sort.md" "b/docs/13_\351\253\230\347\272\247\346\216\222\345\272\217\347\256\227\346\263\225/merge_sort.md" deleted file mode 100644 index d2cdecb..0000000 --- "a/docs/13_\351\253\230\347\272\247\346\216\222\345\272\217\347\256\227\346\263\225/merge_sort.md" +++ /dev/null @@ -1,110 +0,0 @@ -# 分治法 (Divide and Conquer) - -很多有用的算法结构上是递归的,为了解决一个特定问题,算法一次或者多次递归调用其自身以解决若干子问题。 -这些算法典型地遵循分治法的思想:将原问题分解为几个规模较小但是类似于原问题的子问题,递归求解这些子问题, -然后再合并这些问题的解来建立原问题的解。 - -分治法在每层递归时有三个步骤: - -- **分解**原问题为若干子问题,这些子问题是原问题的规模最小的实例 -- **解决**这些子问题,递归地求解这些子问题。当子问题的规模足够小,就可以直接求解 -- **合并**这些子问题的解成原问题的解 - - -# 归并排序 -现在我们就来看下归并排序是是如何利用分治法解决问题的。 - -- **分解**:将待排序的 n 个元素分成各包含 n/2 个元素的子序列 -- **解决**:使用归并排序递归排序两个子序列 -- **合并**:合并两个已经排序的子序列以产生已排序的答案 - -考虑我们排序这个数组:[10,23,51,18,4,31,13,5] ,我们递归地将数组进行分解 - -![](./merge_sort_split.png) - -当数组被完全分隔成只有单个元素的数组时,我们需要把它们合并回去,每次两两合并成一个有序的序列。 - -![](./merge_sort_merge.png) - -用递归代码来描述这个问题: - -```py -def merge_sort(seq): - if len(seq) <= 1: # 只有一个元素是递归出口 - return seq - else: - mid = int(len(seq)/2) - left_half = merge_sort(seq[:mid]) - right_half = merge_sort(seq[mid:]) - - # 合并两个有序的数组 - new_seq = merge_sorted_list(left_half, right_half) - return new_seq -``` - -注意我们这里有一个函数没实现,就是如何合并两个有序数组 merge_sorted_list。其实你在纸上画一画, -合并两个有序数组并不难实现。 - -![](./merge_sorted_array.png) - -![](./merge_sorted_array_2.png) - - -```py -def merge_sorted_list(sorted_a, sorted_b): - """ 合并两个有序序列,返回一个新的有序序列 - - :param sorted_a: - :param sorted_b: - """ - length_a, length_b = len(sorted_a), len(sorted_b) - a = b = 0 - new_sorted_seq = list() - - while a < length_a and b < length_b: - if sorted_a[a] < sorted_b[b]: - new_sorted_seq.append(sorted_a[a]) - a += 1 - else: - new_sorted_seq.append(sorted_b[b]) - b += 1 - - # 最后别忘记把多余的都放到有序数组里 - if a < length_a: - new_sorted_seq.extend(sorted_a[a:]) - else: - new_sorted_seq.extend(sorted_b[b:]) - - return new_sorted_seq -``` - -这样就实现了归并排序,并且你会发现它返回一个新的数组而不是修改原有数组。 - - -# 时间复杂度 -我们来简单看下它归并排序的时间复杂度,假设排序 n 个数字时间复杂度是 T(n),这里为了方便假设 n 是 2 的幂 - -\begin{align} -T(n)= \begin{cases} c, & \text {if $n$ = 1} \\ 2T(n/2)+cn, & \text{if $n$ > 1} \end{cases} -\end{align} - -![](./merge_sort_recursion_tree.png) - -总的代价是 $cnlg(n)+cn$ ,忽略常数项可以认为是 O(nlg(n))。如果这个图看不懂,我们自己求解下也不难,首先我们简化一下, -把常数系数当成 1,得到以下递归式: - -\begin{align} -T(n)= \begin{cases} 1, & \text {if $n$ = 1} \\ 2T(n/2)+n, & \text{if $n$ > 1} \end{cases} -\end{align} - -![](./tn.png) - - -# 思考题 -- 请你完成归并排序的单元测试 -- 这里实现的归并排序是 inplace 的吗? -- 归并排序是稳定的吗?稳定指的是排序前后相同大小的数字依然保持相对顺序。 - -# 延伸阅读 -- 《算法导论》第 2 章和第 4 章,你需要了解下『主定理』,以及如何求解形如 $T(n)=aT(n/b) + f(n)$ 的递归式复杂度 -- 了解算法导论上递归式的三种求解方法:代入法,递归树法,主方法 diff --git "a/docs/13_\351\253\230\347\272\247\346\216\222\345\272\217\347\256\227\346\263\225/merge_sort.py" "b/docs/13_\351\253\230\347\272\247\346\216\222\345\272\217\347\256\227\346\263\225/merge_sort.py" deleted file mode 100644 index a468470..0000000 --- "a/docs/13_\351\253\230\347\272\247\346\216\222\345\272\217\347\256\227\346\263\225/merge_sort.py" +++ /dev/null @@ -1,48 +0,0 @@ -# -*- coding: utf-8 -*- - - -def merge_sort(seq): - if len(seq) <= 1: # 只有一个元素是递归出口 - return seq - else: - mid = int(len(seq)/2) - left_half = merge_sort(seq[:mid]) - right_half = merge_sort(seq[mid:]) - - # 合并两个有序的数组 - new_seq = merge_sorted_list(left_half, right_half) - return new_seq - - -def merge_sorted_list(sorted_a, sorted_b): - """ 合并两个有序序列,返回一个新的有序序列 - - :param sorted_a: - :param sorted_b: - """ - length_a, length_b = len(sorted_a), len(sorted_b) - a = b = 0 - new_sorted_seq = list() - - while a < length_a and b < length_b: - if sorted_a[a] < sorted_b[b]: - new_sorted_seq.append(sorted_a[a]) - a += 1 - else: - new_sorted_seq.append(sorted_b[b]) - b += 1 - - # 如果 a或b 中还有剩余元素,需要放到最后 - if a < length_a: - new_sorted_seq.extend(sorted_a[a:]) - else: - new_sorted_seq.extend(sorted_b[b:]) - - return new_sorted_seq - - -def test_merge_sort(): - import random - seq = list(range(10)) - random.shuffle(seq) - assert merge_sort(seq) == sorted(seq) diff --git "a/docs/13_\351\253\230\347\272\247\346\216\222\345\272\217\347\256\227\346\263\225/merge_sort_merge.png" "b/docs/13_\351\253\230\347\272\247\346\216\222\345\272\217\347\256\227\346\263\225/merge_sort_merge.png" deleted file mode 100644 index f6f9f66e6b42392a70c9b51a54db7360a9331bcf..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 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-和归并排序一样,快排也是一种分而治之(divide and conquer)的策略。归并排序把数组递归成只有单个元素的数组,之后再不断两两 -合并,最后得到一个有序数组。这里的递归基本条件就是只包含一个元素的数组,当数组只包含一个元素的时候,我们可以认为它本来就是有序的(当然空数组也不用排序)。 - -快排的工作过程其实比较简单,三步走: - -- 选择基准值 pivot 将数组分成两个子数组:小于基准值的元素和大于基准值的元素。这个过程称之为 partition - -- 对这两个子数组进行快速排序。 - -- 合并结果 - -![](./quick_sort.png) - -根据这个想法我们可以快速写出快排的代码,简直就是在翻译上边的描述: - -```py -def quicksort(array): - size = len(array) - if not array or size < 2: # NOTE: 递归出口,空数组或者只有一个元素的数组都是有序的 - return array - pivot_idx = 0 - pivot = array[pivot_idx] - less_part = [array[i] for i in range(size) if array[i] <= pivot and pivot_idx != i] - great_part = [array[i] for i in range(size) if array[i] > pivot and pivot_idx != i] - return quicksort(less_part) + [pivot] + quicksort(great_part) - -def test_quicksort(): - import random - seq = list(range(10)) - random.shuffle(seq) - assert quicksort(seq) == sorted(seq) -``` -是不是很简单,下次面试官让你手写快排你再写不出来就有点不太合适啦。 当然这个实现有两个不好的地方: - -- 第一是它需要额外的存储空间,我们想实现 inplace 原地排序。 -- 第二是它的 partition 操作每次都要两次遍历整个数组,我们想改善一下。 - -这里我们就来优化一下它,实现 inplace 排序并且改善下 partition 操作。新的代码大概如下: - -```py -def quicksort_inplace(array, beg, end): # 注意这里我们都用左闭右开区间,end 传入 len(array) - if beg < end: # beg == end 的时候递归出口 - pivot = partition(array, beg, end) - quicksort_inplace(array, beg, pivot) - quicksort_inplace(array, pivot+1, end) -``` - -能否实现只遍历一次数组就可以完成 partition 操作呢?实际上是可以的。我们设置首位俩个指针 left, right,两个指针不断向中间收拢。如果遇到 left 位置的元素大于 pivot 并且 right 指向的元素小于 pivot,我们就交换这俩元素,当 left > right 的时候退出就行了,这样实现了一次遍历就完成了 partition。如果你感觉懵逼,纸上画画就立马明白了。我们来撸代码实现: - -![](./partition.png) - -```py -def partition(array, beg, end): - pivot_index = beg - pivot = array[pivot_index] - left = pivot_index + 1 - right = end - 1 # 开区间,最后一个元素位置是 end-1 [0, end-1] or [0: end),括号表示开区间 - - while True: - # 从左边找到比 pivot 大的 - while left <= right and array[left] < pivot: - left += 1 - - while right >= left and array[right] >= pivot: - right -= 1 - - if left > right: - break - else: - array[left], array[right] = array[right], array[left] - - array[pivot_index], array[right] = array[right], array[pivot_index] - return right # 新的 pivot 位置 - - -def test_partition(): - l = [4, 1, 2, 8] - assert partition(l, 0, len(l)) == 2 - l = [1, 2, 3, 4] - assert partition(l, 0, len(l)) == 0 - l = [4, 3, 2, 1] - assert partition(l, 0, len(l)) -``` - -大功告成,新的快排就实现好了。 - -# 时间复杂度 -在比较理想的情况下,比如数组每次都被 pivot 均分,我们可以得到递归式: - -T(n) = 2T(n/2) + n - -上一节我们讲过通过递归树得到它的时间复杂度是 O(nlog(n))。即便是很坏的情况,比如 pivot 每次都把数组按照 1:9 划分,依然是 O(nlog(n)),感兴趣请阅读算法导论相关章节。 - -![](quicksort_worst.png) - - -# 思考题 -- 请你补充 quicksort_inplace 的单元测试 -- 最坏的情况下快排的时间复杂度是多少?什么时候会发生这种情况? -- 我们实现的快排是稳定的啵? -- 选择基准值如果选不好就可能导致复杂度升高,算导中提到了一种『median of 3』策略,就是说选择 pivot 的时候 从子数组中随机选三个元素,再取它的中位数,你能实现这个想法吗?这里我们的代码很简单地取了第一个元素作为 pivot -- 利用快排中的 partition 操作,我们还能实现另一个算法,nth_element,快速查找一个无序数组中的第 n 大元素,请你尝试实现它并编写单测。其实这个函数是 C++ STL 中的一个函数。 -- 你知道 Python 内置的 sorted 如何实现的吗?请你 Google 相关资料了解下。很多内置的排序都使用了快排的改良版。 - - -# 延伸阅读 -- 《算法导论》第 7 章 -- [《面试必备 | 排序算法的Python实现》](https://zhuanlan.zhihu.com/p/36419582) - -# 总结 - -面试经常问的就是常用排序算法的时间空间复杂,这里列一个表格方便记忆: - -| 排序算法 | 最差时间分析 | 平均时间复杂度 | 稳定度 | 空间复杂度 | -|------------|--------------|----------------|--------|----------------| -| 冒泡排序 | O(n^2) | O(n^2) | 稳定 | O(1) | -| 选择排序 | O(n^2) | O(n^2) | 不稳定 | O(1) | -| 插入排序 | O(n^2) | O(n^2) | 稳定 | O(1) | -| 二叉树排序 | O(n^2) | O(n\*log2n) | 不一顶 | O(n) | -| 快速排序 | O(n^2) | O(n\*log2n) | 不稳定 | O(log2n)\~O(n) | -| 堆排序 | O(n\*log2n) | O(n\*log2n) | 不稳定 | O(1) | - -[数据结构与算法-排序篇-Python描述](https://blog.csdn.net/mrlevo520/article/details/77829204) - -# Leetcode - -无序数组寻找第 k 大的数字,不止一种方法。 -https://leetcode.com/problems/kth-largest-element-in-an-array/description/ diff --git "a/docs/13_\351\253\230\347\272\247\346\216\222\345\272\217\347\256\227\346\263\225/quick_sort.png" "b/docs/13_\351\253\230\347\272\247\346\216\222\345\272\217\347\256\227\346\263\225/quick_sort.png" deleted file mode 100644 index a9f85612e5df4087724eed5ab22021b7f0c2d51d..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 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"a/docs/13_\351\253\230\347\272\247\346\216\222\345\272\217\347\256\227\346\263\225/quicksort.py" +++ /dev/null @@ -1,110 +0,0 @@ -# -*- coding: utf-8 -*- - - -def quicksort(array): - size = len(array) - if not array or size < 2: # NOTE: 递归出口,空数组或者只有一个元素的数组都是有序的 - return array - pivot_idx = 0 - pivot = array[pivot_idx] - less_part = [array[i] for i in range(size) if array[i] <= pivot and pivot_idx != i] - great_part = [array[i] for i in range(size) if array[i] > pivot and pivot_idx != i] - return quicksort(less_part) + [pivot] + quicksort(great_part) - - -def test_quicksort(): - import random - seq = list(range(10)) - random.shuffle(seq) - assert quicksort(seq) == sorted(seq) # 用内置的sorted 『对拍』 - - -def quicksort_inplace(array, beg, end): # 注意这里我们都用左闭右开区间 - if beg < end: # beg == end 的时候递归出口 - pivot = partition(array, beg, end) - quicksort_inplace(array, beg, pivot) - quicksort_inplace(array, pivot + 1, end) - - -def partition(array, beg, end): - """对给定数组执行 partition 操作,返回新的 pivot 位置""" - pivot_index = beg - pivot = array[pivot_index] - left = pivot_index + 1 - right = end - 1 # 开区间,最后一个元素位置是 end-1 [0, end-1] or [0: end),括号表示开区间 - - while True: - # 从左边找到比 pivot 大的 - while left <= right and array[left] < pivot: - left += 1 - - while right >= left and array[right] >= pivot: - right -= 1 - - if left > right: - break - else: - array[left], array[right] = array[right], array[left] - - array[pivot_index], array[right] = array[right], array[pivot_index] - return right # 新的 pivot 位置 - - -def test_partition(): - l = [4, 1, 2, 8] - assert partition(l, 0, len(l)) == 2 - l = [1, 2, 3, 4] - assert partition(l, 0, len(l)) == 0 - l = [4, 3, 2, 1] - assert partition(l, 0, len(l)) == 3 - l = [1] - assert partition(l, 0, len(l)) == 0 - l = [2,1] - assert partition(l, 0, len(l)) == 1 - - -def test_quicksort_inplace(): - import random - seq = list(range(10)) - random.shuffle(seq) - sorted_seq = sorted(seq) - quicksort_inplace(seq, 0, len(seq)) - assert seq == sorted_seq - - -def nth_element(array, beg, end, nth): - """查找一个数组第 n 大元素""" - if beg < end: - pivot_idx = partition(array, beg, end) - if pivot_idx == nth - 1: # 数组小标从 0 开始 - return array[pivot_idx] - elif pivot_idx > nth - 1: - return nth_element(array, beg, pivot_idx, nth) - else: - return nth_element(array, pivot_idx + 1, end, nth) - - -def test_nth_element(): - l1 = [3, 5, 4, 2, 1] - assert nth_element(l1, 0, len(l1), 3) == 3 - assert nth_element(l1, 0, len(l1), 2) == 2 - - l = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] - for i in l: - assert nth_element(l, 0, len(l), i) == i - for i in reversed(l): - assert nth_element(l, 0, len(l), i) == i - - array = [3, 2, 1, 5, 6, 4] - assert nth_element(array, 0, len(array), 2) == 2 - - array = [2,1] - assert nth_element(array, 0, len(array), 1) == 1 - assert nth_element(array, 0, len(array), 2) == 2 - - array = [3,3,3,3,3,3,3,3,3] - assert nth_element(array, 0, len(array), 1) == 3 - - -if __name__ == '__main__': - test_nth_element() diff --git 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-讲完了树之后后面我们开始介绍一种有用的数据结构堆(heap), 以及借助堆来实现的堆排序,相比前两种排序算法要稍难理解和实现一些。 - -# 树 -这里先简单讲讲树的概念。树结构是一种包括节点(nodes)和边(edges)的拥有层级关系的一种结构, 它的形式和家谱树非常类似: - -![](./family_tree.png) - -如果你了解 linux 文件结构(tree 命令),它的结构也是一棵树。我们快速看下树涉及到的一些概念: - -![](./tree.png) - -- 根节点(root): 树的最上层的节点,任何非空的树都有一个节点 -- 路径(path): 从起始节点到终止节点经历过的边 -- 父亲(parent):除了根节点,每个节点的上一层边连接的节点就是它的父亲(节点) -- 孩子(children): 每个节点由边指向的下一层节点 -- 兄弟(siblings): 同一个父亲并且处在同一层的节点 -- 子树(subtree): 每个节点包含它所有的后代组成的子树 -- 叶子节点(leaf node): 没有孩子的节点成为叶子节点 - - -# 二叉树 - -了解完树的结构以后,我们来看树结构里一种简单但是却比较常用的树-二叉树。 -二叉树是一种简单的树,它的每个节点最多只能包含两个孩子,以下都是一些合法的二叉树: - -![](./binary_tree.png) -![](./binary_tree_level.png) - -通过上边这幅图再来看几个二叉树相关的概念: - -- 节点深度(depth): 节点对应的 level 数字 -- 树的高度(height): 二叉树的高度就是 level 数 + 1,因为 level 从 0开始计算的 -- 树的宽度(width): 二叉树的宽度指的是包含最多节点的层级的节点数 -- 树的 size:二叉树的节点总个数。 - -一棵 size 为 n 的二叉树高度最多可以是 n,最小的高度是 $ \lfloor lgn \rfloor + 1 $,这里 log 以 2 为底简写为 -lgn,和算法导论保持一致。这个结果你只需要用高中的累加公式就可以得到。 - -# 一些特殊的二叉树 -在了解了二叉树的术语和概念之后,我们来看看一些特殊的二叉树,后续章节我们会用到: - -### 满二叉树(full binary tree) -如果每个内部节点(非叶节点)都包含两个孩子,就成为满二叉树。下边是一些例子,它可以有多种形状: - -![](./full_binary_tree.png) - -### 完美二叉树(perfect binary tree) -当所有的叶子节点都在同一层就是完美二叉树,毫无间隙填充了 h 层。 - -![](./perfect_binary_tree.png) - -### 完全二叉树(complete binary tree) -当一个高度为 h 的完美二叉树减少到 h-1,并且最底层的槽被毫无间隙地从左到右填充,我们就叫它完全二叉树。 -下图就是完全二叉树的例子: - -![](./complete_binary_tree.png) - -# 二叉树的表示 -说了那么多,那么怎么表示一棵二叉树呢?其实你发现会和链表有一些相似之处,一个节点,然后节点需要保存孩子的指针,我以构造下边这个二叉树为例子: -我们先定义一个类表示节点: - -![](preorder.png) - -```py -class BinTreeNode(object): - def __init__(self, data, left=None, right=None): - self.data, self.left, self.right = data, left, right -``` - -当然和链表类似,root 节点是我们的入口,于是乎定义一个 二叉树: - -```py -class BinTree(object): - def __init__(self, root=None): - self.root = root -``` - -怎么构造上图中的二叉树呢,似乎其他课本没找到啥例子(有些例子是写了一堆嵌套节点来定义,很难搞清楚层次关系),我自己定义了一种方法,首先我们输入节点信息,仔细看下边代码,叶子节点的 left 和 right 都是 None,并且只有一个根节点 A: - -```py -node_list = [ - {'data': 'A', 'left': 'B', 'right': 'C', 'is_root': True}, - {'data': 'B', 'left': 'D', 'right': 'E', 'is_root': False}, - {'data': 'D', 'left': None, 'right': None, 'is_root': False}, - {'data': 'E', 'left': 'H', 'right': None, 'is_root': False}, - {'data': 'H', 'left': None, 'right': None, 'is_root': False}, - {'data': 'C', 'left': 'F', 'right': 'G', 'is_root': False}, - {'data': 'F', 'left': None, 'right': None, 'is_root': False}, - {'data': 'G', 'left': 'I', 'right': 'J', 'is_root': False}, - {'data': 'I', 'left': None, 'right': None, 'is_root': False}, - {'data': 'J', 'left': None, 'right': None, 'is_root': False}, -] -``` - -然后我们给 BinTreeNode 定义一个 build_from 方法,当然你也可以定义一种自己的构造方法: - -```py -class BinTree(object): - def __init__(self, root=None): - self.root = root - - @classmethod - def build_from(cls, node_list): - """通过节点信息构造二叉树 - 第一次遍历我们构造 node 节点 - 第二次遍历我们给 root 和 孩子赋值 - 最后我们用 root 初始化这个类并返回一个对象 - - :param node_list: {'data': 'A', 'left': None, 'right': None, 'is_root': False} - """ - node_dict = {} - for node_data in node_list: - data = node_data['data'] - node_dict[data] = BinTreeNode(data) - for node_data in node_list: - data = node_data['data'] - node = node_dict[data] - if node_data['is_root']: - root = node - node.left = node_dict.get(node_data['left']) - node.right = node_dict.get(node_data['right']) - return cls(root) -btree = BinTree.build_from(node_list) -``` - -大功告成,这样我们就构造了一棵二叉树对象。下边我们看看它的一些常用操作。 - -# 二叉树的遍历 -不知道你有没有发现,二叉树其实是一种递归结构,因为单独拿出来一个 subtree 子树出来,其实它还是一棵树。那遍历它就很方便啦,我们可以直接用递归的方式来遍历它。但是当处理顺序不同的时候,树又分为三种遍历方式: - -- 先(根)序遍历: 先处理根,之后是左子树,然后是右子树 -- 中(根)序遍历: 先处理左子树,之后是根,最后是右子树 -- 后(根)序遍历: 先处理左子树,之后是右子树,最后是根 - -我们来看下实现,其实算是比较直白的递归函数: - -```py -class BinTreeNode(object): - def __init__(self, data, left=None, right=None): - self.data, self.left, self.right = data, left, right - - -class BinTree(object): - def __init__(self, root=None): - self.root = root - - @classmethod - def build_from(cls, node_list): - """通过节点信息构造二叉树 - 第一次遍历我们构造 node 节点 - 第二次遍历我们给 root 和 孩子赋值 - - :param node_list: {'data': 'A', 'left': None, 'right': None, 'is_root': False} - """ - node_dict = {} - for node_data in node_list: - data = node_data['data'] - node_dict[data] = BinTreeNode(data) - for node_data in node_list: - data = node_data['data'] - node = node_dict[data] - if node_data['is_root']: - root = node - node.left = node_dict.get(node_data['left']) - node.right = node_dict.get(node_data['right']) - return cls(root) - - def preorder_trav(self, subtree): - """ 先(根)序遍历 - - :param subtree: - """ - if subtree is not None: - print(subtree.data) # 递归函数里先处理根 - self.preorder_trav(subtree.left) # 递归处理左子树 - self.preorder_trav(subtree.right) # 递归处理右子树 - - -node_list = [ - {'data': 'A', 'left': 'B', 'right': 'C', 'is_root': True}, - {'data': 'B', 'left': 'D', 'right': 'E', 'is_root': False}, - {'data': 'D', 'left': None, 'right': None, 'is_root': False}, - {'data': 'E', 'left': 'H', 'right': None, 'is_root': False}, - {'data': 'H', 'left': None, 'right': None, 'is_root': False}, - {'data': 'C', 'left': 'F', 'right': 'G', 'is_root': False}, - {'data': 'F', 'left': None, 'right': None, 'is_root': False}, - {'data': 'G', 'left': 'I', 'right': 'J', 'is_root': False}, - {'data': 'I', 'left': None, 'right': None, 'is_root': False}, - {'data': 'J', 'left': None, 'right': None, 'is_root': False}, -] -btree = BinTree.build_from(node_list) -btree.preorder_trav(btree.root) # 输出 A, B, D, E, H, C, F, G, I, J -``` -怎么样是不是挺简单的,比较直白的递归函数。如果你不明白,视频里我们会画个图来说明。 - -# 二叉树层序遍历 - -除了递归的方式遍历之外,我们还可以使用层序遍历的方式。层序遍历比较直白,就是从根节点开始按照一层一层的方式遍历节点。 -我们可以从根节点开始,之后把所有当前层的孩子都按照从左到右的顺序放到一个列表里,下一次遍历所有这些孩子就可以了。 - -```py - def layer_trav(self, subtree): - cur_nodes = [subtree] # current layer nodes - next_nodes = [] - while cur_nodes or next_nodes: - for node in cur_nodes: - print(node.data) - if node.left: - next_nodes.append(node.left) - if node.right: - next_nodes.append(node.right) - cur_nodes = next_nodes # 继续遍历下一层 - next_nodes = [] -``` - -还有一种方式就是使用一个队列,之前我们知道队列是一个先进先出结构,如果我们按照一层一层的顺序从左往右把节点放到一个队列里, -也可以实现层序遍历: - -```py - def layer_trav_use_queue(self, subtree): - q = Queue() - q.append(subtree) - while not q.empty(): - cur_node = q.pop() - print(cur_node.data) - if cur_node.left: - q.append(cur_node.left) - if cur_node.right: - q.append(cur_node.right) - - -from collections import deque -class Queue(object): # 借助内置的 deque 我们可以迅速实现一个 Queue - def __init__(self): - self._items = deque() - - def append(self, value): - return self._items.append(value) - - def pop(self): - return self._items.popleft() - - def empty(self): - return len(self._items) == 0 -``` - - -# 反转二叉树 -之所以单拎出来说这个是因为 mac 下著名的 brew 工具作者据说是因为面试 google 白板编程没写出来反转二叉树跪了。然后人家就去了苹果 😂。其实吧和遍历操作相比也没啥太大区别,递归交换就是了: - -```py - def reverse(self, subtree): - if subtree is not None: - subtree.left, subtree.right = subtree.right, subtree.left - self.reverse(subtree.left) - self.reverse(subtree.right) -``` - - -# 练习题 -- 请你完成二叉树的中序遍历和后序遍历以及单元测试 -- 树的遍历我们用了 print,请你尝试换成一个 callback,这样就能自定义处理树节点的方式了。 -- 请问树的遍历操作时间复杂度是多少?假设它的 size 是 n -- 你能用非递归的方式来实现树的遍历吗?我们知道计算机内部使用了 stack,如果我们自己模拟如何实现?请你尝试完成 -- 只根据二叉树的中序遍历和后序遍历能否确定一棵二叉树?你可以举一个反例吗? - - -# 延伸阅读 -- 《Data Structures and Algorithms in Python》 13 章 Binary Trees -- [https://www.geeksforgeeks.org/iterative-preorder-traversal/](https://www.geeksforgeeks.org/iterative-preorder-traversal/) - - -# Leetcode 练习 - -- [leetcode binary-tree-preorder-traversal](https://leetcode.com/problems/binary-tree-preorder-traversal/) -二叉树的先序遍历 - -- [leetcode binary-tree-inorder-traversal/](https://leetcode.com/problems/binary-tree-inorder-traversal/) -二叉树的中序遍历 - -- [leetcode binary-tree-postorder-traversal](https://leetcode.com/problems/binary-tree-postorder-traversal/) -二叉树的后序遍历 - -- [leetcode binary-tree-right-side-view](https://leetcode.com/problems/binary-tree-right-side-view/description/) -使用树的层序遍历我们能实现一个树的左右视图,比如从一个二叉树的左边能看到哪些节点。 请你尝试做这个练习题 - -- [leetcode construct-binary-tree-from-preorder-and-postorder-traversal](https://leetcode.com/problems/construct-binary-tree-from-preorder-and-postorder-traversal/submissions/) -根据二叉树的 前序和后序遍历,返回一颗完整的二叉树。 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"b/docs/15_\345\240\206\344\270\216\345\240\206\346\216\222\345\272\217/lfu.py" deleted file mode 100644 index e9479d6..0000000 --- "a/docs/15_\345\240\206\344\270\216\345\240\206\346\216\222\345\272\217/lfu.py" +++ /dev/null @@ -1,69 +0,0 @@ -""" -https://medium.com/@epicshane/a-python-implementation-of-lfu-least-frequently-used-cache-with-o-1-time-complexity-e16b34a3c49b -https://leetcode.com/problems/lfu-cache/ - -这里学习下 LRU(least frequently used),就是当缓存满了之后剔除一个最少使用的 key。 -""" -from collections import defaultdict, OrderedDict - - -class Node: - __slots__ = 'key', 'val', 'cnt' - - def __init__(self, key, val, cnt=0): - self.key, self.val, self.cnt = key, val, cnt - - -class LFUCache: - def __init__(self, capacity): - self.capacity = capacity - self.cache = {} # type {key: node} - self.cnt2node = defaultdict(OrderedDict) - self.mincnt = 0 - - def get(self, key, default=-1): - if key not in self.cache: - return default - - node = self.cache[key] - del self.cnt2node[node.cnt][key] - - if not self.cnt2node[node.cnt]: - del self.cnt2node[node.cnt] - - node.cnt += 1 - self.cnt2node[node.cnt][key] = node - - if not self.cnt2node[self.mincnt]: - self.mincnt += 1 - return node.val - - def put(self, key, value): - if key in self.cache: - self.cache[key].val = value - self.get(key) - return - if len(self.cache) >= self.capacity: - pop_key, _pop_node = self.cnt2node[self.mincnt].popitem(last=False) - del self.cache[pop_key] - - self.cache[key] = self.cnt2node[1][key] = Node(key, value, 1) - self.mincnt = 1 - - -def test(): - c = LFUCache(2) - c.put(1, 1) - c.put(2, 2) - assert c.get(1) == 1 - c.put(3, 3) - assert c.get(2) == -1 - assert c.get(3) == 3 - c.put(4, 4) - assert c.get(1) == -1 - assert c.get(3) == 3 - assert c.get(4) == 4 - - -if __name__ == '__main__': - test() diff --git "a/docs/15_\345\240\206\344\270\216\345\240\206\346\216\222\345\272\217/siftdown.png" "b/docs/15_\345\240\206\344\270\216\345\240\206\346\216\222\345\272\217/siftdown.png" deleted file mode 100644 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zzxom+qVTZr*2WnE)f4zPWqOCtxqN%>CV`;?Y?}9;M2{M*@44N{`0oUHO?x#xGi|u1WlvQN;RI6 zq51!KzV7>HQ(OGd_ZeTgo~ZmP$L+`8?~`u4kFQf|t646ARai&!&mif7{k>zYtOuZgyGK zWAgqvU9JlIZ$8h}on12{@mg#5 z%V$$pzlaw*yVXX%Cqhv5@nojP!V$x4(d-UfvUf{ccrPjF0hLEX?Oz%mIq`CEcrgWxh_Clx`&HcH4WHP8lwxiaFF zS}2pn3LGgez**-W0Z$f$|G;)BSORCq)sMT*<3gF-=U|cqPA^+2y9ha=%u+iDc=Q3M syH6HQT#h;m-sIp0+LJg$yl~(@d&c~-054@F83rKmboFyt=akR{01z}GCIA2c diff --git "a/docs/15_\345\240\206\344\270\216\345\240\206\346\216\222\345\272\217/topk.py" "b/docs/15_\345\240\206\344\270\216\345\240\206\346\216\222\345\272\217/topk.py" deleted file mode 100644 index 87b3d7b..0000000 --- "a/docs/15_\345\240\206\344\270\216\345\240\206\346\216\222\345\272\217/topk.py" +++ /dev/null @@ -1,43 +0,0 @@ -import heapq - - -class TopK: - """获取大量元素 topk 大个元素,固定内存 - 思路: - 1. 先放入元素前 k 个建立一个最小堆 - 2. 迭代剩余元素: - 如果当前元素小于堆顶元素,跳过该元素(肯定不是前 k 大) - 否则替换堆顶元素为当前元素,并重新调整堆 - """ - - def __init__(self, iterable, k): - self.minheap = [] - self.capacity = k - self.iterable = iterable - - def push(self, val): - if len(self.minheap) >= self.capacity: - min_val = self.minheap[0] - if val < min_val: # 当然你可以直接 if val > min_val操作,这里我只是显示指出跳过这个元素 - pass - else: - heapq.heapreplace(self.minheap, val) # 返回并且pop堆顶最小值,推入新的 val 值并调整堆 - else: - heapq.heappush(self.minheap, val) # 前面 k 个元素直接放入minheap - - def get_topk(self): - for val in self.iterable: - self.push(val) - return self.minheap - - -def test(): - import random - i = list(range(1000)) # 这里可以是一个可迭代元素,节省内存 - random.shuffle(i) - _ = TopK(i, 10) - print(_.get_topk()) # [990, 991, 992, 996, 994, 993, 997, 998, 999, 995] - - -if __name__ == '__main__': - test() diff --git a/docs/163_course.png b/docs/163_course.png deleted file mode 120000 index c4f4aa7..0000000 --- a/docs/163_course.png +++ /dev/null @@ -1 +0,0 @@ -../163_course.png \ No newline at end of file diff --git "a/docs/16_\344\274\230\345\205\210\347\272\247\351\230\237\345\210\227/priority_queue.md" "b/docs/16_\344\274\230\345\205\210\347\272\247\351\230\237\345\210\227/priority_queue.md" deleted file mode 100644 index 5fe3b16..0000000 --- "a/docs/16_\344\274\230\345\205\210\347\272\247\351\230\237\345\210\227/priority_queue.md" +++ /dev/null @@ -1,53 +0,0 @@ -# 优先级队列 -你可能比较奇怪,队列不是早就讲了嘛。这里之所以放到这里讲优先级队列,是因为虽然名字有队列, -但其实是使用堆来实现的。上一章讲完了堆,这一章我们就趁热打铁来实现一个优先级队列。 - - -# 实现优先级队列 -优先级队列(Priority Queue) 顾名思义,就是入队的时候可以给一个优先级,通常是个数字或者时间戳等, -当出队的时候我们希望按照给定的优先级出队,我们按照 TDD(测试驱动开发) 的方式先来写测试代码: - -```py -def test_priority_queue(): - size = 5 - pq = PriorityQueue(size) - pq.push(5, 'purple') # priority, value - pq.push(0, 'white') - pq.push(3, 'orange') - pq.push(1, 'black') - - res = [] - while not pq.is_empty(): - res.append(pq.pop()) - assert res == ['purple', 'orange', 'black', 'white'] -``` - -上边就是期望的行为,写完测试代码后我们来编写优先级队列的代码,按照出队的时候最大优先级先出的顺序: - - -```py -class PriorityQueue(object): - def __init__(self, maxsize): - self.maxsize = maxsize - self._maxheap = MaxHeap(maxsize) - - def push(self, priority, value): - # 注意这里把这个 tuple push 进去,python 比较 tuple 从第一个开始比较 - # 这样就很巧妙地实现了按照优先级排序 - entry = (priority, value) # 入队的时候会根据 priority 维持堆的特性 - self._maxheap.add(entry) - - def pop(self, with_priority=False): - entry = self._maxheap.extract() - if with_priority: - return entry - else: - return entry[1] - - def is_empty(self): - return len(self._maxheap) == 0 -``` - - -# 练习题 -- 请你实现按照小优先级先出队的顺序的优先级队列 diff --git "a/docs/16_\344\274\230\345\205\210\347\272\247\351\230\237\345\210\227/priority_queue.py" "b/docs/16_\344\274\230\345\205\210\347\272\247\351\230\237\345\210\227/priority_queue.py" deleted file mode 100644 index 039885c..0000000 --- "a/docs/16_\344\274\230\345\205\210\347\272\247\351\230\237\345\210\227/priority_queue.py" +++ /dev/null @@ -1,131 +0,0 @@ -# -*- coding:utf-8 -*- - -# 第二章拷贝的 Array 代码 - - -class Array(object): - - def __init__(self, size=32): - self._size = size - self._items = [None] * size - - def __getitem__(self, index): - return self._items[index] - - def __setitem__(self, index, value): - self._items[index] = value - - def __len__(self): - return self._size - - def clear(self, value=None): - for i in range(len(self._items)): - self._items[i] = value - - def __iter__(self): - for item in self._items: - yield item - -##################################################### -# heap 实现 -##################################################### - - -class MaxHeap(object): - """ - Heaps: - 完全二叉树,最大堆的非叶子节点的值都比孩子大,最小堆的非叶子结点的值都比孩子小 - Heap包含两个属性,order property 和 shape property(a complete binary tree),在插入 - 一个新节点的时候,始终要保持这两个属性 - 插入操作:保持堆属性和完全二叉树属性, sift-up 操作维持堆属性 - extract操作:只获取根节点数据,并把树最底层最右节点copy到根节点后,sift-down操作维持堆属性 - - 用数组实现heap,从根节点开始,从上往下从左到右给每个节点编号,则根据完全二叉树的 - 性质,给定一个节点i, 其父亲和孩子节点的编号分别是: - parent = (i-1) // 2 - left = 2 * i + 1 - rgiht = 2 * i + 2 - 使用数组实现堆一方面效率更高,节省树节点的内存占用,一方面还可以避免复杂的指针操作,减少 - 调试难度。 - - """ - - def __init__(self, maxsize=None): - self.maxsize = maxsize - self._elements = Array(maxsize) - self._count = 0 - - def __len__(self): - return self._count - - def add(self, value): - if self._count >= self.maxsize: - raise Exception('full') - self._elements[self._count] = value - self._count += 1 - self._siftup(self._count-1) # 维持堆的特性 - - def _siftup(self, ndx): - if ndx > 0: - parent = int((ndx-1)/2) - if self._elements[ndx] > self._elements[parent]: # 如果插入的值大于 parent,一直交换 - self._elements[ndx], self._elements[parent] = self._elements[parent], self._elements[ndx] - self._siftup(parent) # 递归 - - def extract(self): - if self._count <= 0: - raise Exception('empty') - value = self._elements[0] # 保存 root 值 - self._count -= 1 - self._elements[0] = self._elements[self._count] # 最右下的节点放到root后siftDown - self._siftdown(0) # 维持堆特性 - return value - - def _siftdown(self, ndx): - left = 2 * ndx + 1 - right = 2 * ndx + 2 - # determine which node contains the larger value - largest = ndx - if (left < self._count and # 有左孩子 - self._elements[left] >= self._elements[largest] and - self._elements[left] >= self._elements[right]): # 原书这个地方没写实际上找的未必是largest - largest = left - elif right < self._count and self._elements[right] >= self._elements[largest]: - largest = right - if largest != ndx: - self._elements[ndx], self._elements[largest] = self._elements[largest], self._elements[ndx] - self._siftdown(largest) - - -class PriorityQueue(object): - def __init__(self, maxsize): - self.maxsize = maxsize - self._maxheap = MaxHeap(maxsize) - - def push(self, priority, value): - entry = (priority, value) # 注意这里把这个 tuple push进去,python 比较 tuple 从第一个开始比较 - self._maxheap.add(entry) - - def pop(self, with_priority=False): - entry = self._maxheap.extract() - if with_priority: - return entry - else: - return entry[1] - - def is_empty(self): - return len(self._maxheap) == 0 - - -def test_priority_queue(): - size = 5 - pq = PriorityQueue(size) - pq.push(5, 'purple') - pq.push(0, 'white') - pq.push(3, 'orange') - pq.push(1, 'black') - - res = [] - while not pq.is_empty(): - res.append(pq.pop()) - assert res == ['purple', 'orange', 'black', 'white'] diff --git "a/docs/17_\344\272\214\345\217\211\346\237\245\346\211\276\346\240\221/binary_search_tree.md" "b/docs/17_\344\272\214\345\217\211\346\237\245\346\211\276\346\240\221/binary_search_tree.md" deleted file mode 100644 index 06e9bc7..0000000 --- "a/docs/17_\344\272\214\345\217\211\346\237\245\346\211\276\346\240\221/binary_search_tree.md" +++ /dev/null @@ -1,252 +0,0 @@ -# 二叉查找树(BST) - -二叉树的一种应用就是来实现堆,今天我们再看看用二叉查找树(Binary Search Tree, BST)。 -前面有章节说到了查找操作,包括线性查找、二分查找、哈希查找等,线性查找效率比较低,二分又要求必须是有序的序列, -为了维持有序插入的代价比较高、哈希查找效率很高但是浪费空间。能不能有一种插入和查找都比较快的数据结构呢?二叉查找树就是这样一种结构,可以高效地插入和查询节点。 - -# BST 定义 - -二叉查找树是这样一种二叉树结构,它的每个节点包含一个 key 和它附带的数据,对于每个内部节点 V: - -- 所有 key 小于 V 的都被存储在 V 的左子树 -- 所有 key 大于 V 的都存储在 V 的右子树 - -![](./bst.png) - -注意这个限制条件,可别和堆搞混了。说白了就是对于每个内部节点,左子树的 key 都比它小,右子树都比它大。 -如果中序遍历(二叉树遍历讲过了)这颗二叉树,你会发现输出的顺序正好是有序的。 -我们先来定义一下 BST 的节点结构: - -```py -class BSTNode(object): - def __init__(self, key, value, left=None, right=None): - self.key, self.value, self.left, self.right = key, value, left, right -``` - -# 构造一个 BST -我们还像之前构造二叉树一样,按照上图构造一个 BST 用来演示: - -```py -class BST(object): - def __init__(self, root=None): - self.root = root - - @classmethod - def build_from(cls, node_list): - cls.size = 0 - key_to_node_dict = {} - for node_dict in node_list: - key = node_dict['key'] - key_to_node_dict[key] = BSTNode(key, value=key) # 这里值暂时用 和 key一样的 - - for node_dict in node_list: - key = node_dict['key'] - node = key_to_node_dict[key] - if node_dict['is_root']: - root = node - node.left = key_to_node_dict.get(node_dict['left']) - node.right = key_to_node_dict.get(node_dict['right']) - cls.size += 1 - return cls(root) - - -NODE_LIST = [ - {'key': 60, 'left': 12, 'right': 90, 'is_root': True}, - {'key': 12, 'left': 4, 'right': 41, 'is_root': False}, - {'key': 4, 'left': 1, 'right': None, 'is_root': False}, - {'key': 1, 'left': None, 'right': None, 'is_root': False}, - {'key': 41, 'left': 29, 'right': None, 'is_root': False}, - {'key': 29, 'left': 23, 'right': 37, 'is_root': False}, - {'key': 23, 'left': None, 'right': None, 'is_root': False}, - {'key': 37, 'left': None, 'right': None, 'is_root': False}, - {'key': 90, 'left': 71, 'right': 100, 'is_root': False}, - {'key': 71, 'left': None, 'right': 84, 'is_root': False}, - {'key': 100, 'left': None, 'right': None, 'is_root': False}, - {'key': 84, 'left': None, 'right': None, 'is_root': False}, -] -bst = BST.build_from(NODE_LIST) -``` - - -# BST 操作 - -## 查找 -如何查找一个指定的节点呢,根据定义我们知道每个内部节点左子树的 key 都比它小,右子树的 key 都比它大,所以 -对于带查找的节点 search_key,从根节点开始,如果 search_key 大于当前 key,就去右子树查找,否则去左子树查找。 一直到当前节点是 None 了说明没找到对应 key。 - -![](./bst_search.png) - -好,撸代码: - -```py - def _bst_search(self, subtree, key): - if subtree is None: # 没找到 - return None - elif key < subtree.key: - return self._bst_search(subtree.left, key) - elif key > subtree.key: - return self._bst_search(subtree.right, key) - else: - return subtree - - def get(self, key, default=None): - node = self._bst_search(self.root, key) - if node is None: - return default - else: - return node.value -``` - - -## 获取最大和最小 key 的节点 - -其实还按照其定义,最小值就一直向着左子树找,最大值一直向右子树找,递归查找就行。 - -```py - def _bst_min_node(self, subtree): - if subtree is None: - return None - elif subtree.left is None: # 找到左子树的头 - return subtree - else: - return self._bst_min_node(subtree.left) - - def bst_min(self): - node = self._bst_min_node(self.root) - return node.value if node else None -``` - -## 插入 -插入节点的时候我们需要一直保持 BST 的性质,每次插入一个节点,我们都通过递归比较把它放到正确的位置。 -你会发现新节点总是被作为叶子结点插入。(请你思考这是为什么) - -![](./bst_insert.png) - -```py - def _bst_insert(self, subtree, key, value): - """ 插入并且返回根节点 - - :param subtree: - :param key: - :param value: - """ - if subtree is None: # 插入的节点一定是根节点,包括 root 为空的情况 - subtree = BSTNode(key, value) - elif key < subtree.key: - subtree.left = self._bst_insert(subtree.left, key, value) - elif key > subtree.key: - subtree.right = self._bst_insert(subtree.right, key, value) - return subtree - - def add(self, key, value): - node = self._bst_search(self.root, key) - if node is not None: # 更新已经存在的 key - node.value = value - return False - else: - self.root = self._bst_insert(self.root, key, value) - self.size += 1 - return True -``` - -## 删除节点 -删除操作相比上边的操作要麻烦很多,首先需要定位一个节点,删除节点后,我们需要始终保持 BST 的性质。 -删除一个节点涉及到三种情况: - -- 节点是叶节点 -- 节点有一个孩子 -- 节点有两个孩子 - -我们分别来看看三种情况下如何删除一个节点: - -#### 删除叶节点 -这是最简单的一种情况,只需要把它的父亲指向它的指针设置为 None 就好。 - -![](./bst_remove_leaf.png) - -#### 删除只有一个孩子的节点 -删除有一个孩子的节点时,我们拿掉需要删除的节点,之后把它的父亲指向它的孩子就行,因为根据 BST -左子树都小于节点,右子树都大于节点的特性,删除它之后这个条件依旧满足。 - -![](./bst_remove_node_with_one_child.png) - -#### 删除有两个孩子的内部节点 -假如我们想删除 12 这个节点改怎么做呢?你的第一反应可能是按照下图的方式: - -![](./remove_interior_replace.png) - -但是这种方式可能会影响树的高度,降低查找的效率。这里我们用另一种非常巧妙的方式。 -还记得上边提到的吗,如果你中序遍历 BST 并且输出每个节点的 key,你会发现就是一个有序的数组。 -`[1 4 12 23 29 37 41 60 71 84 90 100]`。这里我们定义两个概念,逻辑前任(predecessor)和后继(successor),请看下图: - -![](./predecessor_successor.png) - -12 在中序遍历中的逻辑前任和后继分别是 4 和 23 节点。于是我们还有一种方法来删除 12 这个节点: - -- 找到待删除节点 N(12) 的后继节点 S(23) -- 复制节点 S 到节点 N -- 从 N 的右子树中删除节点 S,并更新其删除后继节点后的右子树 - -说白了就是找到后继并且替换,这里之所以能保证这种方法是正确的,你会发现替换后依旧是保持了 BST 的性质。 -有个问题是如何找到后继节点呢?待删除节点的右子树的最小的节点不就是后继嘛,上边我们已经实现了找到最小 key 的方法了。 - - -![](./find_successor.png) - -我们开始编写代码实现,和之前的操作类似,我们还是通过辅助函数的形式来实现,这个递归函数会比较复杂,请你仔细理解: - -```py - def _bst_remove(self, subtree, key): - """删除节点并返回根节点""" - if subtree is None: - return None - elif key < subtree.key: - subtree.left = self._bst_remove(subtree.left, key) - return subtree - elif key > subtree.key: - subtree.right = self._bst_remove(subtree.right, key) - return subtree - else: # 找到了需要删除的节点 - if subtree.left is None and subtree.right is None: # 叶节点,返回 None 把其父亲指向它的指针置为 None - return None - elif subtree.left is None or subtree.right is None: # 只有一个孩子 - if subtree.left is not None: - return subtree.left # 返回它的孩子并让它的父亲指过去 - else: - return subtree.right - else: # 俩孩子,寻找后继节点替换,并从待删节点的右子树中删除后继节点 - successor_node = self._bst_min_node(subtree.right) - subtree.key, subtree.value = successor_node.key, successor_node.value - subtree.right = self._bst_remove(subtree.right, successor_node.key) - return subtree - - def remove(self, key): - assert key in self - self.size -= 1 - return self._bst_remove(self.root, key) -``` - -完整代码你可以在本章的 bst.py 找到。 -另外推荐一个可以在线演示过程的网址大家可以手动执行下看看效果: https://www.cs.usfca.edu/~galles/visualization/BST.html - -# 时间复杂度分析 - -上边介绍的操作时间复杂度和二叉树的形状有关。平均来说时间复杂度是和树的高度成正比的,树的高度 h 是 log(n), -但是最坏情况下以上操作的时间复杂度都是 O(n)。为了改善 BST 有很多变种,感兴趣请参考延伸阅读中的内容。 - -![](./bst_worstcase.png) - - -# 练习题: -- 请你实现查找 BST 最大值的函数 - - -# 延伸阅读 -- 《Data Structures and Algorithms in Python》14 章,树的概念和算法还有很多,我们这里介绍最基本的帮你打个基础 -- 了解红黑树。普通二叉查找树有个很大的问题就是难以保证树的平衡,极端情况下某些节点可能会非常深,导致查找复杂度大幅退化。而平衡二叉树就是为了解决这个问题。请搜索对应资料了解下。 -- 了解 mysql 索引使用的 B-Tree 结构(多路平衡查找树),这个是后端面试数据库的常考点。想想为什么?当元素非常多的时候,二叉树的深度会很深,导致多次磁盘查找。[从B树、B+树、B*树谈到R 树](https://blog.csdn.net/v_JULY_v/article/details/6530142) - - -# Leetcode - -验证是否是合法二叉搜索树 [validate-binary-search-tree](https://leetcode.com/problems/validate-binary-search-tree/ diff --git 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zTMzqnc`>nNoOhi{d{#dwHe0v!+bQ>Xr{$ThwET=YaqfOuXMUM~UP`maEtA4|zz%j3HTjS$mbGcCwpw`*6Fu$<@IXHi)@ru4rQoifqIuj#q++4(w?-I}I@ zO%Vo}vNL~EmbsF{*{%Eb#mSr8ob+ME-H!{(+&_H|U9wX8=oD{GrZ&(((Ffq&A&@%@ zlNA&?pC7xCJ~0Sb_Omb^2CXjsvA%%;a_ylZsNr(Vu6qhF>nLzY$N@9u;d)jU24F8o zfTag?sf6LT$R#4+5y%g~LRs>kwgLm_zC?ut9^g1r&&KmgOMx0Wu-Mv?-cu{432eYQ f2pHHf{b!$;9eMQowIvTgS8{r~`njxgN@xNAN_jeX diff --git "a/docs/17_\344\272\214\345\217\211\346\237\245\346\211\276\346\240\221/bst.py" "b/docs/17_\344\272\214\345\217\211\346\237\245\346\211\276\346\240\221/bst.py" deleted file mode 100644 index e1ceef3..0000000 --- "a/docs/17_\344\272\214\345\217\211\346\237\245\346\211\276\346\240\221/bst.py" +++ /dev/null @@ -1,155 +0,0 @@ -# -*- coding: utf-8 -*- - - -class BSTNode(object): - def __init__(self, key, value, left=None, right=None): - self.key, self.value, self.left, self.right = key, value, left, right - - -class BST(object): - def __init__(self, root=None): - self.root = root - - @classmethod - def build_from(cls, node_list): - cls.size = 0 - key_to_node_dict = {} - for node_dict in node_list: - key = node_dict['key'] - key_to_node_dict[key] = BSTNode(key, value=key) # 这里值暂时用 和 key一样的 - - for node_dict in node_list: - key = node_dict['key'] - node = key_to_node_dict[key] - if node_dict['is_root']: - root = node - node.left = key_to_node_dict.get(node_dict['left']) - node.right = key_to_node_dict.get(node_dict['right']) - cls.size += 1 - return cls(root) - - def _bst_search(self, subtree, key): - if subtree is None: # 没找到 - return None - elif key < subtree.key: - return self._bst_search(subtree.left, key) - elif key > subtree.key: - return self._bst_search(subtree.right, key) - else: - return subtree - - def __contains__(self, key): - """实现 in 操作符""" - return self._bst_search(self.root, key) is not None - - def get(self, key, default=None): - node = self._bst_search(self.root, key) - if node is None: - return default - else: - return node.value - - def _bst_min_node(self, subtree): - if subtree is None: - return None - elif subtree.left is None: # 找到左子树的头 - return subtree - else: - return self._bst_min_node(subtree.left) - - def bst_min(self): - node = self._bst_min_node(self.root) - return node.value if node else None - - def _bst_insert(self, subtree, key, value): - """ 插入并且返回根节点 - - :param subtree: - :param key: - :param value: - """ - if subtree is None: # 插入的节点一定是根节点,包括 root 为空的情况 - subtree = BSTNode(key, value) - elif key < subtree.key: - subtree.left = self._bst_insert(subtree.left, key, value) - elif key > subtree.key: - subtree.right = self._bst_insert(subtree.right, key, value) - return subtree - - def add(self, key, value): - node = self._bst_search(self.root, key) - if node is not None: # 更新已经存在的 key - node.value = value - return False - else: - self.root = self._bst_insert(self.root, key, value) - self.size += 1 - return True - - def _bst_remove(self, subtree, key): - """删除节点并返回根节点""" - if subtree is None: - return None - elif key < subtree.key: - subtree.left = self._bst_remove(subtree.left, key) - return subtree - elif key > subtree.key: - subtree.right = self._bst_remove(subtree.right, key) - return subtree - else: # 找到了需要删除的节点 - if subtree.left is None and subtree.right is None: # 叶节点,返回 None 把其父亲指向它的指针置为 None - return None - elif subtree.left is None or subtree.right is None: # 只有一个孩子 - if subtree.left is not None: - return subtree.left # 返回它的孩子并让它的父亲指过去 - else: - return subtree.right - else: # 俩孩子,寻找后继节点替换,并删除其右子树的后继节点,同时更新其右子树 - successor_node = self._bst_min_node(subtree.right) - subtree.key, subtree.value = successor_node.key, successor_node.value - subtree.right = self._bst_remove(subtree.right, successor_node.key) - return subtree - - def remove(self, key): - assert key in self - self.size -= 1 - return self._bst_remove(self.root, key) - - -NODE_LIST = [ - {'key': 60, 'left': 12, 'right': 90, 'is_root': True}, - {'key': 12, 'left': 4, 'right': 41, 'is_root': False}, - {'key': 4, 'left': 1, 'right': None, 'is_root': False}, - {'key': 1, 'left': None, 'right': None, 'is_root': False}, - {'key': 41, 'left': 29, 'right': None, 'is_root': False}, - {'key': 29, 'left': 23, 'right': 37, 'is_root': False}, - {'key': 23, 'left': None, 'right': None, 'is_root': False}, - {'key': 37, 'left': None, 'right': None, 'is_root': False}, - {'key': 90, 'left': 71, 'right': 100, 'is_root': False}, - {'key': 71, 'left': None, 'right': 84, 'is_root': False}, - {'key': 100, 'left': None, 'right': None, 'is_root': False}, - {'key': 84, 'left': None, 'right': None, 'is_root': False}, -] - - -def test_bst_tree(): - bst = BST.build_from(NODE_LIST) - for node_dict in NODE_LIST: - key = node_dict['key'] - assert bst.get(key) == key - assert bst.size == len(NODE_LIST) - assert bst.get(-1) is None # 单例的 None 我们用 is 来比较 - - assert bst.bst_min() == 1 - - bst.add(0, 0) - assert bst.bst_min() == 0 - - bst.remove(12) - assert bst.get(12) is None - - bst.remove(1) - assert bst.get(1) is None - - bst.remove(29) - assert bst.get(29) is None diff --git 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zMz2!z93uE%IW8YtC^fGI*7M=*y!om5)SSu8b?E_7BQ|qujNW5Yd6V&{FG~)C@UtoJ zWdDoF?0~Bp!bMLX?E7>E*e`Wu6kv~o z!@uDF|5U`g8JX9-G>jA zFWdXqGXEX_;cSiY1C!#8vR*p>zgLri-<1RZW?}K)U;3{l7t{Qg)hjEv&bckk@CfiS MyljFl*K>&cKf&_A2LJ#7 diff --git "a/docs/19_python\345\206\205\347\275\256\345\270\270\347\224\250\347\256\227\346\263\225\345\222\214\346\225\260\346\215\256\347\273\223\346\236\204/builtins.md" "b/docs/19_python\345\206\205\347\275\256\345\270\270\347\224\250\347\256\227\346\263\225\345\222\214\346\225\260\346\215\256\347\273\223\346\236\204/builtins.md" deleted file mode 100644 index 40d838b..0000000 --- "a/docs/19_python\345\206\205\347\275\256\345\270\270\347\224\250\347\256\227\346\263\225\345\222\214\346\225\260\346\215\256\347\273\223\346\236\204/builtins.md" +++ /dev/null @@ -1,23 +0,0 @@ -# Python 常用内置算法和数据结构 -相信到这里大家对常用的数据结构和算法及其实现都比较熟悉了。 -之前在每章的数据结构和算法中涉及到的章节我都会提到对应的 python 内置模块,一般如果内置的可以满足需求,我们优先使用内置模块, -因为在性能和容错性方面内置模块要好于我们自己实现(比如有些是 c 实现的)。本章我们不会再对每个模块的原理详细说明,仅列举出一些常见模块供大家参考, -如果有需要最好的学习方式就是参考 Python 的官方文档。很多高级的数据结构我们也可以通过 google 搜索现成的库拿来直接用。 - -- 常用内置数据类型:list, tuple, dict, set, frozenset -- collections -- heapq -- bisect - -下边我列了一个常用的表格,如果有遗漏可以在 issue 中提出。确保你了解这些数据结构和算法的使用以及时间、空间复杂度。 - -| 数据结构/算法 | 语言内置 | 内置库 | -|----------------|---------------------------------|---------------------------------------------------------------| -| 线性结构 | list(列表)/tuple(元祖) | array(数组,不常用)/collections.namedtuple | -| 链式结构 | | collections.deque(双端队列) | -| 字典结构 | dict(字典) | collections.Counter(计数器)/OrderedDict(有序字典)/defaultdict | -| 集合结构 | set(集合)/frozenset(不可变集合) | | -| 排序算法 | sorted | | -| 二分算法 | | bisect模块 | -| 堆算法 | | heapq模块 | -| 缓存算法 | | functools.lru_cache(Least Recent Used, python3) | diff --git "a/docs/20_\351\235\242\350\257\225\346\214\207\345\215\227/interview.md" "b/docs/20_\351\235\242\350\257\225\346\214\207\345\215\227/interview.md" deleted file mode 100644 index 5958458..0000000 --- "a/docs/20_\351\235\242\350\257\225\346\214\207\345\215\227/interview.md" +++ /dev/null @@ -1,43 +0,0 @@ -# 注意事项 - -- 电子简历尽量用 pdf 格式,方便跨平台打开。doc 等格式在不同的电脑上打开会有排版问题,很多后端技术面试官可能使用的是 mac 或者 linux。 -- 提前复习回顾重点知识,防止卡在基础上。比如 mac 下著名的 brew 工具作者面试 google 就因为没写出来反转二叉树被拒,后来去了苹果😂.(这就只能看人品和运气和眼缘了,如果没见到二面面试官或者 hr,大概率是挂了)。(树、链表、哈希表、二分、快排、TCP/UDP、HTTP、数据库ACID、索引优化等常考点)。 -- 白板编程,练习在纸上手写代码。虽然很多求职者都很抵触手写代码,但是白板编程确实是一种比较好的区分方式。你的思考过程、编码习惯、编码规范等都能看出来。 -- 如果被问到工程里不会使用但是比较刁钻的算法题,建议你和面试官沟通的时候问问这个算法或者题目在开发中有哪些实际使用场景,看看对方怎么说😎。少数公司喜欢考一些算法竞赛题,这种对于没有ACM,信息学竞赛背景的开发者来说比较吃力。大部分业务开发岗位应该只会考察基础算法题 -- 面试的时候准备充分,简历要与招聘方需求对等,甚至可以针对不同公司准备不同的简历内容。笔者每次面试都会带上白纸、笔、简历、电脑等,即使面试没过,至少也让面试官感觉我是有诚意的,给对方留下好印象。 -- 加分项:github、个人技术博客、开源项目、技术论坛帐号等,让面试官有更多渠道了解你,有时候仅仅根据几十分钟的面试来评判面试者是有失偏颇的。(比如面试者临场发挥不好;面试官个人偏好;会的都不问,问的都不会等) - - -# 白板编程 -其实我个人是反对出纯算法题目的,尤其是有些比较刁钻的直接出算法竞赛题,这对与很多做工程的同学来说是比较吃亏的。没事的时候可以去 LeetCode 之类的网站刷刷基础题。 -一般来说 web 业务开发者掌握常见的编程语言内置算法和数据结构就够用了。 - -- 练习手写常见的算法,比如快排,二分,归并等,记住常见排序算法时间复杂度 -- 逻辑正确是前提 -- 有图示描述思路最好,如果时间紧代码没写出来,可以直接描述自己的思路。 -- 字不要写太大,尽量工整。每行代码之间留有一定的空隙,方便你修改(甚至笔者之前会带上铅笔和橡皮手写代码) -- 如果实在写不出来可以和面试官交流,很多时候如果给不出最优方案尽量想一个次优方案,别上来就说不会 -- 想不起来的函数名写伪代码,一般面试官不会强制说让你记住每个 api 的名字 -- 如果有多余的时间(一般不会有)注意一些边界条件,防御性编程、代码风格、单元测试等东西,想好一些异常情况(空值、边界值等)的测试用例 - -# 手写代码注意事项 - -这里我就直接引用《剑指offer》里内容,大家写代码的时候可以多加注意,对于应对算法面试,如果准备时间比较多,推荐看下这本书,并且刷一下 -leetcode 上的基础题目练练手感。 - -- 规范性:书写清晰、布局清晰、命令合理 -- 完整性:完成基本功能,考虑边界条件,做好错误处理 -- 鲁棒性:防御式编程,处理无效输入 - - -# 结语 -这套教程列举的算法很有限,包括图算法、贪心,动态规划,分布式,机器学习算法等很多没有涉及到,因为它们确实需要读者更深入的理论基础,而且这套教程的目的也不是针对算法竞赛。 -不过了解了本教程涉及到的大部分算法是可以应付绝大多数的业务开发的。如果读者对算法有兴趣,本教程引用的几本参考书都可以去深入学习。希望本教程能对你学习算法、养成良好的思维方式和编码习惯等有所帮助。 - -# 延伸阅读 -目前市面上有一些专门针对算法面试的书供大家参考,如果你正在准备算法面试,我强烈建议你看看下面的参考资料学习解题技巧: - -- [那些年,我们一起跪过的算法题[视频]](https://zhuanlan.zhihu.com/p/35175401) -- [《程序员面试金典(第5版)》](https://book.douban.com/subject/25753386/) -- [《剑指Offer》](https://book.douban.com/subject/25910559/) -- [python leetCode](https://github.com/HuberTRoy/leetCodek) diff --git a/docs/index.md b/docs/index.md deleted file mode 120000 index 32d46ee..0000000 --- a/docs/index.md +++ /dev/null @@ -1 +0,0 @@ -../README.md \ No newline at end of file From c39c1138093eafae15e6ef2ceff322523ff6785e Mon Sep 17 00:00:00 2001 From: Bowen 404 Date: Fri, 3 Dec 2021 11:35:33 +0800 Subject: [PATCH 2/4] =?UTF-8?q?=E4=BF=AE=E5=A4=8D=E9=A1=B5=E9=9D=A2?= =?UTF-8?q?=E8=B7=AF=E5=BE=84=E5=90=AB=E4=B8=AD=E6=96=87=E5=AF=BC=E8=87=B4?= =?UTF-8?q?404?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 修复页面路径含中文导致404, 删除yml和路径中所有对应中文 --- docs/00/why_and_how_to_learn.md | 30 +++ docs/01/ADT_OOP.md | 52 ++++ docs/01/bag_adt.py | 43 +++ docs/02/array_and_list.md | 54 ++++ docs/02/array_and_list.py | 45 ++++ docs/02/list.png | Bin 0 -> 248185 bytes docs/03/double_link_list.py | 121 +++++++++ docs/03/linked_list.md | 107 ++++++++ docs/03/linked_list.py | 219 ++++++++++++++++ docs/03/lru_cache.py | 134 ++++++++++ docs/04/array_queue.png | Bin 0 -> 109894 bytes docs/04/array_queue.py | 80 ++++++ docs/04/deque.py | 1 + docs/04/queue.md | 88 +++++++ docs/04/queue.py | 182 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docs/17/bst_search.png create mode 100644 docs/17/bst_worstcase.png create mode 100644 docs/17/find_successor.png create mode 100644 docs/17/predecessor_successor.png create mode 100644 docs/17/remove_interior_replace.png create mode 100644 docs/18/bfs.png create mode 100644 docs/18/graph.md create mode 100644 docs/18/graph.py create mode 100644 docs/18/graph_rep.png create mode 100644 docs/18/graph_road.png create mode 100644 docs/19/builtins.md create mode 100644 docs/20/interview.md diff --git a/docs/00/why_and_how_to_learn.md b/docs/00/why_and_how_to_learn.md new file mode 100644 index 0000000..96eb53c --- /dev/null +++ b/docs/00/why_and_how_to_learn.md @@ -0,0 +1,30 @@ +# 什么是算法和数据结构? + +你可能会在一些教材上看到这句话: + +程序 = 算法 + 数据结构 + +算法(Algorithm):是指解题方案的准确而完整的描述,是一系列解决问题的清晰指令,算法代表着用系统的方法描述解决问题的策略机制。也就是说,能够对一定规范的输入,在有限时间内获得所要求的输出。 + +数据结构(Data Structures):是计算机存储和组织数据的一种方式,可以用来高效地处理数据。 + +举个例子:二分查找就是一个非常经典的算法,而二分查找经常需要作用在一个有序数组上。这里二分就是一种折半的算法思想, +而数组是我们最常用的一种数据结构,支持根据下标快速访问。很多算法需要特定的数据结构来实现,所以经常把它们放到一块讲。 + +实际上,在真正的项目开发中,大部分时间都是 从数据库取数据 -> 数据操作和结构化 -> 返回给前端,在数据操作过程中需要合理地抽象, +组织、处理数据,如果选用了错误的数据结构,就会造成代码运行低效。这也是我们需要学习算法和数据结构的原因。 + +# 笨方法学算法 +这里我们用一种很原始的『笨』方法来学习算法:纸笔模拟。 + +- 阅读资料了解算法思想 +- 纸笔模拟尝试理解 +- 用自己熟悉的编程语言来实现 +- 单测 + +# 小问题 + +- 你还知道哪些经典的算法和数据结构? +- 学习算法你觉得需要哪些预备知识? +- 我们的业务代码开发中会涉及到算法吗? +- 你了解 redis 吗,你知道它有哪几个常用的数据结构吗?你知道它的底层实现方式吗? diff --git a/docs/01/ADT_OOP.md b/docs/01/ADT_OOP.md new file mode 100644 index 0000000..31748b5 --- /dev/null +++ b/docs/01/ADT_OOP.md @@ -0,0 +1,52 @@ +# Python 一切皆对象 + +举个例子,在 python 中我们经常使用的 list + +```py +l = list() # 实例化一个 list 对象 l +l.append(1) # 调用 l 的 append 方法 +l.append(2) +l.remove(1) +print(len(l)) # 调用对象的 `__len__` 方法 +``` + +在后面实现新的数据类型时,我们将使用 python 的 class 实现,它包含属性和方法。 +属性一般是使用某种特定的数据类型,而方法一般是对属性的操作。 +这里你只需了解这么多就行了, 我们不会使用继承等特性。 + + +# 什么是抽象数据类型 ADT + +实际上 python 内置的 list 就可以看成一种抽象数据类型。 + +ADT: Abstract Data Type,抽象数据类型,我们在组合已有的数据结构来实现一种新的数据类型, ADT 定义了类型的数据和操作。 + +我们以抽象一个背包(Bag) 数据类型来说明,背包是一种容器类型,我们可以给它添加东西,也可以移除东西,并且我们想知道背包里 +有多少东西。于是我们可以定义一个新的数据类型叫做 Bag. + +```py +class Bag: + """ 背包类型 """ + pass +``` + + +# 实现一个 Bag ADT +视频中我们将使用 python 的 class 来实现一个新的容器类型叫做 Bag。 + + +# 实现 ADT 我们应该注意什么? +- 如何选用恰当的数据结构作为存储? +- 选取的数据结构能否满足 ADT 的功能需求 +- 实现效率如何? + + +# 小问题: +- 你了解 python 的魔术方法吗? 比如 `__len__` ,调用 len(l) 的时候发生了什么? +- 你了解单测吗?我们以后将使用 pytest 运行单元测试,保证我们实现的数据结构和算法是正确的。你可以网上搜索下它的简单用法 + +# 延伸阅读: + +[数据结构与算法--ADT](http://www.atjiang.com/data-structures-using-python-ADT/) + +[http://www.nhu.edu.tw/~chun/CS-ch12-Abstract%20Data%20Types.pdf](http://www.nhu.edu.tw/~chun/CS-ch12-Abstract%20Data%20Types.pdf) diff --git a/docs/01/bag_adt.py b/docs/01/bag_adt.py new file mode 100644 index 0000000..4e5efc8 --- /dev/null +++ b/docs/01/bag_adt.py @@ -0,0 +1,43 @@ +# coding: utf8 + + +class Bag(object): + + def __init__(self, maxsize=10): + self.maxsize = maxsize + self._items = list() + + def add(self, item): + if len(self) >= self.maxsize: + raise Exception('Full') + self._items.append(item) + + def remove(self, item): + self._items.remove(item) + + def __len__(self): + return len(self._items) + + def __iter__(self): + for item in self._items: + yield item + + +def test_bag(): + bag = Bag() + + bag.add(1) + bag.add(2) + bag.add(3) + + assert len(bag) == 3 + + bag.remove(3) + assert len(bag) == 2 + + for i in bag: + print(i) + + +if __name__ == '__main__': + test_bag() diff --git a/docs/02/array_and_list.md b/docs/02/array_and_list.md new file mode 100644 index 0000000..21b5d90 --- /dev/null +++ b/docs/02/array_and_list.md @@ -0,0 +1,54 @@ +# 线性结构 +本节我们从最简单和常用的线性结构开始,并结合 Python 语言本身内置的数据结构和其底层实现方式来讲解。 +虽然本质上数据结构的思想是语言无关的,但是了解 Python 的实现方式有助于你避免一些坑。 + +我们会在代码中注释出操作的时间复杂度。 + + +# 数组 array + +数组是最常用到的一种线性结构,其实 python 内置了一个 array 模块,但是大部人甚至从来没用过它。 +Python 的 array 是内存连续、存储的都是同一数据类型的结构,而且只能存数值和字符。 + +我建议你课下看下 array 的文档:https://docs.python.org/2/library/array.html + +你可能很少会使用到它(我推荐你用 numpy.array),我将在视频里简单介绍下它的使用和工作方式,最常用的还是接下来要说的 list, +本章最后我们会用 list 来实现一个固定长度、并且支持所有 Python 数据类型的数组 Array. + + +# 列表 list +如果你学过 C++,list 其实和 C++ STL(标准模板库)中的 vector 很类似,它可能是你的 Python 学习中使用最频繁的数据结构之一。 +这里我们不再去自己实现 list,因为这是个 Python 提供的非常基础的数据类型,我会在视频中讲解它的工作方式和内存分配策略, +避免使用过程中碰到一些坑。当然如果你有毅力或者兴趣的了解底层是如何实现的,可以看看 cpython 解释器的具体实现。 + + +操作 | 平均时间复杂度 | +--------------------------------------|----------------| +list[index] | O(1) | +list.append | O(1) | +list.insert | O(n) | +list.pop(index), default last element | O(1) | +list.remove | O(n) | + +![](./list.png) + +# 用 list 实现 Array ADT +讲完了 list 让我们来实现一个定长的数组 Array ADT,在其他一些语言中,内置的数组结构就是定长的。 +这里我们会使用 list 作为 Array 的一个成员(代理)。具体请参考视频讲解和代码示例,后边我们会使用到这个 Array 类。 + + +# 小问题 +- 你知道线性结构的查找,删除,访问一个元素的平均时间复杂度吗?(后边我们会介绍这个概念,现在你可以简单地理解为一个操作需要的平均步骤) +- list 内存重新分配的时候为什么要有冗余?不会浪费空间吗? +- 当你频繁的pop list 的第一个元素的时候,会发生什么?如果需要频繁在两头增添元素,你知道更高效的数据结构吗?后边我们会讲到 + + +# 延伸阅读 + +[Python list implementation](https://www.laurentluce.com/posts/python-list-implementation/) + +[https://github.com/python/cpython/blob/master/Objects/listobject.c](https://github.com/python/cpython/blob/master/Objects/listobject.c) + + +# 勘误 +视频里的 Array.clear 方法有误。应该是 `for i in range(len(self._items))`,已经在后续所有使用到 Array 的代码里修正 diff --git a/docs/02/array_and_list.py b/docs/02/array_and_list.py new file mode 100644 index 0000000..11072a4 --- /dev/null +++ b/docs/02/array_and_list.py @@ -0,0 +1,45 @@ +# -*- coding: utf-8 -*- + +# https://docs.python.org/2/library/array.html +from array import array # python 提供的比较原始的 array 类 + + +arr = array('u', 'asdf') + +print(arr[0], arr[1], arr[2], arr[3]) + + +# 实现定长的 Array ADT,省略了边界检查等 + +class Array(object): + + def __init__(self, size=32): + self._size = size + self._items = [None] * size + + def __getitem__(self, index): + return self._items[index] + + def __setitem__(self, index, value): + self._items[index] = value + + def __len__(self): + return self._size + + def clear(self, value=None): + for i in range(len(self._items)): + 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next=None): + self.value, self.prev, self.next = value, prev, next + + +class CircularDoubleLinkedList(object): + """循环双端链表 ADT + 多了个循环其实就是把 root 的 prev 指向 tail 节点,串起来 + """ + + def __init__(self, maxsize=None): + self.maxsize = maxsize + node = Node() + node.next, node.prev = node, node + self.root = node + self.length = 0 + + def __len__(self): + return self.length + + def headnode(self): + return self.root.next + + def tailnode(self): + return self.root.prev + + def append(self, value): # O(1), 你发现一般不用 for 循环的就是 O(1),有限个步骤 + if self.maxsize is not None and len(self) >= self.maxsize: + raise Exception('LinkedList is Full') + node = Node(value=value) + tailnode = self.tailnode() or self.root + + tailnode.next = node + node.prev = tailnode + node.next = self.root + self.root.prev = node + self.length += 1 + + def appendleft(self, value): + if self.maxsize is not None and len(self) >= self.maxsize: + raise Exception('LinkedList is Full') + node = Node(value=value) + if self.root.next is self.root: # empty + node.next = self.root + node.prev = self.root + self.root.next = node + self.root.prev = node + else: + node.prev = self.root + headnode = self.root.next + node.next = headnode + headnode.prev = node + self.root.next = node + self.length += 1 + + def remove(self, node): # O(1),传入node 而不是 value 我们就能实现 O(1) 删除 + """remove + :param node # 在 lru_cache 里实际上根据key 保存了整个node: + """ + if node is self.root: + return + else: # + node.prev.next = node.next + node.next.prev = node.prev + self.length -= 1 + return node + + def iter_node(self): + if self.root.next is self.root: + return + curnode = self.root.next + while curnode.next is not self.root: + yield curnode + curnode = curnode.next + yield curnode + + def __iter__(self): + for node in self.iter_node(): + yield node.value + + def iter_node_reverse(self): + """相比单链表独有的反序遍历""" + if self.root.prev is self.root: + return + curnode = self.root.prev + while curnode.prev is not self.root: + yield curnode + curnode = curnode.prev + yield curnode + + +def test_double_link_list(): + dll = CircularDoubleLinkedList() + assert len(dll) == 0 + + dll.append(0) + dll.append(1) + dll.append(2) + + assert list(dll) == [0, 1, 2] + + assert [node.value for node in dll.iter_node()] == [0, 1, 2] + assert [node.value for node in dll.iter_node_reverse()] == [2, 1, 0] + + headnode = dll.headnode() + assert headnode.value == 0 + dll.remove(headnode) + assert len(dll) == 2 + assert [node.value for node in dll.iter_node()] == [1, 2] + + dll.appendleft(0) + assert [node.value for node in dll.iter_node()] == [0, 1, 2] + + +if __name__ == '__main__': + test_double_link_list() diff --git a/docs/03/linked_list.md b/docs/03/linked_list.md new file mode 100644 index 0000000..9b3f81f --- /dev/null +++ b/docs/03/linked_list.md @@ -0,0 +1,107 @@ +# 链式结构 + +上一节讲到了支持随机访问的线性结构,这次我们开始讲链式结构, 视频里我会说下这两种结构的区别,然后讲解最常见的单链表和双链表。 +之前在专栏文章[那些年,我们一起跪过的算法题[视频]](https://zhuanlan.zhihu.com/p/35175401)里实现过一个 lru_cache, +使用到的就是循环双端链表,如果感觉这篇文章有点难理解,我们这里将会循序渐进地来实现。 +后边讲到哈希表的冲突解决方式的时候,我们会再次提到链表。 + +上一节我们分析了 list 的各种操作是如何实现的,如果你还有印象的话,list +在头部进行插入是个相当耗时的操作(需要把后边的元素一个一个挪个位置)。假如你需要频繁在数组两头增删,list 就不太合适。 +今天我们介绍的链式结构将摆脱这个缺陷,当然了链式结构本身也有缺陷,比如你不能像数组一样随机根据下标访问,你想查找一个元素只能老老实实从头遍历。 +所以嘛,学习和了解数据结构的原理和实现你才能准确地选择到底什么时候该用什么数据结构,而不是瞎选导致代码性能很差。 + + +# 单链表 +和线性结构不同,链式结构内存不连续的,而是一个个串起来的,这个时候就需要每个链接表的节点保存一个指向下一个节点的指针。 +这里可不要混淆了列表和链表(它们的中文发音类似,但是列表 list 底层其实还是线性结构,链表才是真的通过指针关联的链式结构)。 +看到指针你也不用怕,这里我们用的 python,你只需要一个简单赋值操作就能实现,不用担心 c 语言里复杂的指针。 + +先来定义一个链接表的节点,刚才说到有一个指针保存下一个节点的位置,我们叫它 next, 当然还需要一个 value 属性保存值 + +```py +class Node(object): + def __init__(self, value, next=None): + self.value = value + self.next = next +``` +然后就是我们的单链表 LinkedList ADT: + +```py +class LinkedList(object): + """ 链接表 ADT + [root] -> [node0] -> [node1] -> [node2] + """ +``` +实现我们会在视频中用画图来模拟并且手动代码实现,代码里我们会标识每个步骤的时间复杂度。这里请高度集中精力, +虽然链表的思想很简单,但是想要正确写对链表的操作代码可不容易,稍不留神就可能丢失一些步骤。 +这里我们还是会用简单的单测来验证代码是否按照预期工作。 + +来看下时间复杂度: + +链表操作 | 平均时间复杂度 | +------------------------------|----------------| +linked_list.append(value) | O(1) | +linked_list.appendleft(value) | O(1) | +linked_list.find(value) | O(n) | +linked_list.remove(value) | O(n) | + + +# 双链表 +上边我们亲自实现了一个单链表,但是能看到很明显的问题,单链表虽然 append 是 O(1),但是它的 find 和 remove 都是 O(n)的, +因为删除你也需要先查找,而单链表查找只有一个方式就是从头找到尾,中间找到才退出。 +这里我之前提到过如果要实现一个 lru 缓存(访问时间最久的踢出),我们需要在一个链表里能高效的删除元素, +并把它追加到访问表的最后一个位置,这个时候单链表就满足不了了, +因为缓存在 dict 里查找的时间是 O(1),你更新访问顺序就 O(n)了,缓存就没了优势。 + +这里就要使用到双链表了,相比单链表来说,每个节点既保存了指向下一个节点的指针,同时还保存了上一个节点的指针。 + +```py +class Node(object): + # 如果节点很多,我们可以用 __slots__ 来节省内存,把属性保存在一个 tuple 而不是 dict 里 + # 感兴趣可以自行搜索 python __slots__ + __slots__ = ('value', 'prev', 'next') + + def __init__(self, value=None, prev=None, next=None): + self.value, self.prev, self.next = value, prev, next +``` + +对, 就多了 prev,有啥优势嘛? + +- 看似我们反过来遍历双链表了。反过来从哪里开始呢?我们只要让 root 的 prev 指向 tail 节点,不就串起来了吗? +- 直接删除节点,当然如果给的是一个值,我们还是需要查找这个值在哪个节点? - 但是如果给了一个节点,我们把它拿掉,直接让它的前后节点互相指过去不就行了?哇欧,删除就是 O(1) 了,两步操作就行啦 + +好,废话不多说,我们在视频里介绍怎么实现一个双链表 ADT。你可以直接在本项目的 `docs/03_链表/double_link_list.py` 找到代码。 +最后让我们看下它的时间复杂度:(这里 CircularDoubleLinkedList 取大写字母缩写为 cdll) + +循环双端链表操作 | 平均时间复杂度 | +---------------------------------------|----------------| +cdll.append(value) | O(1) | +cdll.appendleft(value) | O(1) | +cdll.remove(node),注意这里参数是 node | O(1) | +cdll.headnode() | O(1) | +cdll.tailnode() | O(1) | + + +# 小问题: +- 这里单链表我没有实现 insert 方法,你能自己尝试实现吗? insert(value, new_value),我想在某个值之前插入一个值。你同样需要先查找,所以这个步骤也不够高效。 +- 你能尝试自己实现个 lru cache 吗?需要使用到我们这里提到的循环双端链表 +- 借助内置的 collections.OrderedDict,它有两个方法 popitem 和 move_to_end,我们可以迅速实现一个 LRU cache。请你尝试用 OrderedDict 来实现。 +- python 内置库的哪些数据结构使用到了本章讲的链式结构? + + +# 相关阅读 + +[那些年,我们一起跪过的算法题- Lru cache[视频]](https://zhuanlan.zhihu.com/p/35175401) + +# 勘误: + +视频中 LinkedList.remove 方法讲解有遗漏, linked_list.py 文件已经修正,请读者注意。具体请参考 [fix linked_list & add gitigonre](https://github.com/PegasusWang/python_data_structures_and_algorithms/pull/3)。视频最后增加了一段勘误说明。 + +# Leetcode + +反转链表 [reverse-linked-list](https://leetcode.com/problems/reverse-linked-list/) + +这里有一道关于 LRU 的练习题你可以尝试下。 +[LRU Cache](https://leetcode.com/problems/lru-cache/description/) + +合并两个有序链表 [merge-two-sorted-lists](/https://leetcode.com/problems/merge-two-sorted-lists/submissions/) diff --git a/docs/03/linked_list.py b/docs/03/linked_list.py new file mode 100644 index 0000000..b64eb1c --- /dev/null +++ b/docs/03/linked_list.py @@ -0,0 +1,219 @@ +# -*- coding: utf-8 -*- + + +class Node(object): + def __init__(self, value=None, next=None): # 这里我们 root 节点默认都是 None,所以都给了默认值 + self.value = value + self.next = next + + def __str__(self): + """方便你打出来调试,复杂的代码可能需要断点调试""" + return ''.format(self.value, self.next) + + __repr__ = __str__ + + +class LinkedList(object): + """ 链接表 ADT + [root] -> [node0] -> [node1] -> [node2] + """ + + def __init__(self, maxsize=None): + """ + :param maxsize: int or None, 如果是 None,无限扩充 + """ + self.maxsize = maxsize + self.root = Node() # 默认 root 节点指向 None + self.tailnode = None + self.length = 0 + + def __len__(self): + return self.length + + def append(self, value): # O(1) + if self.maxsize is not None and len(self) >= self.maxsize: + raise Exception('LinkedList is Full') + node = Node(value) # 构造节点 + tailnode = self.tailnode + if tailnode is None: # 还没有 append 过,length = 0, 追加到 root 后 + self.root.next = node + else: # 否则追加到最后一个节点的后边,并更新最后一个节点是 append 的节点 + tailnode.next = node + self.tailnode = node + self.length += 1 + + def appendleft(self, value): + if self.maxsize is not None and len(self) >= self.maxsize: + raise Exception('LinkedList is Full') + node = Node(value) + if self.tailnode is None: # 如果原链表为空,插入第一个元素需要设置 tailnode + self.tailnode = node + + headnode = self.root.next + self.root.next = node + node.next = headnode + self.length += 1 + + def __iter__(self): + for node in self.iter_node(): + yield node.value + + def iter_node(self): + """遍历 从 head 节点到 tail 节点""" + curnode = self.root.next + while curnode is not self.tailnode: # 从第一个节点开始遍历 + yield curnode + curnode = curnode.next # 移动到下一个节点 + if curnode is not None: + yield curnode + + def remove(self, value): # O(n) + """ 删除包含值的一个节点,将其前一个节点的 next 指向被查询节点的下一个即可 + + :param value: + """ + prevnode = self.root # + for curnode in self.iter_node(): + if curnode.value == value: + prevnode.next = curnode.next + if curnode is self.tailnode: # NOTE: 注意更新 tailnode + if prevnode is self.root: + self.tailnode = None + else: + self.tailnode = prevnode + del curnode + self.length -= 1 + return 1 # 表明删除成功 + else: + prevnode = curnode + return -1 # 表明删除失败 + + def find(self, value): # O(n) + """ 查找一个节点,返回序号,从 0 开始 + + :param value: + """ + index = 0 + for node in self.iter_node(): # 我们定义了 __iter__,这里就可以用 for 遍历它了 + if node.value == value: + return index + index += 1 + return -1 # 没找到 + + def popleft(self): # O(1) + """ 删除第一个链表节点 + """ + if self.root.next is None: + raise Exception('pop from empty LinkedList') + headnode = self.root.next + self.root.next = headnode.next + self.length -= 1 + value = headnode.value + + if self.tailnode is headnode: # 勘误:增加单节点删除 tailnode 处理 + self.tailnode = None + del headnode + return value + + def clear(self): + for node in self.iter_node(): + del node + self.root.next = None + self.length = 0 + self.tailnode = None + + def reverse(self): + """反转链表""" + curnode = self.root.next + self.tailnode = curnode # 记得更新 tailnode,多了这个属性处理起来经常忘记 + prevnode = None + + while curnode: + nextnode = curnode.next + curnode.next = prevnode + + if nextnode is None: + self.root.next = curnode + + prevnode = curnode + curnode = nextnode + + +def test_linked_list(): + ll = LinkedList() + + ll.append(0) + ll.append(1) + ll.append(2) + ll.append(3) + + assert len(ll) == 4 + assert ll.find(2) == 2 + assert ll.find(-1) == -1 + + assert ll.remove(0) == 1 + assert ll.remove(10) == -1 + assert ll.remove(2) == 1 + assert len(ll) == 2 + assert list(ll) == [1, 3] + assert ll.find(0) == -1 + + ll.appendleft(0) + assert list(ll) == [0, 1, 3] + assert len(ll) == 3 + + headvalue = ll.popleft() + assert headvalue == 0 + assert len(ll) == 2 + assert list(ll) == [1, 3] + + assert ll.popleft() == 1 + assert list(ll) == [3] + ll.popleft() + assert len(ll) == 0 + assert ll.tailnode is None + + ll.clear() + assert len(ll) == 0 + assert list(ll) == [] + + +def test_linked_list_remove(): + ll = LinkedList() + ll.append(3) + ll.append(4) + ll.append(5) + ll.append(6) + ll.append(7) + ll.remove(7) + print(list(ll)) + +def test_single_node(): + # https://github.com/PegasusWang/python_data_structures_and_algorithms/pull/21 + ll = LinkedList() + ll.append(0) + ll.remove(0) + ll.appendleft(1) + assert list(ll) == [1] + +def test_linked_list_reverse(): + ll = LinkedList() + n = 10 + for i in range(n): + ll.append(i) + ll.reverse() + assert list(ll) == list(reversed(range(n))) + + +def test_linked_list_append(): + ll = LinkedList() + ll.appendleft(1) + ll.append(2) + assert list(ll) == [1, 2] + + +if __name__ == '__main__': + test_single_node() + test_linked_list() + test_linked_list_append() + test_linked_list_reverse() diff --git a/docs/03/lru_cache.py b/docs/03/lru_cache.py new file mode 100644 index 0000000..426a309 --- /dev/null +++ b/docs/03/lru_cache.py @@ -0,0 +1,134 @@ +""" +python3 only +LRU cache +""" +from collections import OrderedDict +from functools import wraps + + +def fib(n): + if n <= 1: # 0 or 1 + return n + return f(n - 1) + f(n - 2) # 由于涉及到重复计算,这个递归函数在 n 大了以后会非常慢。 O(2^n) + + +""" +下边就来写一个缓存装饰器来优化它。传统方法是用个数组记录之前计算过的值,但是这种方式不够 Pythonic +""" + + +def cache(func): + """先引入一个简单的装饰器缓存,其实原理很简单,就是内部用一个字典缓存已经计算过的结果""" + store = {} + + @wraps(func) + def _(n): # 这里函数没啥意义就随便用下划线命名了 + if n in store: + return store[n] + else: + res = func(n) + store[n] = res + return res + return _ + + +@cache +def f(n): + if n <= 1: # 0 or 1 + return n + return f(n - 1) + f(n - 2) + + +""" +问题来了,假如空间有限怎么办,我们不可能一直向缓存塞东西,当缓存达到一定个数之后,我们需要一种策略踢出一些元素, +用来给新的元素腾出空间。 +一般缓存失效策略有 +- LRU(Least-Recently-Used): 替换掉最近请求最少的对象,实际中使用最广。cpu缓存淘汰和虚拟内存效果好,web应用欠佳 +- LFU(Least-Frequently-Used): 缓存污染问题(一个先前流行的缓存对象会在缓存中驻留很长时间) +- First in First out(FIFO) +- Random Cache: 随机选一个删除 + +LRU 是常用的一个,比如 redis 就实现了这个策略,这里我们来模拟实现一个。 +要想实现一个 LRU,我们需要一种方式能够记录访问的顺序,并且每次访问之后我们要把最新使用到的元素放到最后(表示最新访问)。 +当容量满了以后,我们踢出最早访问的元素。假如用一个链表来表示的话: + +[1] -> [2] -> [3] + +假设最后边是最后访问的,当访问到一个元素以后,我们把它放到最后。当容量满了,我们踢出第一个元素就行了。 +一开始的想法可能是用一个链表来记录访问顺序,但是单链表有个问题就是如果访问了中间一个元素,我们需要拿掉它并且放到链表尾部。 +而单链表无法在O(1)的时间内删除一个节点(必须要先搜索到它),但是双端链表可以,因为一个节点记录了它的前后节点, +只需要把要删除的节点的前后节点链接起来就行了。 +还有个问题是如何把删除后的节点放到链表尾部,如果是循环双端链表就可以啦,我们有个 root 节点链接了首位节点, +只需要让 root 的前一个指向这个被删除节点,然后让之前的最后一个节点也指向它就行了。 + +使用了循环双端链表之后,我们的操作就都是 O(1) 的了。这也就是使用一个 dict 和一个 循环双端链表 实现LRU 的思路。 +不过一般我们使用内置的 OrderedDict(原理和这个类似)就好了,要实现一个循环双端链表是一个不简单的事情,因为链表操作很容易出错。 + +补充:其实 lru 有个缺点就是额外的链表比较占用空间,如果你感兴趣的话可以看看 redis 如何实现的 lru 算法 +""" + + +class LRUCache: + def __init__(self, capacity=128): + self.capacity = capacity + # 借助 OrderedDict 我们可以快速实现一个 LRUCache,OrderedDict 内部其实也是使用循环双端链表实现的 + # OrderedDict 有两个重要的函数用来实现 LRU,一个是 move_to_end,一个是 popitem,请自己看文档 + self.od = OrderedDict() + + def get(self, key, default=None): + val = self.od.get(key, default) # 如果没有返回 default,保持 dict 语义 + self.od.move_to_end(key) # 每次访问就把key 放到最后表示最新访问 + return val + + def add_or_update(self, key, value): + if key in self.od: # update + self.od[key] = value + self.od.move_to_end(key) + else: # insert + self.od[key] = value + if len(self.od) > self.capacity: # full + self.od.popitem(last=False) + + def __call__(self, func): + """ + 一个简单的 LRU 实现。有一些问题需要思考下: + + - 这里为了简化默认参数只有一个数字 n,假如可以传入多个参数,如何确定缓存的key 呢? + - 这里实现没有考虑线程安全的问题,要如何才能实现线程安全的 LRU 呢?当然如果不是多线程环境下使用是不需要考虑的 + - 假如这里没有用内置的 dict,你能使用 redis 来实现这个 LRU 吗,如果使用了 redis,我们可以存储更多数据到服务器。而使用字典实际上是缓存了Python进程里(localCache)。 + - 这里只是实现了 lru 策略,你能同时实现一个超时 timeout 参数吗?比如像是memcache 实现的 lazy expiration 策略 + - LRU有个缺点就是,对于周期性的数据访问会导致命中率迅速下降,有一种优化是 LRU-K,访问了次数达到 k 次才会将数据放入缓存 + """ + def _(n): + if n in self.od: + return self.get(n) + else: + val = func(n) + self.add_or_update(n, val) + return val + return _ + + +@LRUCache(10) +def f_use_lru(n): + if n <= 1: # 0 or 1 + return n + return f_use_lru(n - 1) + f_use_lru(n - 2) + + +def test(): + import time + beg = time.time() + for i in range(34): + print(f(i)) + print(time.time() - beg) + beg = time.time() + for i in range(34): + print(f_use_lru(i)) + print(time.time() - beg) + + +# TODO 要怎么给 lru 写单测? + +if __name__ == '__main__': + test() diff --git a/docs/04/array_queue.png b/docs/04/array_queue.png new file mode 100644 index 0000000000000000000000000000000000000000..25368872ba134e559a6110909dd62486a0079b5f GIT binary patch literal 109894 zcmdpcWmjCwwsr%JyGw9Qf(Iu^LvVKs?(VLQyF0;xYjAf6!6m`n-QD3OJK5*nG4A;V zZ;$%0x@uL`no?7qS)p<=qR0pY2mk;8SzJs=0RRA^0ss)(a4@euge-;Z0017D*{4r( z;-5Z|%Gui(n^_tG0BV7WZm>#N(lyRwqcCboR6FVMn@DjEqxinBkMg190>Wr=KqWZ= zxh@a}Gg-)6@**%qY_=gbmK<;sMPTK`=fr1>w_z-GZEn9Lt+1t}r4SD=n;XY@KRHOnszKV2V$ z+#mqzdz_Q)p|Ku2Snfm`K=xJrv4en`+&s2w4s#$1Uuh<7+)7CR@o-rObsWpjnRvT zX;@enZX_1FP#?lK$U{2c?~Lmq5~O}|FWP%Av^V$I^~@(y(85 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self._size + + def clear(self, value=None): + for i in range(len(self._items)): + self._items[i] = value + + def __iter__(self): + for item in self._items: + yield item + + +class FullError(Exception): + pass + + +class ArrayQueue(object): + def __init__(self, maxsize): + self.maxsize = maxsize + self.array = Array(maxsize) + self.head = 0 + self.tail = 0 + + def push(self, value): + if len(self) >= self.maxsize: + raise FullError('queue full') + self.array[self.head % self.maxsize] = value + self.head += 1 + + def pop(self): + value = self.array[self.tail % self.maxsize] + self.tail += 1 + return value + + def __len__(self): + return self.head - self.tail + + +def test_queue(): + import pytest # pip install pytest + size = 5 + q = ArrayQueue(size) + for i in range(size): + q.push(i) + + with pytest.raises(FullError) as excinfo: # 我们来测试是否真的抛出了异常 + q.push(size) + assert 'full' in str(excinfo.value) + + assert len(q) == 5 + + assert q.pop() == 0 + assert q.pop() == 1 + + q.push(5) + + assert len(q) == 4 + + assert q.pop() == 2 + assert q.pop() == 3 + assert q.pop() == 4 + assert q.pop() == 5 + + assert len(q) == 0 diff --git a/docs/04/deque.py b/docs/04/deque.py new file mode 100644 index 0000000..0cc58f2 --- /dev/null +++ b/docs/04/deque.py @@ -0,0 +1 @@ +# 留给读者练习,如果实在想不出,在第 5 章栈里 stack.py 会有实现 diff --git a/docs/04/queue.md b/docs/04/queue.md new file mode 100644 index 0000000..bae29fb --- /dev/null +++ b/docs/04/queue.md @@ -0,0 +1,88 @@ +# 队列和栈 + +前面讲了线性和链式结构,如果你顺利掌握了,下边的队列和栈就小菜一碟了。因为我们会用前两章讲到的东西来实现队列和栈。 +之所以放到一起讲是因为这两个东西很类似,队列是先进先出结构(FIFO, first in first out), +栈是后进先出结构(LIFO, last in first out)。 + +生活中的数据结构: + +- 队列。没错就是咱平常排队,第一个来的第一个走 + +本章我们详细讲讲常用的队列 + +# 队列 Queue + +这里卖个关子,如果你熟悉了上两节讲的内容,这里你会选取哪个数据结构作为队列的底层存储? +还记得第一章讲的如何实现 ADT 吗?我视频了说了三个注意事项: + +- 1.如何选用恰当的数据结构作为存储? +- 2.选取的数据结构能否满足 ADT 的功能需求 +- 3.实现效率如何? + +我们先来看看 list 可以不?对照这个三个需求,看看能否满足: + +- 1.我们选择了 list +- 2.看起来队列需要从头删除,向尾部增加元素,也就是 list.pop(0) 和 list.append(element) +- 3.嗯,貌似 list.pop(0) 会导致所有其后所有元素向前移动一个位置,O(n)复杂度。append 平均倒是O(1),但是如果内存不够还要重新分配内存。 + +你看,使用了 list 的话频繁 pop(0) 是非常低效的。(当然list 实现还有另一种方式就是插入用 list.insert(0, item),删除用list.pop()) + +脑子再转转, 我们第二章实现了 链表 LinkedList,看看能否满足要求: + +- 1.这里选择 LinkedList +- 2.删除头元素 LinkedList.popleft(),追加 append(element)。都可以满足 +- 3.哇欧,这两个操作都是 O(1) 的,完美。 + +好, 就用 LinkedList 了,我们开始实现,具体看视频。这次实现我们还将演示自定义异常和测试异常。 + + +# 用数组实现队列 + +难道用数组就不能实现队列了吗?其实还是可以的。只不过数组是预先分配固定内存的,所以如果你知道了队列的最大长度,也是 +可以用数组来实现的。 + +想象一下,队列就俩操作,进进出出,一进一出,pop 和 push 操作。 +似乎只要两个下标 head, tail 就可以了。 当我们 push 的时候赋值并且前移 head,pop 的时候前移 tail 就可以了。你可以在纸上 +模拟下试试。列队的长度就是 head-pop,这个长度必须不能大于初始化的最大程度。 + +如果 head 先到了数组末尾咋办?重头来呗,只要我们保证 tail 不会超过 head 就行。 + +head = 0,1,2,3,4 ... 0,1,2,3,4 ... + +重头再来,循环往复,仿佛一个轮回。。。。 +怎么重头来呢?看上边数组的规律你如果还想不起来用取模,估计小学数学是体育老师教的。 + +```py +maxsize = 5 +for i in range(100): + print(i % maxsize) +``` + +![](./array_queue.png) + +我们来实现一个空间有限的循环队列。ArrayQueue,它的实现很简单,但是缺点是需要预先知道队列的长度来分配内存。 + + +# 双端队列 Double ended Queue +看了视频相信你已经会实现队列了,你可能还听过双端队列。上边讲到的队列 队头出,尾尾进,我们如果想头部和尾巴都能进能出呢? +这就是双端队列了,如果你用过 collections.deque 模块,就是这个东西。他能高效在两头操作。 + +假如让你实现你能想起来嘛? +似乎我们需要一个能 append() appendleft() popleft() pop() 都是 O(1) 的数据结构。 + +上边我们实现 队列的 LinkedList 可以吗?貌似就差一个 pop() 最后边的元素无法实现了。 +对,我们还有双端链表。它有这几个方法: + +- append +- appendleft +- headnode() +- tailnode() +- remove(node) # O(1) + +啊哈,似乎删除头尾都可以啦,而且都是 O(1) 的,完美。 +交给你一个艰巨的任务,实现双端队列 Deque() ADT。你可以参考前几章的任何代码,挑战一下这个任务,别忘记写单元测试呦。当然如果没想出来也没关系,后边我们实现栈的时候还会用到它,那里我们会实现这个代码。 + + +# 思考题 +- 你能用 python 的 deque 来实现 queue ADT 吗? +- 哪些经典算法里用到了队列呢? diff --git a/docs/04/queue.py b/docs/04/queue.py new file mode 100644 index 0000000..4ca6fed --- /dev/null +++ b/docs/04/queue.py @@ -0,0 +1,182 @@ +# -*- coding: utf-8 -*- + +from collections import deque + +# NOTE:注意这里是第三章 linked_list.py 里的内容,为了使文件自包含,我直接拷贝过来的 + + +class Node(object): + def __init__(self, value=None, next=None): # 这里我们 root 节点默认都是 None,所以都给了默认值 + self.value = value + self.next = next + + def __str__(self): + """方便你打出来调试,复杂的代码可能需要断点调试""" + return ''.format(self.value, self.next) + + __repr__ = __str__ + + +class LinkedList(object): + """ 链接表 ADT + [root] -> [node0] -> [node1] -> [node2] + """ + + def __init__(self, maxsize=None): + """ + :param maxsize: int or None, 如果是 None,无限扩充 + """ + self.maxsize = maxsize + self.root = Node() # 默认 root 节点指向 None + self.tailnode = None + self.length = 0 + + def __len__(self): + return self.length + + def append(self, value): # O(1) + if self.maxsize is not None and len(self) >= self.maxsize: + raise Exception('LinkedList is Full') + node = Node(value) # 构造节点 + tailnode = self.tailnode + if tailnode is None: # 还没有 append 过,length = 0, 追加到 root 后 + self.root.next = node + else: # 否则追加到最后一个节点的后边,并更新最后一个节点是 append 的节点 + tailnode.next = node + self.tailnode = node + self.length += 1 + + def appendleft(self, value): + headnode = self.root.next + node = Node(value) + self.root.next = node + node.next = headnode + self.length += 1 + + def __iter__(self): + for node in self.iter_node(): + yield node.value + + def iter_node(self): + """遍历 从 head 节点到 tail 节点""" + curnode = self.root.next + while curnode is not self.tailnode: # 从第一个节点开始遍历 + yield curnode + curnode = curnode.next # 移动到下一个节点 + yield curnode + + def remove(self, value): # O(n) + """ 删除包含值的一个节点,将其前一个节点的 next 指向被查询节点的下一个即可 + + :param value: + """ + prevnode = self.root # + curnode = self.root.next + for curnode in self.iter_node(): + if curnode.value == value: + prevnode.next = curnode.next + del curnode + self.length -= 1 + return 1 # 表明删除成功 + else: + prevnode = curnode + return -1 # 表明删除失败 + + def find(self, value): # O(n) + """ 查找一个节点,返回序号,从 0 开始 + + :param value: + """ + index = 0 + for node in self.iter_node(): # 我们定义了 __iter__,这里就可以用 for 遍历它了 + if node.value == value: + return index + index += 1 + return -1 # 没找到 + + def popleft(self): # O(1) + """ 删除第一个链表节点 + """ + if self.root.next is None: + raise Exception('pop from empty LinkedList') + headnode = self.root.next + self.root.next = headnode.next + self.length -= 1 + value = headnode.value + del headnode + return value + + def clear(self): + for node in self.iter_node(): + del node + self.root.next = None + self.length = 0 + +###################################################### +# 下边是 Queue 实现 +###################################################### + + +class EmptyError(Exception): + """自定义异常""" + pass + + +class Queue(object): + def __init__(self, maxsize=None): + self.maxsize = maxsize + self._item_link_list = LinkedList() + + def __len__(self): + return len(self._item_link_list) + + def push(self, value): # O(1) + """ 队尾添加元素 """ + return self._item_link_list.append(value) + + def pop(self): + """队列头部删除元素""" + if len(self) <= 0: + raise EmptyError('empty queue') + return self._item_link_list.popleft() + + +def test_queue(): + q = Queue() + q.push(0) + q.push(1) + q.push(2) + + assert len(q) == 3 + + assert q.pop() == 0 + assert q.pop() == 1 + assert q.pop() == 2 + + import pytest # pip install pytest + with pytest.raises(EmptyError) as excinfo: # 我们来测试是否真的抛出了异常 + q.pop() # 继续调用会抛出异常 + assert 'empty queue' == str(excinfo.value) + + +class MyQueue: + """ + 使用 collections.deque 可以迅速实现一个队列 + """ + def __init__(self): + self.items = deque() + + def append(self, val): + return self.items.append(val) + + def pop(self): + return self.items.popleft() + + def __len__(self): + return len(self.items) + + def empty(self): + return len(self.items) == 0 + + def front(self): + return self.items[0] diff --git a/docs/05/stack.md b/docs/05/stack.md new file mode 100644 index 0000000..e220d8a --- /dev/null +++ b/docs/05/stack.md @@ -0,0 +1,63 @@ +# 栈 + +栈这个词实际上在计算机科学里使用很多,除了数据结构外,还有内存里的栈区 (和堆对应),熟悉 C 系语言的话应该不会陌生。 +上一章我们讲到了先进先出 queue,其实用 python 的内置类型 collections.deque 或者我们自己实现的 LinkedList 来实现它都很简单。 +本章我们讲讲 后进先出的栈。 + +生活中的数据结构: + +- 栈。好比在桶里头放盘子,先放的盘子放在了底下,后来的盘子放在上边。你要拿的时候,也是先拿最上边的。 + +栈其实也很简单,因为基础操作就俩,一个 push 和一个 pop,咦,咋和队列一样的? +确实方法名字一样,但是得到的结果可是不同的。 + + +# 栈 ADT + +上一章我介绍了我们怎样选取恰到的数据结构来实现新的 ADT?你能想到这里我们应该使用之前提到的哪个数据结构来实现吗? +你的大脑可能开始高(gui)速(su)旋转了,上几章学过的 array, list, deque, LinkedList, CircularDoubleLinkedList, queue +等在大脑里呼啸而过,这个时候可能已经一脸愁容了,到底该选啥? + +还用问嘛,当然是时间复杂度最小的啦,大部分情况下空间都是够用的。 +其实你会发现栈比队列还简单,因为它只在顶上操作(想象装着盘子的桶),如果有一种数据结构能方便在尾部增减元素不就满足需求了吗。 +这个时候如果你忘记了,可以翻翻前几章,看看哪个数据结构符合要求。 + +想一下,似乎 CircularDoubleLinkedList 循环双端队列是满足的,因为增删最后一个元素都是 O(1)。 +不过看了下示例代码,似乎没有 pop() 方法,对,因为我已经把实现 deque 作为思考题了。😂 +如果之前你没写出来也没关系,这里我们会再实现它。 + + +视频里我们将借助 CircularDoubleLinkedList 实现 双端队列 Deque ,并且用 Deque 实现 Stack。 + + +# Stack over flow 什么鬼? +嗯,stackoverflow 不是一个程序员问答网站吗?没错。 +函数的临时变量是存储在栈区的,如果你不幸写了一个没有出口的递归函数,就会这个错。不信你试试: + + +```py +def infinite_fib(n): + return infinite_fib(n-1) + infinite_fib(n-2) +infinite_fib(10) +``` + +一大段输出之后就会出现异常: RecursionError: maximum recursion depth exceeded。 +后边会讲到递归,递归是初学者比较难理解的概念,在树的遍历等地方还会看到它。 + + +# 数据结构头脑风暴法 + +当我们不知道使用什么数据结构来解决问题的时候,《程序员面试金典》这本书的第六章提到了一种方式叫做『数据结构头脑风暴法』。 +这种笨方法就是快速过一遍数据结构的列表,然后逐一尝试各种数据结构看看哪个最适合。 + +在你实现一个更高级的数据结构的时候,如果脑子没有思路,不妨尝试下这个方法,迅速过一遍你所知道的数据结构,看看哪种最适合。(从每个操作的时间复杂度和空间复杂度分析寻找最优解) + +# 思考题 +- 上一章我们用数组实现了队列,其实也能用数组来实现栈,你能自己用数组来实现一个栈的 ADT 吗? +- 实际上借助 python 内置的 list/collections.deque 结构就很容易实现一个栈,请你尝试实现,本章我们全部使用自己编写的数据结构而没用到 python 内置的数据结构。 +- 这里我们自己实现了 Deque,你能用 python 内置的 collections.deque 实现栈吗?有轮子能直接用的话看起来就简单多了,这里我们为了学习数据结构的实现就避免了直接使用内置结构 +- 哪些经典算法里使用到了栈呢? + +# Leetcode 练习 + +https://leetcode.com/problems/implement-queue-using-stacks/ diff --git a/docs/05/stack.py b/docs/05/stack.py new file mode 100644 index 0000000..8cd8970 --- /dev/null +++ b/docs/05/stack.py @@ -0,0 +1,188 @@ +# -*- coding: utf-8 -*- + +# NOTE: 这里拷贝的 double_link_list.py 里的代码 + +from collections import deque + + +class Node(object): + + def __init__(self, value=None, prev=None, next=None): + self.value, self.prev, self.next = value, prev, next + + +class CircularDoubleLinkedList(object): + """循环双端链表 ADT + 多了个循环其实就是把 root 的 prev 指向 tail 节点,串起来 + """ + + def __init__(self, maxsize=None): + self.maxsize = maxsize + node = Node() + node.next, node.prev = node, node + self.root = node + self.length = 0 + + def __len__(self): + return self.length + + def headnode(self): + return self.root.next + + def tailnode(self): + return self.root.prev + + def append(self, value): # O(1), 你发现一般不用 for 循环的就是 O(1),有限个步骤 + if self.maxsize is not None and len(self) >= self.maxsize: + raise Exception('LinkedList is Full') + node = Node(value=value) + tailnode = self.tailnode() or self.root + + tailnode.next = node + node.prev = tailnode + node.next = self.root + self.root.prev = node + self.length += 1 + + def appendleft(self, value): + if self.maxsize is not None and len(self) >= self.maxsize: + raise Exception('LinkedList is Full') + node = Node(value=value) + if self.root.next is self.root: # empty + node.next = self.root + node.prev = self.root + self.root.next = node + self.root.prev = node + else: + node.prev = self.root + headnode = self.root.next + node.next = headnode + headnode.prev = node + self.root.next = node + self.length += 1 + + def remove(self, node): # O(1),传入node 而不是 value 我们就能实现 O(1) 删除 + """remove + :param node # 在 lru_cache 里实际上根据key 保存了整个node: + """ + if node is self.root: + return + else: # + node.prev.next = node.next + node.next.prev = node.prev + self.length -= 1 + return node + + def iter_node(self): + if self.root.next is self.root: + return + curnode = self.root.next + while curnode.next is not self.root: + yield curnode + curnode = curnode.next + yield curnode + + def __iter__(self): + for node in self.iter_node(): + yield node.value + + def iter_node_reverse(self): + """相比单链表独有的反序遍历""" + if self.root.prev is self.root: + return + curnode = self.root.prev + while curnode.prev is not self.root: + yield curnode + curnode = curnode.prev + yield curnode + + +############################################################ +# 分割线,下边是本章 内容实现 +############################################################ + + +class Deque(CircularDoubleLinkedList): # 注意这里我们用到了继承,嗯,貌似我说过不会用啥 OOP 特性的,抱歉 + + def pop(self): + """删除尾节点""" + if len(self) == 0: + raise Exception('empty') + tailnode = self.tailnode() + value = tailnode.value + self.remove(tailnode) + return value + + def popleft(self): + if len(self) == 0: + raise Exception('empty') + headnode = self.headnode() + value = headnode.value + self.remove(headnode) + return value + + +def test_deque(): + dq = Deque() + dq.append(1) + + dq.append(2) + assert list(dq) == [1, 2] + + dq.appendleft(0) + assert list(dq) == [0, 1, 2] + + dq.pop() + assert list(dq) == [0, 1] + + dq.popleft() + assert list(dq) == [1] + + dq.pop() + assert len(dq) == 0 + + +class Stack(object): + def __init__(self): + self.deque = Deque() # 你可以很容易替换为 python 内置的 collections.deque + + def push(self, value): + self.deque.append(value) + + def pop(self): + return self.deque.pop() + + +class Stack2(object): + + def __init__(self): + self._deque = deque() + + def push(self, value): + return self._deque.append(value) + + def pop(self): + return self._deque.pop() + + def empty(self): + return len(self._deque) == 0 + + +def test_stack(): + s = Stack() + s.push(0) + s.push(1) + s.push(2) + + assert s.pop() == 2 + assert s.pop() == 1 + assert s.pop() == 0 + + import pytest # pip install pytest + with pytest.raises(Exception) as excinfo: # 我们来测试是否真的抛出了异常 + s.pop() + assert 'empty' in str(excinfo.value) + + +if __name__ == '__main__': + test_stack() diff --git a/docs/06/big_o.md b/docs/06/big_o.md new file mode 100644 index 0000000..5f26aed --- /dev/null +++ b/docs/06/big_o.md @@ -0,0 +1,122 @@ +# 算法复杂度分析 +前面我们说了很多次时间复杂度是 O(1), O(n) 啥的,并没有仔细讲解这个 O 符号究竟是什么。 +你可以大概理解为操作的次数和数据个数的比例关系。比如 O(1) 就是有限次数操作,O(n) 就是操作正比于你的元素个数。 +这一章我们用更严谨的方式来定义它。 + + +# 大 O 表示法 +我们从一个计算矩阵的例子来引入,这里我参考了 [《Data Structures and Algorithms in Python》]( +https://book.douban.com/subject/10607365/) 中给的一个例子: + +考虑计算一个 n * n 矩阵所有元素的和(如果你不知道矩阵,就理解为一个二维数组): + +$$ + \begin{bmatrix} + 0 & 1 & 2 \\ + 3 & 4 & 5 \\ + 6 & 7 & 8 \\ + \end{bmatrix} +$$ + +这里列举两种方式: + +```py +# version1 +total_sum = 0 +for i in range(n): + row_sum[i] = 0 + for j in range(n): + row_sum[i] = row_sum[i] + matrix[i, j] + total_sum = total_sum + matrix[i, j] + +# version2 +total_sum = 0 +for i in range(n): + row_sum[i] = 0 + for j in range(n): + row_sum[i] = row_sum[i] + matrix[i, j] + total_sum = total_sum + row_sum[i] # 注意这里和上边的不同 +``` + +v1 版本的关键操作在 j 循环里,两步加法操作,由于嵌套在第一个循环里,操作步骤是 $ (2n) * n = 2n^2 $。 + +v2 版本的 total_sum 只有 n 次操作,它的操作次数是 $ n + n*n = n^2 + n $。 + + +这里你可能还感觉不到它们有多大差别,因为计算机执行的太快了,但是当 n 增长特别快的时候,总的操作次数差距就很明显了: + +n | $ 2n^2 $ | $ n^2 +n $ | +-------|----------------|----------------| +10 | 200 | 110 | +100 | 20,000 | 10,100 | +1000 | 2,000,000 | 1,001,000 | +10000 | 200,000,000 | 100,010,000 | +100000 | 20,000,000,000 | 10,000,100,000 | + +通常我们不太关注每个算法具体执行了多少次,而更关心随着输入规模 n 的增加,算法运行时间将以什么速度增加。为此计算机科学家定义了一个符号, +用来表示在最糟糕的情况下算法的运行时间,大 O 符号,在数学上称之为渐进上界(《算法导论》)。 + +# 如何计算时间复杂度 +上边我们列举了两个版本的计算矩阵和的代码,你看到了两个公式: + +- v1: $ 2n*n = 2n^2 $ +- v2: $ n + n*n = n + n^2 $ + +当 n 非常大的时候,$ n^2 $ 的数值这里将占主导,我们可以忽略 n 的影响 + +- v1: $ 2n*n = 2n^2 $ +- v2: $ n + n*n = n + n^2 \leq 2n^2 $ + +这里我们可以认为两个算法的时间复杂度均为 $ O(n^2) $ + +# 常用时间复杂度 +这里我们列举一些常用的时间复杂度,按照增长速度排序,日常我们的业务代码中最常用的是指数之前的复杂度,指数和阶乘的增长速度非常快, +当输入比较大的时候用在业务代码里是不可接受的。 + +O | 名称 | 举例 | +----------|--------------|--------------------| +1 | 常量时间 | 一次赋值 | +$\log n$ | 对数时间 | 折半查找 | +$n$ | 线性时间 | 线性查找 | +n$\log n$ | 对数线性时间 | 快速排序 | +$n^2$ | 平方 | 两重循环 | +$n^3$ | 立方 | 三重循环 | +$2^n$ | 指数 | 递归求斐波那契数列 | +$n!$ | 阶乘 | 旅行商问题 | + + +# 空间复杂度 +相比时间复杂度,空间复杂度讨论比较少。因为用户老爷等不及,况且现在存储越来越白菜价了,更多时候我们为了提升响应速度宁可多 使用点空间。 +空间复杂度相对好算一些,就是每个元素的空间占用乘以总的元素数,有些算法需要额外的空间存储,有些可以本地解决。 +如果能本地搞定的我们成为 in place 的,原地操作,比如交换一个 数组中的某两个位置的元素。但是有些操作可能就需要申请额外的空间 +来完成算法了,后边我们介绍排序算法的时候会讲到。 + + +# 常见复杂度增长趋势图 +为了让你有个直观的感觉,我们来看看一些经典的时间复杂度和对应的增长趋势图,不同函数在输入规模增长的时候很快就会有巨大的增长差异 + +![函数增长趋势图](./function_growth.png) + + +# 时间换空间,空间换时间 +有一些时候时间和空间两者不可兼得,我们会牺牲其中之一来换取另一个。 + +空间换时间:比如典型的就是 python 中的集合(后面会讲到它的实现原理),虽然它比较浪费空间,但是却能用 O(1) +的时间复杂度来判重。 + +时间换空间:当我们空间不够用,典型的就是缓存失效算法,我们不可能缓存下无限容量的数据,就会使用一些缓存淘汰算法来保证空间可用。 + + +# 思考题 +- 回头看看前几章我们讲到的数据结构,以及每个操作的时间复杂度,你能理解了吗? +- 二分查找是针对有序元素的一种经典的查找算法,你知道的它的时间复杂度吗?你能简单证明下吗。 +- 斐波那契数列你肯定很熟悉,它的公式是 F(n) = F(n-1) + F(n-2),你知道计算一个斐波那契数 F(n) + 的时间复杂度吗?你会用数学公式证明吗? +- 你能指出时间和空间权衡的例子吗?往往很多高效的数据结构能同时兼顾时间和空间复杂度,但是有时候我们却得做出一定的权衡 + + +# 参考资料 +如果你对数学感兴趣,建议你阅读《算法导论》『函数的增长』这一节 和《Data Structures and Algorithms in Python》第4章。 + + +(本章我用了 [MathJax](https://www.zybuluo.com/codeep/note/163962) 来书写一些简单的数学公式,使用 "$"包含起来的就是数学公式) diff --git a/docs/06/function_growth.png b/docs/06/function_growth.png new file mode 100644 index 0000000000000000000000000000000000000000..e6a2280dfb8cb10bbbfea6a54c2fd64cef645050 GIT binary patch literal 145936 zcmce-V|Zpu(>5I2wrxyo+s?$cZJQI@n%K6JiEZ0<@@Dpa?gz*B|GR#zuCA_HRb5@x z-RHW(ZSp|GF;003Ym#Dx_B0D#5-0DxT}z`iNlC)KY202qYkLPGKqLP7-c4z?!d zR>lAT>cL6w;L0d6^)6GBVCu=FN0|xxFuxopG5x7}p$5ELvRu7qu9 z#m&lWwci5(qTmN}1LPW|rqUXpoke+%wKhhW5nYwKq1FAW-~4KxaeEVX2L#X{TAt$? 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z^v0I|igt(tTdvo~7M~Qf;d^U2SA?rb74ab<&kqlwg9WtNl^`t)&7bW_O z-5s{SS@T=$_)T0i(niRU+ZmG^zrk!i3fgnr1(M3mj=SK0F_(o*kkiwsB+oQuJ#ml3 zpF(c@hv-vVoQ$ujC@=hHJf8mh$CT;52ALw}7IzYx^G{PsnDNtX^t)+o(*M#}5x<^^ za&m#-L2Uf2q*K`R?h<_Pn+($yufl5+L(O(PTCAnjiw0WP7-@a$^RsEi0=cEOwXqX2 zMVgcB3+buiPm4TZ#-AcNL-5b-t(QVZ3N#HW{!BLK@wuGin}n(@oK)r3Y{J+P^m+h2nc7 zo)!1BG$(Ghh1a{UhDudAjZ0xf+?q`JFYuEzRy0yM$!m)ev2*h(%v=67kzZ=iVZej9 zcVN5QR+0q`Zid#^qgB)I5-&e?5UEHOgSv-mc6rZRU)Z#^SGPt1B{6y_ z>unThqd*%4+9+^GQ9$5CduVhR*24ka{H2WNJF0zh5t`-|gnp3&V8_I_u@iVgVj~2O kJF2;DjcKF6e~beE4=xrKQ^VLf8~^|S07*qoM6N<$f`q4e)c^nh literal 0 HcmV?d00001 diff --git a/docs/07/hashtable.md b/docs/07/hashtable.md new file mode 100644 index 0000000..efccb63 --- /dev/null +++ b/docs/07/hashtable.md @@ -0,0 +1,181 @@ +# 哈希表 +不知道你有没有好奇过为什么 Python 里的 dict 和 set 查找速度这么快呢,用了什么黑魔法吗? +经常听别人说哈希表(也叫做散列表),究竟什么是哈希表呢?这一章我们来介绍哈希表,后续章节我们会看到 Python 中的字典和集合是如何实现的。 + +# 哈希表的工作过程 +前面我们已经讲到了数组和链表,数组能通过下标 O(1) 访问,但是删除一个中间元素却要移动其他元素,时间 O(n)。 +循环双端链表倒是可以在知道一个节点的情况下迅速删除它,但是吧查找又成了 O(n)。 + +难道就没有一种方法可以快速定位和删除元素吗?似乎想要快速找到一个元素除了知道下标之外别无他法,于是乎聪明的计算机科学家又想到了一种方法。 +能不能给每个元素一种『逻辑下标』,然后直接找到它呢,哈希表就是这种实现。它通过一个哈希函数来计算一个元素应该放在数组哪个位置,当然对于一个 +特定的元素,哈希函数每次计算的下标必须要一样才可以,而且范围不能超过给定的数组长度。 + +我们还是以书中的例子说明,假如我们有一个数组 T,包含 M=13 个元素,我们可以定义一个简单的哈希函数 h + +``` +h(key) = key % M +``` + +这里取模运算使得 h(key) 的结果不会超过数组的长度下标。我们来分别插入以下元素: + +765, 431, 96, 142, 579, 226, 903, 388 + +先来计算下它们应用哈希函数后的结果: + +``` +M = 13 +h(765) = 765 % M = 11 +h(431) = 431 % M = 2 +h(96) = 96 % M = 5 +h(142) = 142 % M = 12 +h(579) = 579 % M = 7 +h(226) = 226 % M = 5 +h(903) = 903 % M = 6 +h(388) = 388 % M = 11 +``` +下边我画个图演示整个插入过程(纯手工绘制,原谅我字写得不太优雅): + +![](./insert_hash.png) + + +# 哈希冲突 (collision) +这里到插入 226 这个元素的时候,不幸地发现 h(226) = h(96) = 5,不同的 key 通过我们的哈希函数计算后得到的下标一样, +这种情况成为哈希冲突。怎么办呢?聪明的计算机科学家又想到了办法,其实一种直观的想法是如果冲突了我能不能让数组中 +对应的槽变成一个链式结构呢?这就是其中一种解决方法,叫做 **链接法(chaining)**。如果我们用链接法来处理冲突,后边的插入是这样的: + +![](./insert_hash_chaining.png) + +这样就用链表解决了冲突问题,但是如果哈希函数选不好的话,可能就导致冲突太多一个链变得太长,这样查找就不再是 O(1) 的了。 +还有一种叫做开放寻址法(open addressing),它的基本思想是当一个槽被占用的时候,采用一种方式来寻找下一个可用的槽。 +(这里槽指的是数组中的一个位置),根据找下一个槽的方式不同,分为: + +- 线性探查(linear probing): 当一个槽被占用,找下一个可用的槽。 $ h(k, i) = (h^\prime(k) + i) \% m, i = 0,1,...,m-1 $ +- 二次探查(quadratic probing): 当一个槽被占用,以二次方作为偏移量。 $ h(k, i) = (h^\prime(k) + c_1 + c_2i^2) \% m , i=0,1,...,m-1 $ +- 双重散列(double hashing): 重新计算 hash 结果。 $ h(k,i) = (h_1(k) + ih_2(k)) \% m $ + +我们选一个简单的二次探查函数 $ h(k, i) = (home + i^2) \% m $,它的意思是如果 +遇到了冲突,我们就在原始计算的位置不断加上 i 的平方。我写了段代码来模拟整个计算下标的过程: + +```py +inserted_index_set = set() +M = 13 + +def h(key, M=13): + return key % M + +to_insert = [765, 431, 96, 142, 579, 226, 903, 388] +for number in to_insert: + index = h(number) + first_index = index + i = 1 + while index in inserted_index_set: # 如果计算发现已经占用,继续计算得到下一个可用槽的位置 + print('\th({number}) = {number} % M = {index} collision'.format(number=number, index=index)) + index = (first_index + i*i) % M # 根据二次方探查的公式重新计算下一个需要插入的位置 + i += 1 + else: + print('h({number}) = {number} % M = {index}'.format(number=number, index=index)) + inserted_index_set.add(index) +``` +这段代码输出的结果如下: + +``` +h(765) = 765 % M = 11 +h(431) = 431 % M = 2 +h(96) = 96 % M = 5 +h(142) = 142 % M = 12 +h(579) = 579 % M = 7 + h(226) = 226 % M = 5 collision +h(226) = 226 % M = 6 + h(903) = 903 % M = 6 collision + h(903) = 903 % M = 7 collision +h(903) = 903 % M = 10 + h(388) = 388 % M = 11 collision + h(388) = 388 % M = 12 collision + h(388) = 388 % M = 2 collision + h(388) = 388 % M = 7 collision +h(388) = 388 % M = 1 +``` + +遇到冲突之后会重新计算,每个待插入元素最终的下标就是: + +![](quadratic_hash.png) + +![](quadratic_result.png) + + +# Cpython 如何解决哈希冲突 +如果你对 cpython 解释器的实现感兴趣,可以参考下这个文件 [dictobject.c](https://github.com/python/cpython/blob/master/Objects/dictobject.c#L165)。 +不同 cpython 版本实现的探查方式是不同的,后边我们自己实现 HashTable ADT 的时候会模仿这个探查方式来解决冲突。 + + +``` +The first half of collision resolution is to visit table indices via this +recurrence: + + j = ((5*j) + 1) mod 2**i + +For any initial j in range(2**i), repeating that 2**i times generates each +int in range(2**i) exactly once (see any text on random-number generation for +proof). By itself, this doesn't help much: like linear probing (setting +j += 1, or j -= 1, on each loop trip), it scans the table entries in a fixed +order. This would be bad, except that's not the only thing we do, and it's +actually *good* in the common cases where hash keys are consecutive. In an +example that's really too small to make this entirely clear, for a table of +size 2**3 the order of indices is: + + 0 -> 1 -> 6 -> 7 -> 4 -> 5 -> 2 -> 3 -> 0 [and here it's repeating] +``` + +# 哈希函数 +到这里你应该明白哈希表插入的工作原理了,不过有个重要的问题之前没提到,就是 hash 函数怎么选? +当然是散列得到的冲突越来越小就好啦,也就是说每个 key 都能尽量被等可能地散列到 m 个槽中的任何一个,并且与其他 key 被散列到哪个槽位无关。 +如果你感兴趣,可以阅读后边提到的一些参考资料。视频里我们使用二次探查函数,它相比线性探查得到的结果冲突会更少。 + + +# 装载因子(load factor) +如果继续往我们的哈希表里塞东西会发生什么?空间不够用。这里我们定义一个负载因子的概念(load factor),其实很简单,就是已经使用的槽数比哈希表大小。 +比如我们上边的例子插入了 8 个元素,哈希表总大小是 13, 它的 load factor 就是 $ 8/13 \approx 0.62 $。当我们继续往哈希表插入数据的时候,很快就不够用了。 +通常当负载因子开始超过 0.8 的时候,就要新开辟空间并且重新进行散列了。 + + +# 重哈希(Rehashing) +当负载因子超过 0.8 的时候,需要进行 rehashing 操作了。步骤就是重新开辟一块新的空间,开多大呢?感兴趣的话可以看下 cpython 的 dictobject.c 文件然后搜索 +GROWTH_RATE 这个关键字,你会发现不同版本的 cpython 使用了不同的策略。python3.3 的策略是扩大为已经使用的槽数目的两倍。开辟了新空间以后,会把原来哈希表里 +不为空槽的数据重新插入到新的哈希表里,插入方式和之前一样。这就是 rehashing 操作。 + +# HashTable ADT +实践是检验真理的唯一标准,这里我们来实现一个简化版的哈希表 ADT,主要是为了让你更好地了解它的工作原理,有了它,后边实现起 dict 和 set 来就小菜一碟了。 +这里我们使用到了定长数组,还记得我们在数组和列表章节里实现的 Array 吧,这里要用上了。 + +解决冲突我们使用二次探查法,模拟 cpython 二次探查函数的实现。我们来实现三个哈希表最常用的基本操作,这实际上也是使用字典的时候最常用的操作。 + +- add(key, value) +- get(key, default) +- remove(key) + +```py +class Slot(object): + """定义一个 hash 表 数组的槽 + 注意,一个槽有三种状态,看你能否想明白 + 1.从未使用 HashMap.UNUSED。此槽没有被使用和冲突过,查找时只要找到 UNUSED 就不用再继续探查了 + 2.使用过但是 remove 了,此时是 HashMap.EMPTY,该探查点后边的元素扔可能是有key + 3.槽正在使用 Slot 节点 + """ + def __init__(self, key, value): + self.key, self.value = key, value + +class HashTable(object): + pass +``` + +具体的实现和代码编写在视频里讲解。这个代码可不太好实现,稍不留神就会有错,我们还是通过编写单元测试验证代码的正确性。 + +# 思考题 +- 请你分析下哈希表插入和删除元素的平均时间复杂度是多少?我们都实现代码了,相信这个问题你可以回答上来 +- Slot 在二次探查法里为什么不能直接删除?为什么我们要给它定义几个状态? + +# 延伸阅读 +- 《Data Structures and Algorithms in Python》11 章 Hash Tables +- 《算法导论》第三版 11 章散列表,了解几种哈希冲突的解决方式,以及为什么我们选择二次探查而不是线性探查法? +- 介绍 c 解释器如何实现的 python dict对象:[Python dictionary implementation](http://www.laurentluce.com/posts/python-dictionary-implementation/) +- [Python hash function implement](https://stackoverflow.com/questions/2070276/where-can-i-find-source-or-algorithm-of-pythons-hash-function) diff --git a/docs/07/hashtable.py b/docs/07/hashtable.py new file mode 100644 index 0000000..f1df5e2 --- /dev/null +++ b/docs/07/hashtable.py @@ -0,0 +1,176 @@ +# -*- coding: utf-8 -*- + +# 从数组和列表章复制的代码 + + +class Array(object): + + def __init__(self, size=32, init=None): + self._size = size + self._items = [init] * size + + def __getitem__(self, index): + return self._items[index] + + def __setitem__(self, index, value): + self._items[index] = value + + def __len__(self): + return self._size + + def clear(self, value=None): + for i in range(len(self._items)): + self._items[i] = value + + def __iter__(self): + for item in self._items: + yield item + + +class Slot(object): + """定义一个 hash 表数组的槽(slot 这里指的就是数组的一个位置) + hash table 就是一个 数组,每个数组的元素(也叫slot槽)是一个对象,对象包含两个属性 key 和 value。 + + 注意,一个槽有三种状态,看你能否想明白。相比链接法解决冲突,探查法删除一个 key 的操作稍微复杂。 + 1.从未使用 HashMap.UNUSED。此槽没有被使用和冲突过,查找时只要找到 UNUSED 就不用再继续探查了 + 2.使用过但是 remove 了,此时是 HashMap.EMPTY,该探查点后边的元素仍然可能是有key的,需要继续查找 + 3.槽正在使用 Slot 节点 + """ + + def __init__(self, key, value): + self.key, self.value = key, value + + +class HashTable(object): + + UNUSED = None # 没被使用过 + EMPTY = Slot(None, None) # 使用却被删除过 + + def __init__(self): + self._table = Array(8, init=HashTable.UNUSED) # 保持 2*i 次方 + self.length = 0 + + @property + def _load_factor(self): + # load_factor 超过 0.8 重新分配 + return self.length / float(len(self._table)) + + def __len__(self): + return self.length + + def _hash(self, key): + return abs(hash(key)) % len(self._table) + + def _find_key(self, key): + """ + 解释一个 slot 为 UNUSED 和 EMPTY 的区别 + 因为使用的是二次探查的方式,假如有两个元素 A,B 冲突了,首先A hash 得到是 slot 下标5,A 放到了第5个槽,之后插入 B 因为冲突了,所以继续根据二次探查方式放到了 slot8。 + 然后删除 A,槽 5 被置为 EMPTY。然后我去查找 B,第一次 hash 得到的是 槽5,但是这个时候我还是需要第二次计算 hash 才能找到 B。但是如果槽是 UNUSED 我就不用继续找了,我认为 B 就是不存在的元素。这个就是 UNUSED 和 EMPTY 的区别。 + """ + origin_index = index = self._hash(key) # origin_index 判断是否又走到了起点,如果查找一圈了都找不到则无此元素 + _len = len(self._table) + while self._table[index] is not HashTable.UNUSED: + if self._table[index] is HashTable.EMPTY: # 注意如果是 EMPTY,继续寻找下一个槽 + index = (index * 5 + 1) % _len + if index == origin_index: + break + continue + if self._table[index].key == key: # 找到了key + return index + else: + index = (index * 5 + 1) % _len # 没有找到继续找下一个位置 + if index == origin_index: + break + + return None + + def _find_slot_for_insert(self, key): + index = self._hash(key) + _len = len(self._table) + while not self._slot_can_insert(index): # 直到找到一个可以用的槽 + index = (index * 5 + 1) % _len + return index + + def _slot_can_insert(self, index): + return (self._table[index] is HashTable.EMPTY or self._table[index] is HashTable.UNUSED) + + def __contains__(self, key): # in operator,实现之后可以使用 in 操作符判断 + index = self._find_key(key) + return index is not None + + def add(self, key, value): + if key in self: # update + index = self._find_key(key) + self._table[index].value = value + return False + else: + index = self._find_slot_for_insert(key) + self._table[index] = Slot(key, value) + self.length += 1 + if self._load_factor >= 0.8: + self._rehash() + return True + + def _rehash(self): + old_table = self._table + newsize = len(self._table) * 2 + self._table = Array(newsize, HashTable.UNUSED) + + self.length = 0 + + for slot in old_table: + if slot is not HashTable.UNUSED and slot is not HashTable.EMPTY: + index = self._find_slot_for_insert(slot.key) + self._table[index] = slot + self.length += 1 + + def get(self, key, default=None): + index = self._find_key(key) + if index is None: + return default + else: + return self._table[index].value + + def remove(self, key): + index = self._find_key(key) + if index is None: + raise KeyError() + value = self._table[index].value + self.length -= 1 + self._table[index] = HashTable.EMPTY + return value + + def __iter__(self): + for slot in self._table: + if slot not in (HashTable.EMPTY, HashTable.UNUSED): + yield 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in (HashTable.EMPTY, HashTable.UNUSED): + yield slot.key + +######################################### +# 上边是从 哈希表章 拷贝过来的代码,我们会直接继承 HashTable 实现 dict +######################################### + + +class DictADT(HashTable): + + def _iter_slot(self): + for slot in self._table: + if slot not in (HashTable.EMPTY, HashTable.UNUSED): + yield slot + + def __setitem__(self, key, value): + self.add(key, value) + + def __getitem__(self, key): + if key not in self: + raise KeyError() + else: + return self.get(key) + + def items(self): + for slot in self._iter_slot(): + yield (slot.key, slot.value) + + def keys(self): + for slot in self._iter_slot(): + yield slot.key + + def values(self): + for slot in self._iter_slot(): + yield slot.value + + +def test_dict_adt(): + import random + d = DictADT() + + d['a'] = 1 + assert d['a'] == 1 + d.remove('a') + + l = list(range(30)) + random.shuffle(l) + for i in l: + d.add(i, i) + + for i in range(30): + assert d.get(i) == i + + assert sorted(list(d.keys())) == sorted(l) + + +test_dict_adt() diff --git a/docs/09/set.md b/docs/09/set.md new file mode 100644 index 0000000..6a45488 --- /dev/null +++ b/docs/09/set.md @@ -0,0 +1,52 @@ +# 集合 set + +集合是一种不包含重复元素的数据结构,经常用来判断是否重复这种操作,或者集合中是否存在一个元素。 +这一章讲集合,实际上它的底层也是哈希表实现的,所以像实现 DictADT 一样,借助 HashTable 实现它也比较简单。 + + +# 集合操作 +集合可能最常用的就是去重,判断是否存在一个元素等,但是 set 相比 dict 有更丰富的操作,主要是数学概念上的。 +如果你学过《离散数学》中集合相关的概念,基本上是一致的。 python 的 set 提供了如下基本的集合操作, +假设有两个集合 A,B,有以下操作: + +- 交集: A & B,表示同时在 A 和 B 中的元素。 python 中重载 `__and__` 实现 +- 并集: A | B,表示在 A 或者 B 中的元素,两个集合相加。python 中重载 `__or__` 实现 +- 差集: A - B,表示在 A 中但是不在 B 中的元素。 python 中重载 `__sub__` 实现 +- 对称差: A ^ B,返回在 A 或 B 但是不在 A、B 中都出现的元素。其实就是 (A|B) - (A&B), python 中重载 `__xor__` 实现 + +这里使用的 &, |, -, ^ 在 python 内置的 set 实现中都是重载了内置的运算符。这里我们也用这种方式实现, +具体实现我会在视频里演示。python 同样实现了 intersection, union, difference, symmetric_difference 这四个方法, +和使用运算符的功能是一样的。 + +![](./set.png) + +# python frozenset +在 python 里还有一个 frozenset,看它的名字就知道这种也是集合,但是它的内容是无法变动的。一般我们使用 +它的常见就是用一个可迭代对象初始化它,然后只用来判重等操作。 + + +# 实现一个 set ADT +如何实现一个集合的 ADT 呢,其实还是个哈希表,哈希表不是有 key 和 value 嘛,咱把 value 置为 1 不就行了。 + +```py +class SetADT(HashTable): + + def add(self, key): + # 集合其实就是一个 dict,只不过我们把它的 value 设置成 1 + return super(SetADT, self).add(key, True) +``` + +当然其它数学上的操作就麻烦点了,不过也很容易实现。 + + +# 思考题 +- 集合判断一个元素是否存在的时间复杂度是多少? +- 集合的元素 key 需要满足什么概念?可变对象可以吗? +- 请你在 SetADT 基础上实现集合的 remove 操作和 pop 操作 +- 你能尝试实现对称差操作吗?这里我没有实现,留给你作为练习 +- 你知道如何重载 python 的内置运算符吗?这里我们实现 set 的集合操作就是用到了重载,请阅读相关 python 文档。 +- 当元素个数不多的时候,我们可以用 set 来判重,但是如果是大量元素会非常耗费内存。请你了解下 Bloom Filter + + +# 延伸阅读 +阅读 python 文档关于 set 的相关章节,了解 set 还有哪些操作?比如比较运算符的概念,比较两个集合意味着什么。 diff --git a/docs/09/set.png b/docs/09/set.png new file mode 100644 index 0000000000000000000000000000000000000000..5124a1ee3e32639d950f93ae84891c8b8ae39281 GIT binary patch literal 82401 zcmZU(18`ofItCp z6Jy$^&A-K{4#w4H%^Z(@N2AZqor7cBv+Ew#t+$!p7XP0ReX zbRb6HNR;ykAbZKaS84#>K?DYz(*n3hcqm~dCdA6CL)kkBzb3G`Rb(Wj<{)%Lnu}L_ 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z;p}i>1GW+FJ^PDgcz-9qa{_=5MFITa|8GuI%N;^P3$vTfx2}puu+M8L1<6uzgMj}4 DLxfdV literal 0 HcmV?d00001 diff --git a/docs/09/set_adt.py b/docs/09/set_adt.py new file mode 100644 index 0000000..3efbe78 --- /dev/null +++ b/docs/09/set_adt.py @@ -0,0 +1,191 @@ +# -*- coding: utf-8 -*- + +# 从数组和列表章复制的代码 + + +class Array(object): + + def __init__(self, size=32, init=None): + self._size = size + self._items = [init] * size + + def __getitem__(self, index): + return self._items[index] + + def __setitem__(self, index, value): + self._items[index] = value + + def __len__(self): + return self._size + + def clear(self, value=None): + for i in range(len(self._items)): + self._items[i] = value + + def __iter__(self): + for item in self._items: + yield item + + +class Slot(object): + """定义一个 hash 表 数组的槽 + 注意,一个槽有三种状态,看你能否想明白。相比链接法解决冲突,二次探查法删除一个 key 的操作稍微复杂。 + 1.从未使用 HashMap.UNUSED。此槽没有被使用和冲突过,查找时只要找到 UNUSED 就不用再继续探查了 + 2.使用过但是 remove 了,此时是 HashMap.EMPTY,该探查点后边的元素扔可能是有key + 3.槽正在使用 Slot 节点 + """ + + def __init__(self, key, value): + self.key, self.value = key, value + + +class HashTable(object): + + UNUSED = None # 没被使用过 + EMPTY = Slot(None, None) # 使用却被删除过 + + def __init__(self): + self._table = Array(8, init=HashTable.UNUSED) # 保持 2*i 次方 + self.length = 0 + + @property + def _load_factor(self): + # load_factor 超过 0.8 重新分配 + return self.length / float(len(self._table)) + + def __len__(self): + return self.length + + def _hash(self, key): + return abs(hash(key)) % len(self._table) + + def _find_key(self, key): + index = self._hash(key) + _len = len(self._table) + while self._table[index] is not HashTable.UNUSED: + if self._table[index] is HashTable.EMPTY: + index = (index*5 + 1) % _len + continue + elif self._table[index].key == key: + return index + else: + index = (index*5 + 1) % _len + return None + + def _find_slot_for_insert(self, key): + index = self._hash(key) + _len = len(self._table) + while not self._slot_can_insert(index): + index = (index*5 + 1) % _len + return index + + def _slot_can_insert(self, index): + return (self._table[index] is HashTable.EMPTY or self._table[index] is HashTable.UNUSED) + + def __contains__(self, key): # in operator + index = self._find_key(key) + return index is not None + + def add(self, key, value): + if key in self: + index = self._find_key(key) + self._table[index].value = value + return False + else: + index = self._find_slot_for_insert(key) + self._table[index] = Slot(key, value) + self.length += 1 + if self._load_factor >= 0.8: + self._rehash() + return True + + def _rehash(self): + old_table = self._table + newsize = len(self._table) * 2 + self._table = Array(newsize, HashTable.UNUSED) + + self.length = 0 + + for slot in old_table: + if slot is not HashTable.UNUSED and slot is not HashTable.EMPTY: + index = self._find_slot_for_insert(slot.key) + self._table[index] = slot + self.length += 1 + + def get(self, key, default=None): + index = self._find_key(key) + if index is None: + return default + else: + return self._table[index].value + + def remove(self, key): + index = self._find_key(key) + if index is None: + raise KeyError() + value = self._table[index].value + self.length -= 1 + self._table[index] = HashTable.EMPTY + return value + + def __iter__(self): + for slot in self._table: + if slot not in (HashTable.EMPTY, HashTable.UNUSED): + yield slot.key + +######################################### +# 上边是从 哈希表章 拷贝过来的代码,我们会直接继承 HashTable 实现 集合 set +######################################### + + +class SetADT(HashTable): + + def add(self, key): + # 集合其实就是一个 dict,只不过我们把它的 value 设置成 1 + return super(SetADT, self).add(key, True) + + def __and__(self, other_set): + """交集 A&B""" + new_set = SetADT() + for element_a in self: + if element_a in other_set: + new_set.add(element_a) + return new_set + + def __sub__(self, other_set): + """差集 A-B""" + new_set = SetADT() + for element_a in self: + if element_a not in other_set: + new_set.add(element_a) + return new_set + + def __or__(self, other_set): + """并集 A|B""" + new_set = SetADT() + for element_a in self: + new_set.add(element_a) + for element_b in other_set: + new_set.add(element_b) + return new_set + + +def test_set_adt(): + sa = SetADT() + sa.add(1) + sa.add(2) + sa.add(3) + assert 1 in sa # 测试 __contains__ 方法,实现了 add 和 __contains__,集合最基本的功能就实现啦 + + sb = SetADT() + sb.add(3) + sb.add(4) + sb.add(5) + + assert sorted(list(sa & sb)) == [3] + assert sorted(list(sa - sb)) == [1, 2] + assert sorted(list(sa | sb)) == [1, 2, 3, 4, 5] + + +if __name__ == '__main__': + test_set_adt() diff --git a/docs/10/fact.png b/docs/10/fact.png new file mode 100644 index 0000000000000000000000000000000000000000..63ee8774ef15b65662a560d3207e5dc9a2072ee0 GIT binary patch literal 12812 zcmdscWmFx_wk;N%;Oh4i<&5Be~kVJyVg9ifxLz0#fQvm}5=K{4IV4*>GkKm0&FfbfaD^XD; zX;D!UC1(e7D_b)#Fs;xOPZ%`}`9`P|8QefnQ#q1MSR6$ON=->v zsb2_bLZN;8jSRMz>VK)lBshf3LU2-qbpI7rT#XH-`tnfm20EY_a&8qJ4Xq^< z8-@P-nb_~`gE%kO-}i@7T!f$A2yk)Fw}=J@WKWWKIu5b8 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+递归用一种通俗的话来说就是自己调用自己,但是需要分解它的参数,让它解决一个更小一点的问题,当问题小到一定规模的时候,需要一个递归出口返回。 +这里举一个和其他很多老套的教科书一样喜欢举的例子,阶乘函数,我觉得用来它演示再直观不过。它的定义是这样的: + +![](./fact.png) + +我们很容易根据它的定义写出这样一个递归函数,因为它本身就是递归定义的。 + +```py +def fact(n): + if n == 0: + return 1 + else: + return n * fact(n-1) +``` +看吧,几乎完全是按照定义来写的。我们来看下递归函数的几个特点: + +- 递归必须包含一个基本的出口(base case),否则就会无限递归,最终导致栈溢出。比如这里就是 n == 0 返回 1 +- 递归必须包含一个可以分解的问题(recursive case)。 要想求得 fact(n),就需要用 n * fact(n-1) +- 递归必须必须要向着递归出口靠近(toward the base case)。 这里每次递归调用都会 n-1,向着递归出口 n == 0 靠近 + + +# 调用栈 +看了上一个例子你可能觉得递归好简单,先别着急,我们再举个简单的例子,上边我们并没有讲递归如何工作的。 +假如让你输出从 1 到 10 这十个数字,如果你是个正常人的话,我想你的第一反应都是这么写: + +```py +def print_num(n): + for i in range(1, n + 1): # 注意很多编程语言使用的都是 从 0 开始的左闭右开区间, python 也不例外 + print(i) + + +if __name__ == '__main__': + print_num(10) +``` + +我们尝试写一个递归版本,不就是自己调用自己嘛: + +```py +def print_num_recursive(n): + if n > 0: + print_num_recursive(n-1) + print(n) +``` + +你猜下它的输出?然后我们调换下 print 顺序,你再猜下它的输出 + +```py +def print_num_recursive_revserve(n): + if n > 0: + print(n) + print_num_recursive_revserve(n-1) +``` +你能明白是为什么吗?我建议你运行下这几个小例子,它们很简单但是却能说明问题。 +计算机内部使用调用栈来实现递归,这里的栈一方面指的是内存中的栈区,一方面栈又是之前讲到的后进先出这种数据结构。 +每当进入递归函数的时候,系统都会为当前函数开辟内存保存当前变量值等信息,每个调用栈之间的数据互不影响,新调用的函数 +入栈的时候会放在栈顶。视频里我们会画图来演示这个过程。 + +递归只用大脑不用纸笔模拟的话很容易晕,因为明明是同一个变量名字,但是在不同的调用栈里它是不同的值,所以我建议 +你最好手动画画这个过程。 + +![](./print_rec.png) + +# 用栈模拟递归 +刚才说到了调用栈,我们就用栈来模拟一把。之前栈这一章我们讲了如何自己实现栈,不过这里为了不拷贝太多代码,我们直接用 collections.deque 就可以 +快速实现一个简单的栈。 + +```py +from collections import deque + + +class Stack(object): + def __init__(self): + self._deque = deque() + + def push(self, value): + return self._deque.append(value) + + def pop(self): + return self._deque.pop() + + def is_empty(self): + return len(self._deque) == 0 + + +def print_num_use_stack(n): + s = Stack() + while n > 0: # 不断将参数入栈 + s.push(n) + n -= 1 + + while not s.is_empty(): # 参数弹出 + print(s.pop()) +``` +这里结果也是输出 1 到 10,只不过我们是手动模拟了入栈和出栈的过程,帮助你理解计算机是如何实现递归的,是不是挺简单!现在你能明白为什么上边 print_num_recursive print_num_recursive_revserve 两个函数输出的区别了吗? + + +# 尾递归 +上边的代码示例(麻雀虽小五脏俱全)中实际上包含了两种形式的递归,一种是普通的递归,还有一种叫做尾递归: + +```py +def print_num_recursive(n): + if n > 0: + print_num_recursive(n-1) + print(n) + + +def print_num_recursive_revserve(n): + if n > 0: + print(n) + print_num_recursive_revserve(n-1) # 尾递归 +``` + +概念上它很简单,就是递归调用放在了函数的最后。有什么用呢? +普通的递归, 每一级递归都产生了新的局部变量, 必须创建新的调用栈, 随着递归深度的增加, 创建的栈越来越多, 造成爆栈。虽然尾递归调用也会创建新的栈, +但是我们可以优化使得尾递归的每一级调用共用一个栈!, 如此便可解决爆栈和递归深度限制的问题! +不幸的是 python 默认不支持尾递归优化(见延伸阅读),不过一般尾递归我们可以用一个迭代来优化它。 + + +# 汉诺塔问题 + +有三根杆子A,B,C。A杆上有N个(N>1)穿孔圆盘,盘的尺寸由下到上依次变小。要求按下列规则将所有圆盘移至C杆: +但是有两个条件: + +- 每次只能移动一个圆盘; +- 大盘不能叠在小盘上面。 + +> 最早发明这个问题的人是法国数学家爱德华·卢卡斯。 +> 传说越南河内某间寺院有三根银棒,上串64个金盘。寺院里的僧侣依照一个古老的预言,以上述规则移动这些盘子;预言说当这些盘子移动完毕,世界就会灭亡。 +> 这个传说叫做梵天寺之塔问题(Tower of Brahma puzzle)。但不知道是卢卡斯自创的这个传说,还是他受他人启发。 + + + +![五个盘子的汉诺塔问题](./hanoi_tower.png) + +理解这个问题需要我们一些思维上的转换,因为我们正常的思维可能都是从上边最小的盘子开始移动,但是这里我们从移动最底下的盘子开始思考。 +假设我们已经知道了如何移动上边的四个盘子到 B(pole2),现在把最大的盘子从 A -> C 就很简单了。当把最大的盘子移动到 +C 之后,只需要把 B 上的 4 个盘子从 B -> C 就行。(这里的 pole1, 2, 3 分别就是 A, B, C 杆) + +![](./hanoi_four_disks.png) + +问题是仍要想办法如何移动上边的 4 个盘子,我们可以同样的方式来移动上边的 4 个盘子,这就是一种递归的解法。 +给定 n 个盘子和三个杆分别是 源杆(Source), 目标杆(Destination),和中介杆(Intermediate),我们可以定义如下递归操作: + +- 把上边的 n-1 个盘子从 S 移动到 I,借助 D 杆 +- 把最底下的盘子从 S 移动到 D +- 把 n-1 个盘子从 I 移动到 D,借助 S + +我们把它转换成代码: + +```py +def hanoi_move(n, source, dest, intermediate): + if n >= 1: # 递归出口,只剩一个盘子 + hanoi_move(n-1, source, intermediate, dest) + print("Move %s -> %s" % (source, dest)) + hanoi_move(n-1, intermediate, dest, source) +hanoi_move(3, 'A', 'C', 'B') + +# 输出,建议你手动模拟下。三个盘子 A(Source), B(intermediate), C(Destination) +""" +Move A -> C +Move A -> B +Move C -> B +Move A -> C +Move B -> A +Move B -> C +Move A -> C +""" +``` + +
    +![三个盘子的汉诺塔解法](./hanoi.gif) +
    + +是不是很神奇,但是老实说这个过程仅凭大脑空想是比较难以想象出来的。人的大脑『栈』深度很有限,因为你甚至都没法同时记住超过 8 个以上的 +无意义数字,所以用大脑模拟不如用纸笔来模拟下。(不排除有些聪明的同学能迅速在脑瓜里完成这个过程) + +# 延伸阅读 +递归是个非常重要的概念,我们后边的数据结构和算法中还会多次碰到它,我建议你多阅读一些资料加深理解: + +- 《算法图解》第三章 递归 +- 《Data Structures and Algorithms in Python》 第 10 章 Recursion +- [《Python开启尾递归优化!》](https://segmentfault.com/a/1190000007641519) +- [尾调用优化](http://www.ruanyifeng.com/blog/2015/04/tail-call.html) +- [汉诺塔](https://zh.wikipedia.org/wiki/%E6%B1%89%E8%AF%BA%E5%A1%94) + +# 思考题 +- 你能举出其他一些使用到递归的例子吗? +- 实现一个 flatten 函数,把嵌套的列表扁平化,你需要用递归函数来实现。比如 [[1,2], [1,2,3] -> [1,2,1,2,3] +- 使用递归和循环各有什么优缺点,你能想到吗?怎么把一个尾递归用迭代替换? +- 递归有时候虽然很优雅直观,但是时间复杂度却不理想,比如斐波那契数列,它的表达式是 F(n) = F(n-1) + F(n-2),你能计算它的时间复杂度吗?请你画个树来表示它的计算过程,为什么这个时间复杂度很不理想?我们怎样去优化它。 +- python 内置的 dict 只能用 dict['key'] 的形式访问比较麻烦,我们想用 dict.key 的形式访问。tornado web 框架中提供了一个 ObjectDict,请你实现一个递归函数接收一个字典,并返回一个可以嵌套访问的 ObjectDict diff --git a/docs/10/recursion.py b/docs/10/recursion.py new file mode 100644 index 0000000..86d2f90 --- /dev/null +++ b/docs/10/recursion.py @@ -0,0 +1,72 @@ +# -*- coding: utf-8 -*- + + +def fact(n): + if n == 0: + return 1 + else: + return n * fact(n - 1) + + +def print_num(n): + for i in range(1, n + 1): # 注意很多编程语言使用的都是 从 0 开始的左闭右开区间, python 也不例外 + print(i) + + +def print_num_recursive(n): + if n > 0: + print_num_recursive(n - 1) + print(n) + + +def print_num_recursive_revserve(n): + if n > 0: + print(n) + print_num_recursive_revserve(n - 1) + + +from collections import deque + + +class Stack(object): + def __init__(self): + self._deque = deque() + + def push(self, value): + return self._deque.append(value) + + def pop(self): + return self._deque.pop() + + def is_empty(self): + return len(self._deque) == 0 + + +def print_num_use_stack(n): + s = Stack() + while n > 0: # 不断将参数入栈 + s.push(n) + n -= 1 + + while not s.is_empty(): # 参数弹出 + print(s.pop()) + + +def hanoi_move(n, source, dest, intermediate): + if n >= 1: # 递归出口,只剩一个盘子 + hanoi_move(n - 1, source, intermediate, dest) + print("Move %s -> %s" % (source, dest)) + hanoi_move(n - 1, intermediate, dest, source) + + +def flatten(rec_list): + for i in rec_list: + if isinstance(i, list): + for i in flatten(i): + yield i + else: + yield i + + +def test_flatten(): + assert list(flatten([[[1], 2, 3], [1, 2, 3]])) == [1, 2, 3, 1, 2, 3] diff --git a/docs/11/search.md b/docs/11/search.md new file mode 100644 index 0000000..8ead740 --- /dev/null +++ b/docs/11/search.md @@ -0,0 +1,118 @@ +# 查找 + +查找可以说是我们业务代码里用得最多的操作,比如我们经常需要在一个列表里找到我们需要的一个元素,然后返回它的位置。 +其实之前我们介绍的哈希表就是非常高效率的查找数据结构,很明显地它是用空间换时间。这一节介绍两个基本的基于线性结构的查找。 + +# 线性查找 +线性查找就是从头找到尾,直到符合条件了就返回。比如在一个 list 中找到一个等于 5 的元素并返回下标: + +```py +number_list = [0, 1, 2, 3, 4, 5, 6, 7] + + +def linear_search(value, iterable): + for index, val in enumerate(iterable): + if val == value: + return index + return -1 + + +assert linear_search(5, number_list) == 5 + +``` +是不是 so easy。当然我们需要来一点花样,比如传一个谓词进去,你要知道,在 python 里一切皆对象,所以我们可以把函数当成一个参数传给另一个函数。 + +```py +def linear_search_v2(predicate, iterable): + for index, val in enumerate(iterable): + if predicate(val): + return index + return -1 + + +assert linear_search_v2(lambda x: x == 5, number_list) == 5 +``` + +效果是一样的,但是传入一个谓词函数进去更灵活一些,比如我们可以找到第一个大于或者小于 5 的,从而控制函数的行为。 +还能玩出什么花样呢?前面我们刚学习了递归,能不能发挥自虐精神没事找事用递归来实现呢? + +```py +def linear_search_recusive(array, value): + if len(array) == 0: + return -1 + index = len(array)-1 + if array[index] == value: + return index + return linear_search_recusive(array[0:index], value) + + +assert linear_search_recusive(number_list, 5) == 5 +assert linear_search_recusive(number_list, 8) == -1 +assert linear_search_recusive(number_list, 7) == 7 +assert linear_search_recusive(number_list, 0) == 0 +``` +这里的 assert 我多写了几个,包括正常情况、异常情况和边界值等,因为递归比较容易出错。注意这里的两个递归出口。 +当然业务代码里如果碰到这种问题我们肯定是选上边最直白的方式来实现,要不你的同事肯定想打你。 + +# 二分查找 +上一小节说的线性查找针对的是无序序列,假如一个序列已经有序了呢,我们还需要从头找到尾吗?当然不用,折半(二分)是一种经典思想。日常生活中还有哪些经典的二分思想呢? + +- 猜数字游戏 +- 一尺之棰,日取其半,万世不竭 +- 有些民间股神,告诉一堆人某个股票会涨,告诉另一半人会跌。后来真涨了,慢慢又告诉信了他的一半人另一个股票会涨,另一半说会跌。就这样韭菜多了总有一些人信奉他为股神。。。 + +其实之前写过博客[《抱歉,我是开发,你居然让我写单测[视频]》](https://zhuanlan.zhihu.com/p/35352024)讲过二分查找,当时主要是为了引入单元测试这个概念的,因为很多不正规的项目代码很糙,更别说写单测了。这里我就直接贴代码啦 + +```py +def binary_search(sorted_array, val): + if not sorted_array: + return -1 + + beg = 0 + end = len(sorted_array) - 1 + + while beg <= end: + mid = int((beg + end) / 2) # beg + (end-beg)/2, 为了屏蔽 python 2/3 差异我用了强转 + if sorted_array[mid] == val: + return mid + elif sorted_array[mid] > val: + end = mid - 1 + else: + beg = mid + 1 + return -1 + + +def test_binary_search(): + a = list(range(10)) + + # 正常值 + assert binary_search(a, 1) == 1 + assert binary_search(a, -1) == -1 + + # 异常值 + assert binary_search(None, 1) == -1 + + # 边界值 + assert binary_search(a, 0) == 0 +``` + + +# 思考题 +- 给你个挑战,用递归来实现本章的二分查找。你要十分注意边界条件,注意用单测测试呦,在你写代码的时候,可能会碰到边界问题或者无穷递归等。 如果你想不起来,可以看看本章的代码示例 +- 二分查找有一个变形,比如我们想在一个有序数组中插入一个值之后,数组仍保持有序,请你找出这个位置。(bisect 模块) + + +# 延伸阅读 +这里没给链接,请善用 google 等搜索引擎和 Dash(mac) 等文档查询工具,在你学习代码的过程中你会非常频繁地使用它们。 +或者如果你有时间也可以跳转到这些模块的源码,看看它们的实现方式。标准库都是些高手写的,肯定能学到一些姿势。 + +- 阅读 python 文档关于二分的 bisect 模块。 +- 阅读 python 文档 itertools 相关模块和常见的几个函数 takewhile, dropwhile, from_iterable, count, tee 等用法 +- [每个程序员都应该会点形式化证明](https://zhuanlan.zhihu.com/p/35364999?group_id=967109293607129088) + + +# Leetcode + +[找旋转过的排序数组中最小的数 find-minimum-in-rotated-sorted-array](https://leetcode.com/problems/find-minimum-in-rotated-sorted-array/description/) + +[已排序的数组中找到第一和最后一个元素 find-first-and-last-position-of-element-in-sorted-array/](https://leetcode.com/problems/find-first-and-last-position-of-element-in-sorted-array/submissions/) diff --git a/docs/11/search.py b/docs/11/search.py new file mode 100644 index 0000000..6a9e708 --- /dev/null +++ b/docs/11/search.py @@ -0,0 +1,60 @@ +# -*- coding: utf-8 -*- + +number_list = [0, 1, 2, 3, 4, 5, 6, 7] + + +def linear_search(value, iterable): + for index, val in enumerate(iterable): + if val == value: + return index + return -1 + + +assert linear_search(5, number_list) == 5 + + +def linear_search_v2(predicate, iterable): + for index, val in enumerate(iterable): + if predicate(val): + return index + return -1 + + +assert linear_search_v2(lambda x: x == 5, number_list) == 5 + + +def linear_search_recusive(array, value): + if len(array) == 0: + return -1 + index = len(array)-1 + if array[index] == value: + return index + return linear_search_recusive(array[0:index], value) + + +assert linear_search_recusive(number_list, 5) == 5 +assert linear_search_recusive(number_list, 8) == -1 +assert linear_search_recusive(number_list, 7) == 7 +assert linear_search_recusive(number_list, 0) == 0 + + +def binary_search_recursive(sorted_array, beg, end, val): + if beg >= end: + return -1 + mid = int((beg + end) / 2) # beg + (end-beg)/2 + if sorted_array[mid] == val: + return mid + elif sorted_array[mid] > val: + return binary_search_recursive(sorted_array, beg, mid, val) # 注意我依然假设 beg, end 区间是左闭右开的 + else: + return binary_search_recursive(sorted_array, mid+1, end, val) + + +def test_binary_search_recursive(): + # 我们测试所有值和边界条件 + a = list(range(10)) + for i in a: + assert binary_search_recursive(a, 0, len(a), i) == i + + assert binary_search_recursive(a, 0, len(a), -1) == -1 + assert binary_search_recursive(a, 0, len(a), 10) == -1 diff --git a/docs/12/basic_sort.md b/docs/12/basic_sort.md new file mode 100644 index 0000000..9804502 --- /dev/null +++ b/docs/12/basic_sort.md @@ -0,0 +1,145 @@ +# 基本排序算法 +从本章开始讲常见的基于比较的排序算法,先讲三个简单的但是时间复杂度却不太理想的排序算法,包括冒泡排序、选择排序和插入排序。 + + +# 冒泡排序 +bubble sort 可以说是最简单的一种排序算法了,它的思想如下。对一个数组进行 n-1 轮迭代,每次比较相邻两个元素, +如果相邻的元素前者大于后者,就交换它们。因为直接在元素上操作而不是返回新的数组,所以是一个 inplace 的操作。 +这里冒泡的意思其实就是每一轮冒泡一个最大的元素就会通过不断比较和交换相邻元素使它转移到最右边。 + +你可以想象假如有 10 个小盆友从左到右站成一排,个头不等。老师想让他们按照个头从低到高站好,于是他开始喊口号。 +每喊一次,从第一个小盆友开始,相邻的小朋友如果身高不是正序就会两两调换,就这样第一轮个头最高的排到了最右边。(冒泡到最右边) +第二轮依次这么来,从第一个小朋友开始两两交换,这样次高的小盆友又排到了倒数第二个位置。依次类推。 + + +我们在视频里手动模拟下它的过程。 + + +```py +import random + + +def bubble_sort(seq): # O(n^2), n(n-1)/2 = 1/2(n^2 + n) + n = len(seq) + for i in range(n-1): + print(seq) # 我打印出来让你看清楚每一轮最高、次高、次次高...的小朋友会冒泡到右边 + for j in range(n-1-i): # 这里之所以 n-1 还需要 减去 i 是因为每一轮冒泡最大的元素都会冒泡到最后,无需再比较 + if seq[j] > seq[j+1]: + seq[j], seq[j+1] = seq[j+1], seq[j] + print(seq) + + +def test_bubble_sort(): + seq = list(range(10)) # 注意 python3 返回迭代器,所以我都用 list 强转了,python2 range 返回的就是 list + random.shuffle(seq) # shuffle inplace 操作,打乱数组 + bubble_sort(seq) + assert seq == sorted(seq) # 注意呦,内置的 sorted 就不是 inplace 的,它返回一个新的数组,不影响传入的参数 + +""" 我打印出来让你看到每次从最高到次高的小盆友就这么排好序了,因为是随机数,你第一个没有排序的数组应该和我的不一样 +[3, 4, 5, 0, 9, 1, 7, 8, 6, 2] +[3, 4, 0, 5, 1, 7, 8, 6, 2, 9] +[3, 0, 4, 1, 5, 7, 6, 2, 8, 9] +[0, 3, 1, 4, 5, 6, 2, 7, 8, 9] +[0, 1, 3, 4, 5, 2, 6, 7, 8, 9] +[0, 1, 3, 4, 2, 5, 6, 7, 8, 9] +[0, 1, 3, 2, 4, 5, 6, 7, 8, 9] +[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] +[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] +[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] +""" +``` + + + +# 选择排序 +刚才看到冒泡是每轮迭代中,如果相邻的两个元素前者大于后者了就交换两个相邻元素(假设正序排序)。其实还有一种思路就是, +每次我们找到最小的元素插入迭代的起始位置,这样每个位置从它自己的位置开始它就是最小的了,一圈下来数组就有序了。 +选择可以理解为 一个 0 到 n-1 的迭代,每次向后查找选择一个最小的元素。 + +同样小盆友又来啦,这次我们从第一个开始,从头到尾找一个个头最小的小盆友,然后把它和第一个小盆友交换。 +然后从第二个小盆友开始采取同样的策略,这样一圈下来小盆友就有序了。 + +```py +def select_sort(seq): + n = len(seq) + for i in range(n-1): + min_idx = i # 我们假设当前下标的元素是最小的 + for j in range(i+1, n): # 从 i 的后边开始找到最小的元素,得到它的下标 + if seq[j] < seq[min_idx]: + min_idx = j # 一个 j 循环下来之后就找到了最小的元素它的下标 + if min_idx != i: # swap + seq[i], seq[min_idx] = seq[min_idx], seq[i] + + +def test_select_sort(): + seq = list(range(10)) + random.shuffle(seq) + select_sort(seq) + assert seq == sorted(seq) + +""" +[4, 7, 5, 3, 6, 0, 2, 9, 8, 1] +[0, 7, 5, 3, 6, 4, 2, 9, 8, 1] +[0, 1, 5, 3, 6, 4, 2, 9, 8, 7] +[0, 1, 2, 3, 6, 4, 5, 9, 8, 7] +[0, 1, 2, 3, 6, 4, 5, 9, 8, 7] +[0, 1, 2, 3, 4, 6, 5, 9, 8, 7] +[0, 1, 2, 3, 4, 5, 6, 9, 8, 7] +[0, 1, 2, 3, 4, 5, 6, 9, 8, 7] +[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] +[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] + +""" +``` + + +# 插入排序 +插入排序很多教科书都是用扑克牌的例子讲的,想象你手里有一些扑克牌,它们顺序是散乱的,现在需要你把它们整理成有序的,你会怎么做呢? +首先拿最顶上的一张,然后拿第二张,第二张点数大,你就把第二张放在第一张的下边,否则放在第一张上边。 +当你拿第三张的时候,你同样会找到适合它大小的位置插入进去。 + +换成小朋友一样,第一个小盆友只有一个人我们假设是有序的,然后第二个小盆友会跟第一个比,如果第一个高就交换位置。 +接下来第三个小盆友从第二个位置开始比较,如果没第二个高就交换位置,然后没第一个高也交换位置,保持前边三个小盆友身高有序就好。 +依次类推,等到最后一个小盆友也转移到合适的位置,整个队列就是有序的了。 + +插入排序就是这个道理, 每次挑选下一个元素插入已经排序的数组中,初始时已排序数组只有一个元素。我们就直接上代码吧。 + + +```py +def insertion_sort(seq): + """ 每次挑选下一个元素插入已经排序的数组中,初始时已排序数组只有一个元素""" + n = len(seq) + print(seq) + for i in range(1, n): + value = seq[i] # 保存当前位置的值,因为转移的过程中它的位置可能被覆盖 + # 找到这个值的合适位置,使得前边的数组有序 [0,i] 有序 + pos = i + while pos > 0 and value < seq[pos-1]: + seq[pos] = seq[pos-1] # 如果前边的元素比它大,就让它一直前移 + pos -= 1 + seq[pos] = value # 找到了合适的位置赋值就好 + print(seq) + + +""" 不断把新元素放到已经有序的数组中 +[1, 7, 3, 0, 9, 4, 8, 2, 6, 5] +[1, 7, 3, 0, 9, 4, 8, 2, 6, 5] +[1, 3, 7, 0, 9, 4, 8, 2, 6, 5] +[0, 1, 3, 7, 9, 4, 8, 2, 6, 5] +[0, 1, 3, 7, 9, 4, 8, 2, 6, 5] +[0, 1, 3, 4, 7, 9, 8, 2, 6, 5] +[0, 1, 3, 4, 7, 8, 9, 2, 6, 5] +[0, 1, 2, 3, 4, 7, 8, 9, 6, 5] +[0, 1, 2, 3, 4, 6, 7, 8, 9, 5] +[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] +""" +``` + + +# 思考题 +- 本章介绍的几个排序算法平均时间复杂度是多少? +- 请你补充插入排序的单元测试代码 + + +# 延伸阅读 +- 《Data Structures and Algorithms in Python》第5章 diff --git a/docs/12/basic_sort.py b/docs/12/basic_sort.py new file mode 100644 index 0000000..d25682d --- /dev/null +++ b/docs/12/basic_sort.py @@ -0,0 +1,56 @@ +# -*- coding: utf-8 -*- + + +import random + + +def bubble_sort(seq): # O(n^2), n(n-1)/2 = 1/2(n^2 + n) + n = len(seq) + for i in range(n-1): + print(seq) # 我打印出来让你看清楚每一轮最高、次高、次次高...的小朋友会冒泡到右边 + for j in range(n-1-i): # 这里之所以 n-1 还需要 减去 i 是因为每一轮冒泡最大的元素都会冒泡到最后,无需再比较 + if seq[j] > seq[j+1]: + seq[j], seq[j+1] = seq[j+1], seq[j] + print(seq) + + +def test_bubble_sort(): + seq = list(range(10)) # 注意 python3 返回迭代器,所以我都用 list 强转了,python2 range 返回的就是 list + random.shuffle(seq) # shuffle inplace 操作,打乱数组 + sorted_seq = sorted(seq) # 注意呦,内置的 sorted 就不是 inplace 的,它返回一个新的数组,不影响传入的参数 + bubble_sort(seq) + assert seq == sorted_seq + + +def select_sort(seq): + n = len(seq) + for i in range(n-1): + min_idx = i # 我们假设当前下标的元素是最小的 + for j in range(i+1, n): # 从 i 的后边开始找到最小的元素,得到它的下标 + if seq[j] < seq[min_idx]: + min_idx = j # 一个 j 循环下来之后就找到了最小的元素它的下标 + if min_idx != i: # swap + seq[i], seq[min_idx] = seq[min_idx], seq[i] + + +def test_select_sort(): + seq = list(range(10)) + random.shuffle(seq) + sorted_seq = sorted(seq) + select_sort(seq) + assert seq == sorted_seq + + +def insertion_sort(seq): + """ 每次挑选下一个元素插入已经排序的数组中,初始时已排序数组只有一个元素""" + n = len(seq) + print(seq) + for i in range(1, n): + value = seq[i] # 保存当前位置的值,因为转移的过程中它的位置可能被覆盖 + # 找到这个值的合适位置,使得前边的数组有序 [0,i] 有序 + pos = i + while pos > 0 and value < seq[pos-1]: + seq[pos] = seq[pos-1] # 如果前边的元素比它大,就让它一直前移 + pos -= 1 + seq[pos] = value # 找到了合适的位置赋值就好 + print(seq) diff --git a/docs/13/advanced_sorting.md b/docs/13/advanced_sorting.md new file mode 100644 index 0000000..2f3c835 --- /dev/null +++ b/docs/13/advanced_sorting.md @@ -0,0 +1,10 @@ +# 高级排序算法 + +本章开始讲几个高级一些的排序算法,因为涉及到分治、递归和一些高级数据结构等,所以比前一章节的基本排序要稍微难理解一些。包括: + +- [分治法与归并排序](./merge_sort.md) +- [快速排序](./quick_sort.md) + +在讲完二叉树之后,我们会看下它的应用: + +- 堆和堆排序 diff --git a/docs/13/merge_sort.md b/docs/13/merge_sort.md new file mode 100644 index 0000000..d2cdecb --- /dev/null +++ b/docs/13/merge_sort.md @@ -0,0 +1,110 @@ +# 分治法 (Divide and Conquer) + +很多有用的算法结构上是递归的,为了解决一个特定问题,算法一次或者多次递归调用其自身以解决若干子问题。 +这些算法典型地遵循分治法的思想:将原问题分解为几个规模较小但是类似于原问题的子问题,递归求解这些子问题, +然后再合并这些问题的解来建立原问题的解。 + +分治法在每层递归时有三个步骤: + +- **分解**原问题为若干子问题,这些子问题是原问题的规模最小的实例 +- **解决**这些子问题,递归地求解这些子问题。当子问题的规模足够小,就可以直接求解 +- **合并**这些子问题的解成原问题的解 + + +# 归并排序 +现在我们就来看下归并排序是是如何利用分治法解决问题的。 + +- **分解**:将待排序的 n 个元素分成各包含 n/2 个元素的子序列 +- **解决**:使用归并排序递归排序两个子序列 +- **合并**:合并两个已经排序的子序列以产生已排序的答案 + +考虑我们排序这个数组:[10,23,51,18,4,31,13,5] ,我们递归地将数组进行分解 + +![](./merge_sort_split.png) + +当数组被完全分隔成只有单个元素的数组时,我们需要把它们合并回去,每次两两合并成一个有序的序列。 + +![](./merge_sort_merge.png) + +用递归代码来描述这个问题: + +```py +def merge_sort(seq): + if len(seq) <= 1: # 只有一个元素是递归出口 + return seq + else: + mid = int(len(seq)/2) + left_half = merge_sort(seq[:mid]) + right_half = merge_sort(seq[mid:]) + + # 合并两个有序的数组 + new_seq = merge_sorted_list(left_half, right_half) + return new_seq +``` + +注意我们这里有一个函数没实现,就是如何合并两个有序数组 merge_sorted_list。其实你在纸上画一画, +合并两个有序数组并不难实现。 + +![](./merge_sorted_array.png) + +![](./merge_sorted_array_2.png) + + +```py +def merge_sorted_list(sorted_a, sorted_b): + """ 合并两个有序序列,返回一个新的有序序列 + + :param sorted_a: + :param sorted_b: + """ + length_a, length_b = len(sorted_a), len(sorted_b) + a = b = 0 + new_sorted_seq = list() + + while a < length_a and b < length_b: + if sorted_a[a] < sorted_b[b]: + new_sorted_seq.append(sorted_a[a]) + a += 1 + else: + new_sorted_seq.append(sorted_b[b]) + b += 1 + + # 最后别忘记把多余的都放到有序数组里 + if a < length_a: + new_sorted_seq.extend(sorted_a[a:]) + else: + new_sorted_seq.extend(sorted_b[b:]) + + return new_sorted_seq +``` + +这样就实现了归并排序,并且你会发现它返回一个新的数组而不是修改原有数组。 + + +# 时间复杂度 +我们来简单看下它归并排序的时间复杂度,假设排序 n 个数字时间复杂度是 T(n),这里为了方便假设 n 是 2 的幂 + +\begin{align} +T(n)= \begin{cases} c, & \text {if $n$ = 1} \\ 2T(n/2)+cn, & \text{if $n$ > 1} \end{cases} +\end{align} + +![](./merge_sort_recursion_tree.png) + +总的代价是 $cnlg(n)+cn$ ,忽略常数项可以认为是 O(nlg(n))。如果这个图看不懂,我们自己求解下也不难,首先我们简化一下, +把常数系数当成 1,得到以下递归式: + +\begin{align} +T(n)= \begin{cases} 1, & \text {if $n$ = 1} \\ 2T(n/2)+n, & \text{if $n$ > 1} \end{cases} +\end{align} + +![](./tn.png) + + +# 思考题 +- 请你完成归并排序的单元测试 +- 这里实现的归并排序是 inplace 的吗? +- 归并排序是稳定的吗?稳定指的是排序前后相同大小的数字依然保持相对顺序。 + +# 延伸阅读 +- 《算法导论》第 2 章和第 4 章,你需要了解下『主定理』,以及如何求解形如 $T(n)=aT(n/b) + f(n)$ 的递归式复杂度 +- 了解算法导论上递归式的三种求解方法:代入法,递归树法,主方法 diff --git a/docs/13/merge_sort.py b/docs/13/merge_sort.py new file mode 100644 index 0000000..a468470 --- /dev/null +++ b/docs/13/merge_sort.py @@ -0,0 +1,48 @@ +# -*- coding: utf-8 -*- + + +def merge_sort(seq): + if len(seq) <= 1: # 只有一个元素是递归出口 + return seq + else: + mid = int(len(seq)/2) + left_half = merge_sort(seq[:mid]) + right_half = merge_sort(seq[mid:]) + + # 合并两个有序的数组 + new_seq = merge_sorted_list(left_half, right_half) + return new_seq + + +def merge_sorted_list(sorted_a, sorted_b): + """ 合并两个有序序列,返回一个新的有序序列 + + :param sorted_a: + :param sorted_b: + """ + length_a, length_b = len(sorted_a), len(sorted_b) + a = b = 0 + new_sorted_seq = list() + + while a < length_a and b < length_b: + if sorted_a[a] < sorted_b[b]: + new_sorted_seq.append(sorted_a[a]) + a += 1 + else: + new_sorted_seq.append(sorted_b[b]) + b += 1 + + # 如果 a或b 中还有剩余元素,需要放到最后 + if a < length_a: + new_sorted_seq.extend(sorted_a[a:]) + else: + new_sorted_seq.extend(sorted_b[b:]) + + return new_sorted_seq + + +def test_merge_sort(): + import random + seq = list(range(10)) + random.shuffle(seq) + assert merge_sort(seq) == sorted(seq) diff --git a/docs/13/merge_sort_merge.png b/docs/13/merge_sort_merge.png new file mode 100644 index 0000000000000000000000000000000000000000..f6f9f66e6b42392a70c9b51a54db7360a9331bcf GIT binary patch literal 54590 zcmZU4WmsHIwl(hV?ivUZTpMWI-6eQ%cMI+w+}&M+28ZD8?oM#$>&)Cc^WNv{U#F|K 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z-J?_c9%Zw8aI>-c@(qE5Ip3wNXQ9ov7GZ7N7wF*A$6A6uo*9qaHaRMxhs3)2v+k@$ zS>ZKYrw#)H3|<#J2->C>lr@3!gC%LcP?U-eQ{*cQdrGND5zm2(GW-pTG7}b)Mxw!bY8u4!&}*%fc8jM7 zQ!1%m{eJk(W<07l-KS5;)2gei)l35`4S~W^4Xa3jTLC!*2SBf;P>>?d6!eVK|86-$ fp!Q~|TyqtW4d5M3GDd}gg52a~RHSPqO@jXq(D5r* literal 0 HcmV?d00001 diff --git a/docs/13/quick_sort.md b/docs/13/quick_sort.md new file mode 100644 index 0000000..33e7b6e --- /dev/null +++ b/docs/13/quick_sort.md @@ -0,0 +1,131 @@ +# 快速排序 + +快速排序名字可不是盖的,很多程序语言标准库实现的内置排序都有它的身影,我们就直奔主题吧。 +和归并排序一样,快排也是一种分而治之(divide and conquer)的策略。归并排序把数组递归成只有单个元素的数组,之后再不断两两 +合并,最后得到一个有序数组。这里的递归基本条件就是只包含一个元素的数组,当数组只包含一个元素的时候,我们可以认为它本来就是有序的(当然空数组也不用排序)。 + +快排的工作过程其实比较简单,三步走: + +- 选择基准值 pivot 将数组分成两个子数组:小于基准值的元素和大于基准值的元素。这个过程称之为 partition + +- 对这两个子数组进行快速排序。 + +- 合并结果 + +![](./quick_sort.png) + +根据这个想法我们可以快速写出快排的代码,简直就是在翻译上边的描述: + +```py +def quicksort(array): + size = len(array) + if not array or size < 2: # NOTE: 递归出口,空数组或者只有一个元素的数组都是有序的 + return array + pivot_idx = 0 + pivot = array[pivot_idx] + less_part = [array[i] for i in range(size) if array[i] <= pivot and pivot_idx != i] + great_part = [array[i] for i in range(size) if array[i] > pivot and pivot_idx != i] + return quicksort(less_part) + [pivot] + quicksort(great_part) + +def test_quicksort(): + import random + seq = list(range(10)) + random.shuffle(seq) + assert quicksort(seq) == sorted(seq) +``` +是不是很简单,下次面试官让你手写快排你再写不出来就有点不太合适啦。 当然这个实现有两个不好的地方: + +- 第一是它需要额外的存储空间,我们想实现 inplace 原地排序。 +- 第二是它的 partition 操作每次都要两次遍历整个数组,我们想改善一下。 + +这里我们就来优化一下它,实现 inplace 排序并且改善下 partition 操作。新的代码大概如下: + +```py +def quicksort_inplace(array, beg, end): # 注意这里我们都用左闭右开区间,end 传入 len(array) + if beg < end: # beg == end 的时候递归出口 + pivot = partition(array, beg, end) + quicksort_inplace(array, beg, pivot) + quicksort_inplace(array, pivot+1, end) +``` + +能否实现只遍历一次数组就可以完成 partition 操作呢?实际上是可以的。我们设置首位俩个指针 left, right,两个指针不断向中间收拢。如果遇到 left 位置的元素大于 pivot 并且 right 指向的元素小于 pivot,我们就交换这俩元素,当 left > right 的时候退出就行了,这样实现了一次遍历就完成了 partition。如果你感觉懵逼,纸上画画就立马明白了。我们来撸代码实现: + +![](./partition.png) + +```py +def partition(array, beg, end): + pivot_index = beg + pivot = array[pivot_index] + left = pivot_index + 1 + right = end - 1 # 开区间,最后一个元素位置是 end-1 [0, end-1] or [0: end),括号表示开区间 + + while True: + # 从左边找到比 pivot 大的 + while left <= right and array[left] < pivot: + left += 1 + + while right >= left and array[right] >= pivot: + right -= 1 + + if left > right: + break + else: + array[left], array[right] = array[right], array[left] + + array[pivot_index], array[right] = array[right], array[pivot_index] + return right # 新的 pivot 位置 + + +def test_partition(): + l = [4, 1, 2, 8] + assert partition(l, 0, len(l)) == 2 + l = [1, 2, 3, 4] + assert partition(l, 0, len(l)) == 0 + l = [4, 3, 2, 1] + assert partition(l, 0, len(l)) +``` + +大功告成,新的快排就实现好了。 + +# 时间复杂度 +在比较理想的情况下,比如数组每次都被 pivot 均分,我们可以得到递归式: + +T(n) = 2T(n/2) + n + +上一节我们讲过通过递归树得到它的时间复杂度是 O(nlog(n))。即便是很坏的情况,比如 pivot 每次都把数组按照 1:9 划分,依然是 O(nlog(n)),感兴趣请阅读算法导论相关章节。 + +![](quicksort_worst.png) + + +# 思考题 +- 请你补充 quicksort_inplace 的单元测试 +- 最坏的情况下快排的时间复杂度是多少?什么时候会发生这种情况? +- 我们实现的快排是稳定的啵? +- 选择基准值如果选不好就可能导致复杂度升高,算导中提到了一种『median of 3』策略,就是说选择 pivot 的时候 从子数组中随机选三个元素,再取它的中位数,你能实现这个想法吗?这里我们的代码很简单地取了第一个元素作为 pivot +- 利用快排中的 partition 操作,我们还能实现另一个算法,nth_element,快速查找一个无序数组中的第 n 大元素,请你尝试实现它并编写单测。其实这个函数是 C++ STL 中的一个函数。 +- 你知道 Python 内置的 sorted 如何实现的吗?请你 Google 相关资料了解下。很多内置的排序都使用了快排的改良版。 + + +# 延伸阅读 +- 《算法导论》第 7 章 +- [《面试必备 | 排序算法的Python实现》](https://zhuanlan.zhihu.com/p/36419582) + +# 总结 + +面试经常问的就是常用排序算法的时间空间复杂,这里列一个表格方便记忆: + +| 排序算法 | 最差时间分析 | 平均时间复杂度 | 稳定度 | 空间复杂度 | +|------------|--------------|----------------|--------|----------------| +| 冒泡排序 | O(n^2) | O(n^2) | 稳定 | O(1) | +| 选择排序 | O(n^2) | O(n^2) | 不稳定 | O(1) | +| 插入排序 | O(n^2) | O(n^2) | 稳定 | O(1) | +| 二叉树排序 | O(n^2) | O(n\*log2n) | 不一顶 | O(n) | +| 快速排序 | O(n^2) | O(n\*log2n) | 不稳定 | O(log2n)\~O(n) | +| 堆排序 | O(n\*log2n) | O(n\*log2n) | 不稳定 | O(1) | + +[数据结构与算法-排序篇-Python描述](https://blog.csdn.net/mrlevo520/article/details/77829204) + +# Leetcode + +无序数组寻找第 k 大的数字,不止一种方法。 +https://leetcode.com/problems/kth-largest-element-in-an-array/description/ diff --git a/docs/13/quick_sort.png b/docs/13/quick_sort.png new file mode 100644 index 0000000000000000000000000000000000000000..a9f85612e5df4087724eed5ab22021b7f0c2d51d GIT binary patch literal 94479 zcmeFYgLh`n(l(q-Y-?iM6HIK|wr$(CIk6|UolHEjZQIFv&-~7F&Uv5jKls*rueI0S zsoh=OUERB@`noFNax$WD&|jf}fPmn{#e@`qfW9mM0Ritof&(PuD3M!0K$wK)f`W45 zf`SBc4z?!dR>nX;YQf3w5K5@h^)6GB;A$zPhgpexunCTnSpI-V`EVkBVMIAlB{_b% zUOr?-!Y~S=Qa|AMA4X`Xa-e(g{F|4)m%dZn%~R>y%O~X-CGF+yC73`Kh7QOG6q)ij0?IY@Aafw6Lz`{pOPB`<@P5#?x!aVRLt`&tj-?Mljv6s+i0xx zW=Vd1GR-uUt9z$BQ6-8A|dLRFQS7lLdVN*y&pUx#qB)x0)Z2sTg+X+FCvg4 zOmOpYN4BRm{V4ND*m|}xdPD)Q`2ki^F889y@6O|&M?yntw)O|2R|&R-UH38Y<2cCo zrdl;Y80MKE%mFL~@78dxR{Em36r!`dYaQNB_b%)`8$5X~?OFOH?G06~*3A8Zw66Et z%sjo|3!JyGx{(o$&xWr=U&U*(-Tj6^#D`Ki-?o#l%p_NFZOpzgfN}6)*ac$l-GJ&7 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递归出口,空数组或者只有一个元素的数组都是有序的 + return array + pivot_idx = 0 + pivot = array[pivot_idx] + less_part = [array[i] for i in range(size) if array[i] <= pivot and pivot_idx != i] + great_part = [array[i] for i in range(size) if array[i] > pivot and pivot_idx != i] + return quicksort(less_part) + [pivot] + quicksort(great_part) + + +def test_quicksort(): + import random + seq = list(range(10)) + random.shuffle(seq) + assert quicksort(seq) == sorted(seq) # 用内置的sorted 『对拍』 + + +def quicksort_inplace(array, beg, end): # 注意这里我们都用左闭右开区间 + if beg < end: # beg == end 的时候递归出口 + pivot = partition(array, beg, end) + quicksort_inplace(array, beg, pivot) + quicksort_inplace(array, pivot + 1, end) + + +def partition(array, beg, end): + """对给定数组执行 partition 操作,返回新的 pivot 位置""" + pivot_index = beg + pivot = array[pivot_index] + left = pivot_index + 1 + right = end - 1 # 开区间,最后一个元素位置是 end-1 [0, end-1] or [0: end),括号表示开区间 + + while True: + # 从左边找到比 pivot 大的 + while left <= right and array[left] < pivot: + left += 1 + + while right >= left and array[right] >= pivot: + right -= 1 + + if left > right: + break + else: + array[left], array[right] = array[right], array[left] + + array[pivot_index], array[right] = array[right], array[pivot_index] + return right # 新的 pivot 位置 + + +def test_partition(): + l = [4, 1, 2, 8] + assert partition(l, 0, len(l)) == 2 + l = [1, 2, 3, 4] + assert partition(l, 0, len(l)) == 0 + l = [4, 3, 2, 1] + assert partition(l, 0, len(l)) == 3 + l = [1] + assert partition(l, 0, len(l)) == 0 + l = [2,1] + assert partition(l, 0, len(l)) == 1 + + +def test_quicksort_inplace(): + import random + seq = list(range(10)) + random.shuffle(seq) + sorted_seq = sorted(seq) + quicksort_inplace(seq, 0, len(seq)) + assert seq == sorted_seq + + +def nth_element(array, beg, end, nth): + """查找一个数组第 n 大元素""" + if beg < end: + pivot_idx = partition(array, beg, end) + if pivot_idx == nth - 1: # 数组小标从 0 开始 + return array[pivot_idx] + elif pivot_idx > nth - 1: + return nth_element(array, beg, pivot_idx, nth) + else: + return nth_element(array, pivot_idx + 1, end, nth) + + +def test_nth_element(): + l1 = [3, 5, 4, 2, 1] + assert nth_element(l1, 0, len(l1), 3) == 3 + assert nth_element(l1, 0, len(l1), 2) == 2 + + l = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] + for i in l: + assert nth_element(l, 0, len(l), i) == i + for i in reversed(l): + assert nth_element(l, 0, len(l), i) == i + + array = [3, 2, 1, 5, 6, 4] + assert nth_element(array, 0, len(array), 2) == 2 + + array = [2,1] + assert nth_element(array, 0, len(array), 1) == 1 + assert nth_element(array, 0, len(array), 2) == 2 + + array = [3,3,3,3,3,3,3,3,3] + assert nth_element(array, 0, len(array), 1) == 3 + + +if __name__ == '__main__': + test_nth_element() diff --git a/docs/13/quicksort_worst.png b/docs/13/quicksort_worst.png new file mode 100644 index 0000000000000000000000000000000000000000..d25640ccfb7039b0a8a70e83560de862d0fe8a8b GIT binary patch literal 67090 zcmeEtV{~TE(r;`V6Wh)tnHUq>wr$%^X2OY`Ol)If+qP}zKAHb{&pGe9-|wfJwO3Z} 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a/docs/13/test.sh b/docs/13/test.sh new file mode 100644 index 0000000..fd4895a --- /dev/null +++ b/docs/13/test.sh @@ -0,0 +1,7 @@ + + +#!/usr/bin/env bash + +# pip install when-changed, 监控文件变动并且文件修改之后自动执行 pytest 单测,方便我们边修改边跑测试 + when-changed -v -r -1 -s ./ "py.test -s $1" + diff --git a/docs/13/tn.png b/docs/13/tn.png new file mode 100644 index 0000000000000000000000000000000000000000..a41193dc9220c05888c67303b60c8163aff4c409 GIT binary patch literal 173846 zcmZU419T_L_H{C`&51p+ZQHhOJDJ$V#1q@L?TPJVVq4$u-v4{|&RbtrXRS(iS9P7j zIs5Eg373}@gM-F~1_AUJ;5E7D?5E3GgceFFL zur>h#QV&k{fKW!2sdt^80#{EVJ<3enhfQ#r!uk#aYj&g@Pa49hhsBic)+0*Dut!SQ`_>Szoqfdlz&&?_j$JLkuV^}g^k2k-2!Pbn2T=pphTCW>)!W2A|A z{902O?-)11>WC|bmuQH_<%`(hi}3NvTki+2Xi++_|`joobO%Pdp3Dugs;^aBa<*>A^VpG3)~|_isQA2oOQiDR&1fPLp;^ty6b8^5r7^lupPV>kjg)km$J+rrHvH{`h$ zgP;kkNfEQkUqJkTbYmt?M+?h8h>JJRWr?dDCI0nkjCq?n$mjHYd1Q0_I?Zq~7M>^k ztb?BI_k8-c@Y0Nt_4$L?Zohi8n`MXkiEJH=&%b?#wFS$*|8X0=1){4y_|vp+_+0ww 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linux 文件结构(tree 命令),它的结构也是一棵树。我们快速看下树涉及到的一些概念: + +![](./tree.png) + +- 根节点(root): 树的最上层的节点,任何非空的树都有一个节点 +- 路径(path): 从起始节点到终止节点经历过的边 +- 父亲(parent):除了根节点,每个节点的上一层边连接的节点就是它的父亲(节点) +- 孩子(children): 每个节点由边指向的下一层节点 +- 兄弟(siblings): 同一个父亲并且处在同一层的节点 +- 子树(subtree): 每个节点包含它所有的后代组成的子树 +- 叶子节点(leaf node): 没有孩子的节点成为叶子节点 + + +# 二叉树 + +了解完树的结构以后,我们来看树结构里一种简单但是却比较常用的树-二叉树。 +二叉树是一种简单的树,它的每个节点最多只能包含两个孩子,以下都是一些合法的二叉树: + +![](./binary_tree.png) +![](./binary_tree_level.png) + +通过上边这幅图再来看几个二叉树相关的概念: + +- 节点深度(depth): 节点对应的 level 数字 +- 树的高度(height): 二叉树的高度就是 level 数 + 1,因为 level 从 0开始计算的 +- 树的宽度(width): 二叉树的宽度指的是包含最多节点的层级的节点数 +- 树的 size:二叉树的节点总个数。 + +一棵 size 为 n 的二叉树高度最多可以是 n,最小的高度是 $ \lfloor lgn \rfloor + 1 $,这里 log 以 2 为底简写为 +lgn,和算法导论保持一致。这个结果你只需要用高中的累加公式就可以得到。 + +# 一些特殊的二叉树 +在了解了二叉树的术语和概念之后,我们来看看一些特殊的二叉树,后续章节我们会用到: + +### 满二叉树(full binary tree) +如果每个内部节点(非叶节点)都包含两个孩子,就成为满二叉树。下边是一些例子,它可以有多种形状: + +![](./full_binary_tree.png) + +### 完美二叉树(perfect binary tree) +当所有的叶子节点都在同一层就是完美二叉树,毫无间隙填充了 h 层。 + +![](./perfect_binary_tree.png) + +### 完全二叉树(complete binary tree) +当一个高度为 h 的完美二叉树减少到 h-1,并且最底层的槽被毫无间隙地从左到右填充,我们就叫它完全二叉树。 +下图就是完全二叉树的例子: + +![](./complete_binary_tree.png) + +# 二叉树的表示 +说了那么多,那么怎么表示一棵二叉树呢?其实你发现会和链表有一些相似之处,一个节点,然后节点需要保存孩子的指针,我以构造下边这个二叉树为例子: +我们先定义一个类表示节点: + +![](preorder.png) + +```py +class BinTreeNode(object): + def __init__(self, data, left=None, right=None): + self.data, self.left, self.right = data, left, right +``` + +当然和链表类似,root 节点是我们的入口,于是乎定义一个 二叉树: + +```py +class BinTree(object): + def __init__(self, root=None): + self.root = root +``` + +怎么构造上图中的二叉树呢,似乎其他课本没找到啥例子(有些例子是写了一堆嵌套节点来定义,很难搞清楚层次关系),我自己定义了一种方法,首先我们输入节点信息,仔细看下边代码,叶子节点的 left 和 right 都是 None,并且只有一个根节点 A: + +```py +node_list = [ + {'data': 'A', 'left': 'B', 'right': 'C', 'is_root': True}, + {'data': 'B', 'left': 'D', 'right': 'E', 'is_root': False}, + {'data': 'D', 'left': None, 'right': None, 'is_root': False}, + {'data': 'E', 'left': 'H', 'right': None, 'is_root': False}, + {'data': 'H', 'left': None, 'right': None, 'is_root': False}, + {'data': 'C', 'left': 'F', 'right': 'G', 'is_root': False}, + {'data': 'F', 'left': None, 'right': None, 'is_root': False}, + {'data': 'G', 'left': 'I', 'right': 'J', 'is_root': False}, + {'data': 'I', 'left': None, 'right': None, 'is_root': False}, + {'data': 'J', 'left': None, 'right': None, 'is_root': False}, +] +``` + +然后我们给 BinTreeNode 定义一个 build_from 方法,当然你也可以定义一种自己的构造方法: + +```py +class BinTree(object): + def __init__(self, root=None): + self.root = root + + @classmethod + def build_from(cls, node_list): + """通过节点信息构造二叉树 + 第一次遍历我们构造 node 节点 + 第二次遍历我们给 root 和 孩子赋值 + 最后我们用 root 初始化这个类并返回一个对象 + + :param node_list: {'data': 'A', 'left': None, 'right': None, 'is_root': False} + """ + node_dict = {} + for node_data in node_list: + data = node_data['data'] + node_dict[data] = BinTreeNode(data) + for node_data in node_list: + data = node_data['data'] + node = node_dict[data] + if node_data['is_root']: + root = node + node.left = node_dict.get(node_data['left']) + node.right = node_dict.get(node_data['right']) + return cls(root) +btree = BinTree.build_from(node_list) +``` + +大功告成,这样我们就构造了一棵二叉树对象。下边我们看看它的一些常用操作。 + +# 二叉树的遍历 +不知道你有没有发现,二叉树其实是一种递归结构,因为单独拿出来一个 subtree 子树出来,其实它还是一棵树。那遍历它就很方便啦,我们可以直接用递归的方式来遍历它。但是当处理顺序不同的时候,树又分为三种遍历方式: + +- 先(根)序遍历: 先处理根,之后是左子树,然后是右子树 +- 中(根)序遍历: 先处理左子树,之后是根,最后是右子树 +- 后(根)序遍历: 先处理左子树,之后是右子树,最后是根 + +我们来看下实现,其实算是比较直白的递归函数: + +```py +class BinTreeNode(object): + def __init__(self, data, left=None, right=None): + self.data, self.left, self.right = data, left, right + + +class BinTree(object): + def __init__(self, root=None): + self.root = root + + @classmethod + def build_from(cls, node_list): + """通过节点信息构造二叉树 + 第一次遍历我们构造 node 节点 + 第二次遍历我们给 root 和 孩子赋值 + + :param node_list: {'data': 'A', 'left': None, 'right': None, 'is_root': False} + """ + node_dict = {} + for node_data in node_list: + data = node_data['data'] + node_dict[data] = BinTreeNode(data) + for node_data in node_list: + data = node_data['data'] + node = node_dict[data] + if node_data['is_root']: + root = node + node.left = node_dict.get(node_data['left']) + node.right = node_dict.get(node_data['right']) + return cls(root) + + def preorder_trav(self, subtree): + """ 先(根)序遍历 + + :param subtree: + """ + if subtree is not None: + print(subtree.data) # 递归函数里先处理根 + self.preorder_trav(subtree.left) # 递归处理左子树 + self.preorder_trav(subtree.right) # 递归处理右子树 + + +node_list = [ + {'data': 'A', 'left': 'B', 'right': 'C', 'is_root': True}, + {'data': 'B', 'left': 'D', 'right': 'E', 'is_root': False}, + {'data': 'D', 'left': None, 'right': None, 'is_root': False}, + {'data': 'E', 'left': 'H', 'right': None, 'is_root': False}, + {'data': 'H', 'left': None, 'right': None, 'is_root': False}, + {'data': 'C', 'left': 'F', 'right': 'G', 'is_root': False}, + {'data': 'F', 'left': None, 'right': None, 'is_root': False}, + {'data': 'G', 'left': 'I', 'right': 'J', 'is_root': False}, + {'data': 'I', 'left': None, 'right': None, 'is_root': False}, + {'data': 'J', 'left': None, 'right': None, 'is_root': False}, +] +btree = BinTree.build_from(node_list) +btree.preorder_trav(btree.root) # 输出 A, B, D, E, H, C, F, G, I, J +``` +怎么样是不是挺简单的,比较直白的递归函数。如果你不明白,视频里我们会画个图来说明。 + +# 二叉树层序遍历 + +除了递归的方式遍历之外,我们还可以使用层序遍历的方式。层序遍历比较直白,就是从根节点开始按照一层一层的方式遍历节点。 +我们可以从根节点开始,之后把所有当前层的孩子都按照从左到右的顺序放到一个列表里,下一次遍历所有这些孩子就可以了。 + +```py + def layer_trav(self, subtree): + cur_nodes = [subtree] # current layer nodes + next_nodes = [] + while cur_nodes or next_nodes: + for node in cur_nodes: + print(node.data) + if node.left: + next_nodes.append(node.left) + if node.right: + next_nodes.append(node.right) + cur_nodes = next_nodes # 继续遍历下一层 + next_nodes = [] +``` + +还有一种方式就是使用一个队列,之前我们知道队列是一个先进先出结构,如果我们按照一层一层的顺序从左往右把节点放到一个队列里, +也可以实现层序遍历: + +```py + def layer_trav_use_queue(self, subtree): + q = Queue() + q.append(subtree) + while not q.empty(): + cur_node = q.pop() + print(cur_node.data) + if cur_node.left: + q.append(cur_node.left) + if cur_node.right: + q.append(cur_node.right) + + +from collections import deque +class Queue(object): # 借助内置的 deque 我们可以迅速实现一个 Queue + def __init__(self): + self._items = deque() + + def append(self, value): + return self._items.append(value) + + def pop(self): + return self._items.popleft() + + def empty(self): + return len(self._items) == 0 +``` + + +# 反转二叉树 +之所以单拎出来说这个是因为 mac 下著名的 brew 工具作者据说是因为面试 google 白板编程没写出来反转二叉树跪了。然后人家就去了苹果 😂。其实吧和遍历操作相比也没啥太大区别,递归交换就是了: + +```py + def reverse(self, subtree): + if subtree is not None: + subtree.left, subtree.right = subtree.right, subtree.left + self.reverse(subtree.left) + self.reverse(subtree.right) +``` + + +# 练习题 +- 请你完成二叉树的中序遍历和后序遍历以及单元测试 +- 树的遍历我们用了 print,请你尝试换成一个 callback,这样就能自定义处理树节点的方式了。 +- 请问树的遍历操作时间复杂度是多少?假设它的 size 是 n +- 你能用非递归的方式来实现树的遍历吗?我们知道计算机内部使用了 stack,如果我们自己模拟如何实现?请你尝试完成 +- 只根据二叉树的中序遍历和后序遍历能否确定一棵二叉树?你可以举一个反例吗? + + +# 延伸阅读 +- 《Data Structures and Algorithms in Python》 13 章 Binary Trees +- [https://www.geeksforgeeks.org/iterative-preorder-traversal/](https://www.geeksforgeeks.org/iterative-preorder-traversal/) + + +# Leetcode 练习 + +- [leetcode binary-tree-preorder-traversal](https://leetcode.com/problems/binary-tree-preorder-traversal/) +二叉树的先序遍历 + +- [leetcode binary-tree-inorder-traversal/](https://leetcode.com/problems/binary-tree-inorder-traversal/) +二叉树的中序遍历 + +- [leetcode binary-tree-postorder-traversal](https://leetcode.com/problems/binary-tree-postorder-traversal/) +二叉树的后序遍历 + +- [leetcode binary-tree-right-side-view](https://leetcode.com/problems/binary-tree-right-side-view/description/) +使用树的层序遍历我们能实现一个树的左右视图,比如从一个二叉树的左边能看到哪些节点。 请你尝试做这个练习题 + +- [leetcode construct-binary-tree-from-preorder-and-postorder-traversal](https://leetcode.com/problems/construct-binary-tree-from-preorder-and-postorder-traversal/submissions/) +根据二叉树的 前序和后序遍历,返回一颗完整的二叉树。 diff --git a/docs/14/tree.png b/docs/14/tree.png new file mode 100644 index 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0000000..e9479d6 --- /dev/null +++ b/docs/15/lfu.py @@ -0,0 +1,69 @@ +""" +https://medium.com/@epicshane/a-python-implementation-of-lfu-least-frequently-used-cache-with-o-1-time-complexity-e16b34a3c49b +https://leetcode.com/problems/lfu-cache/ + +这里学习下 LRU(least frequently used),就是当缓存满了之后剔除一个最少使用的 key。 +""" +from collections import defaultdict, OrderedDict + + +class Node: + __slots__ = 'key', 'val', 'cnt' + + def __init__(self, key, val, cnt=0): + self.key, self.val, self.cnt = key, val, cnt + + +class LFUCache: + def __init__(self, capacity): + self.capacity = capacity + self.cache = {} # type {key: node} + self.cnt2node = defaultdict(OrderedDict) + self.mincnt = 0 + + def get(self, key, default=-1): + if key not in self.cache: + return default + + node = self.cache[key] + del self.cnt2node[node.cnt][key] + + if not self.cnt2node[node.cnt]: + del self.cnt2node[node.cnt] + + node.cnt += 1 + self.cnt2node[node.cnt][key] = node + + if not self.cnt2node[self.mincnt]: + self.mincnt += 1 + return node.val + + def put(self, key, value): + if key in self.cache: + self.cache[key].val = value + self.get(key) + return + if len(self.cache) >= self.capacity: + pop_key, _pop_node = self.cnt2node[self.mincnt].popitem(last=False) + del self.cache[pop_key] + + self.cache[key] = self.cnt2node[1][key] = Node(key, value, 1) + self.mincnt = 1 + + +def test(): + c = LFUCache(2) + c.put(1, 1) + c.put(2, 2) + assert c.get(1) == 1 + c.put(3, 3) + assert c.get(2) == -1 + assert c.get(3) == 3 + c.put(4, 4) + assert c.get(1) == -1 + assert c.get(3) == 3 + assert c.get(4) == 4 + + +if __name__ == '__main__': + test() diff --git a/docs/15/siftdown.png b/docs/15/siftdown.png new file mode 100644 index 0000000000000000000000000000000000000000..980316a854f861e2e08a872ed36976bc4556ce98 GIT binary patch literal 117076 zcmeFYWmH`4vMr1>&;)BdxVr{-*WeN~SP1U!uEB!_clY3~5eV*X!QEZ%;@$h6v(Nc{ zfB!CHFld%MQmaBdn?OOyd`U`1P*+0v@){5ySiYtD*TX@ zQ8$AIwMC{&`95R|N(R$3yzfx}?_I3}@*|Qz^ml)-Gj!&#EUrgzFesoa`s@KJP3PpB 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zzxom+qVTZr*2WnE)f4zPWqOCtxqN%>CV`;?Y?}9;M2{M*@44N{`0oUHO?x#xGi|u1WlvQN;RI6 zq51!KzV7>HQ(OGd_ZeTgo~ZmP$L+`8?~`u4kFQf|t646ARai&!&mif7{k>zYtOuZgyGK zWAgqvU9JlIZ$8h}on12{@mg#5 z%V$$pzlaw*yVXX%Cqhv5@nojP!V$x4(d-UfvUf{ccrPjF0hLEX?Oz%mIq`CEcrgWxh_Clx`&HcH4WHP8lwxiaFF zS}2pn3LGgez**-W0Z$f$|G;)BSORCq)sMT*<3gF-=U|cqPA^+2y9ha=%u+iDc=Q3M syH6HQT#h;m-sIp0+LJg$yl~(@d&c~-054@F83rKmboFyt=akR{01z}GCIA2c literal 0 HcmV?d00001 diff --git a/docs/15/topk.py b/docs/15/topk.py new file mode 100644 index 0000000..87b3d7b --- /dev/null +++ b/docs/15/topk.py @@ -0,0 +1,43 @@ +import heapq + + +class TopK: + """获取大量元素 topk 大个元素,固定内存 + 思路: + 1. 先放入元素前 k 个建立一个最小堆 + 2. 迭代剩余元素: + 如果当前元素小于堆顶元素,跳过该元素(肯定不是前 k 大) + 否则替换堆顶元素为当前元素,并重新调整堆 + """ + + def __init__(self, iterable, k): + self.minheap = [] + self.capacity = k + self.iterable = iterable + + def push(self, val): + if len(self.minheap) >= self.capacity: + min_val = self.minheap[0] + if val < min_val: # 当然你可以直接 if val > min_val操作,这里我只是显示指出跳过这个元素 + pass + else: + heapq.heapreplace(self.minheap, val) # 返回并且pop堆顶最小值,推入新的 val 值并调整堆 + else: + heapq.heappush(self.minheap, val) # 前面 k 个元素直接放入minheap + + def get_topk(self): + for val in self.iterable: + self.push(val) + return self.minheap + + +def test(): + import random + i = list(range(1000)) # 这里可以是一个可迭代元素,节省内存 + random.shuffle(i) + _ = TopK(i, 10) + print(_.get_topk()) # [990, 991, 992, 996, 994, 993, 997, 998, 999, 995] + + +if __name__ == '__main__': + test() diff --git a/docs/16/priority_queue.md b/docs/16/priority_queue.md new file mode 100644 index 0000000..5fe3b16 --- /dev/null +++ b/docs/16/priority_queue.md @@ -0,0 +1,53 @@ +# 优先级队列 +你可能比较奇怪,队列不是早就讲了嘛。这里之所以放到这里讲优先级队列,是因为虽然名字有队列, +但其实是使用堆来实现的。上一章讲完了堆,这一章我们就趁热打铁来实现一个优先级队列。 + + +# 实现优先级队列 +优先级队列(Priority Queue) 顾名思义,就是入队的时候可以给一个优先级,通常是个数字或者时间戳等, +当出队的时候我们希望按照给定的优先级出队,我们按照 TDD(测试驱动开发) 的方式先来写测试代码: + +```py +def test_priority_queue(): + size = 5 + pq = PriorityQueue(size) + pq.push(5, 'purple') # priority, value + pq.push(0, 'white') + pq.push(3, 'orange') + pq.push(1, 'black') + + res = [] + while not pq.is_empty(): + res.append(pq.pop()) + assert res == ['purple', 'orange', 'black', 'white'] +``` + +上边就是期望的行为,写完测试代码后我们来编写优先级队列的代码,按照出队的时候最大优先级先出的顺序: + + +```py +class PriorityQueue(object): + def __init__(self, maxsize): + self.maxsize = maxsize + self._maxheap = MaxHeap(maxsize) + + def push(self, priority, value): + # 注意这里把这个 tuple push 进去,python 比较 tuple 从第一个开始比较 + # 这样就很巧妙地实现了按照优先级排序 + entry = (priority, value) # 入队的时候会根据 priority 维持堆的特性 + self._maxheap.add(entry) + + def pop(self, with_priority=False): + entry = self._maxheap.extract() + if with_priority: + return entry + else: + return entry[1] + + def is_empty(self): + return len(self._maxheap) == 0 +``` + + +# 练习题 +- 请你实现按照小优先级先出队的顺序的优先级队列 diff --git a/docs/16/priority_queue.py b/docs/16/priority_queue.py new file mode 100644 index 0000000..039885c --- /dev/null +++ b/docs/16/priority_queue.py @@ -0,0 +1,131 @@ +# -*- coding:utf-8 -*- + +# 第二章拷贝的 Array 代码 + + +class Array(object): + + def __init__(self, size=32): + self._size = size + self._items = [None] * size + + def __getitem__(self, index): + return self._items[index] + + def __setitem__(self, index, value): + self._items[index] = value + + def __len__(self): + return self._size + + def clear(self, value=None): + for i in range(len(self._items)): + self._items[i] = value + + def __iter__(self): + for item in self._items: + yield item + +##################################################### +# heap 实现 +##################################################### + + +class MaxHeap(object): + """ + Heaps: + 完全二叉树,最大堆的非叶子节点的值都比孩子大,最小堆的非叶子结点的值都比孩子小 + Heap包含两个属性,order property 和 shape property(a complete binary tree),在插入 + 一个新节点的时候,始终要保持这两个属性 + 插入操作:保持堆属性和完全二叉树属性, sift-up 操作维持堆属性 + extract操作:只获取根节点数据,并把树最底层最右节点copy到根节点后,sift-down操作维持堆属性 + + 用数组实现heap,从根节点开始,从上往下从左到右给每个节点编号,则根据完全二叉树的 + 性质,给定一个节点i, 其父亲和孩子节点的编号分别是: + parent = (i-1) // 2 + left = 2 * i + 1 + rgiht = 2 * i + 2 + 使用数组实现堆一方面效率更高,节省树节点的内存占用,一方面还可以避免复杂的指针操作,减少 + 调试难度。 + + """ + + def __init__(self, maxsize=None): + self.maxsize = maxsize + self._elements = Array(maxsize) + self._count = 0 + + def __len__(self): + return self._count + + def add(self, value): + if self._count >= self.maxsize: + raise Exception('full') + self._elements[self._count] = value + self._count += 1 + self._siftup(self._count-1) # 维持堆的特性 + + def _siftup(self, ndx): + if ndx > 0: + parent = int((ndx-1)/2) + if self._elements[ndx] > self._elements[parent]: # 如果插入的值大于 parent,一直交换 + self._elements[ndx], self._elements[parent] = self._elements[parent], self._elements[ndx] + self._siftup(parent) # 递归 + + def extract(self): + if self._count <= 0: + raise Exception('empty') + value = self._elements[0] # 保存 root 值 + self._count -= 1 + self._elements[0] = self._elements[self._count] # 最右下的节点放到root后siftDown + self._siftdown(0) # 维持堆特性 + return value + + def _siftdown(self, ndx): + left = 2 * ndx + 1 + right = 2 * ndx + 2 + # determine which node contains the larger value + largest = ndx + if (left < self._count and # 有左孩子 + self._elements[left] >= self._elements[largest] and + self._elements[left] >= self._elements[right]): # 原书这个地方没写实际上找的未必是largest + largest = left + elif right < self._count and self._elements[right] >= self._elements[largest]: + largest = right + if largest != ndx: + self._elements[ndx], self._elements[largest] = self._elements[largest], self._elements[ndx] + self._siftdown(largest) + + +class PriorityQueue(object): + def __init__(self, maxsize): + self.maxsize = maxsize + self._maxheap = MaxHeap(maxsize) + + def push(self, priority, value): + entry = (priority, value) # 注意这里把这个 tuple push进去,python 比较 tuple 从第一个开始比较 + self._maxheap.add(entry) + + def pop(self, with_priority=False): + entry = self._maxheap.extract() + if with_priority: + return entry + else: + return entry[1] + + def is_empty(self): + return len(self._maxheap) == 0 + + +def test_priority_queue(): + size = 5 + pq = PriorityQueue(size) + pq.push(5, 'purple') + pq.push(0, 'white') + pq.push(3, 'orange') + pq.push(1, 'black') + + res = [] + while not pq.is_empty(): + res.append(pq.pop()) + assert res == ['purple', 'orange', 'black', 'white'] diff --git a/docs/17/binary_search_tree.md b/docs/17/binary_search_tree.md new file mode 100644 index 0000000..06e9bc7 --- /dev/null +++ b/docs/17/binary_search_tree.md @@ -0,0 +1,252 @@ +# 二叉查找树(BST) + +二叉树的一种应用就是来实现堆,今天我们再看看用二叉查找树(Binary Search Tree, BST)。 +前面有章节说到了查找操作,包括线性查找、二分查找、哈希查找等,线性查找效率比较低,二分又要求必须是有序的序列, +为了维持有序插入的代价比较高、哈希查找效率很高但是浪费空间。能不能有一种插入和查找都比较快的数据结构呢?二叉查找树就是这样一种结构,可以高效地插入和查询节点。 + +# BST 定义 + +二叉查找树是这样一种二叉树结构,它的每个节点包含一个 key 和它附带的数据,对于每个内部节点 V: + +- 所有 key 小于 V 的都被存储在 V 的左子树 +- 所有 key 大于 V 的都存储在 V 的右子树 + +![](./bst.png) + +注意这个限制条件,可别和堆搞混了。说白了就是对于每个内部节点,左子树的 key 都比它小,右子树都比它大。 +如果中序遍历(二叉树遍历讲过了)这颗二叉树,你会发现输出的顺序正好是有序的。 +我们先来定义一下 BST 的节点结构: + +```py +class BSTNode(object): + def __init__(self, key, value, left=None, right=None): + self.key, self.value, self.left, self.right = key, value, left, right +``` + +# 构造一个 BST +我们还像之前构造二叉树一样,按照上图构造一个 BST 用来演示: + +```py +class BST(object): + def __init__(self, root=None): + self.root = root + + @classmethod + def build_from(cls, node_list): + cls.size = 0 + key_to_node_dict = {} + for node_dict in node_list: + key = node_dict['key'] + key_to_node_dict[key] = BSTNode(key, value=key) # 这里值暂时用 和 key一样的 + + for node_dict in node_list: + key = node_dict['key'] + node = key_to_node_dict[key] + if node_dict['is_root']: + root = node + node.left = key_to_node_dict.get(node_dict['left']) + node.right = key_to_node_dict.get(node_dict['right']) + cls.size += 1 + return cls(root) + + +NODE_LIST = [ + {'key': 60, 'left': 12, 'right': 90, 'is_root': True}, + {'key': 12, 'left': 4, 'right': 41, 'is_root': False}, + {'key': 4, 'left': 1, 'right': None, 'is_root': False}, + {'key': 1, 'left': None, 'right': None, 'is_root': False}, + {'key': 41, 'left': 29, 'right': None, 'is_root': False}, + {'key': 29, 'left': 23, 'right': 37, 'is_root': False}, + {'key': 23, 'left': None, 'right': None, 'is_root': False}, + {'key': 37, 'left': None, 'right': None, 'is_root': False}, + {'key': 90, 'left': 71, 'right': 100, 'is_root': False}, + {'key': 71, 'left': None, 'right': 84, 'is_root': False}, + {'key': 100, 'left': None, 'right': None, 'is_root': False}, + {'key': 84, 'left': None, 'right': None, 'is_root': False}, +] +bst = BST.build_from(NODE_LIST) +``` + + +# BST 操作 + +## 查找 +如何查找一个指定的节点呢,根据定义我们知道每个内部节点左子树的 key 都比它小,右子树的 key 都比它大,所以 +对于带查找的节点 search_key,从根节点开始,如果 search_key 大于当前 key,就去右子树查找,否则去左子树查找。 一直到当前节点是 None 了说明没找到对应 key。 + +![](./bst_search.png) + +好,撸代码: + +```py + def _bst_search(self, subtree, key): + if subtree is None: # 没找到 + return None + elif key < subtree.key: + return self._bst_search(subtree.left, key) + elif key > subtree.key: + return self._bst_search(subtree.right, key) + else: + return subtree + + def get(self, key, default=None): + node = self._bst_search(self.root, key) + if node is None: + return default + else: + return node.value +``` + + +## 获取最大和最小 key 的节点 + +其实还按照其定义,最小值就一直向着左子树找,最大值一直向右子树找,递归查找就行。 + +```py + def _bst_min_node(self, subtree): + if subtree is None: + return None + elif subtree.left is None: # 找到左子树的头 + return subtree + else: + return self._bst_min_node(subtree.left) + + def bst_min(self): + node = self._bst_min_node(self.root) + return node.value if node else None +``` + +## 插入 +插入节点的时候我们需要一直保持 BST 的性质,每次插入一个节点,我们都通过递归比较把它放到正确的位置。 +你会发现新节点总是被作为叶子结点插入。(请你思考这是为什么) + +![](./bst_insert.png) + +```py + def _bst_insert(self, subtree, key, value): + """ 插入并且返回根节点 + + :param subtree: + :param key: + :param value: + """ + if subtree is None: # 插入的节点一定是根节点,包括 root 为空的情况 + subtree = BSTNode(key, value) + elif key < subtree.key: + subtree.left = self._bst_insert(subtree.left, key, value) + elif key > subtree.key: + subtree.right = self._bst_insert(subtree.right, key, value) + return subtree + + def add(self, key, value): + node = self._bst_search(self.root, key) + if node is not None: # 更新已经存在的 key + node.value = value + return False + else: + self.root = self._bst_insert(self.root, key, value) + self.size += 1 + return True +``` + +## 删除节点 +删除操作相比上边的操作要麻烦很多,首先需要定位一个节点,删除节点后,我们需要始终保持 BST 的性质。 +删除一个节点涉及到三种情况: + +- 节点是叶节点 +- 节点有一个孩子 +- 节点有两个孩子 + +我们分别来看看三种情况下如何删除一个节点: + +#### 删除叶节点 +这是最简单的一种情况,只需要把它的父亲指向它的指针设置为 None 就好。 + +![](./bst_remove_leaf.png) + +#### 删除只有一个孩子的节点 +删除有一个孩子的节点时,我们拿掉需要删除的节点,之后把它的父亲指向它的孩子就行,因为根据 BST +左子树都小于节点,右子树都大于节点的特性,删除它之后这个条件依旧满足。 + +![](./bst_remove_node_with_one_child.png) + +#### 删除有两个孩子的内部节点 +假如我们想删除 12 这个节点改怎么做呢?你的第一反应可能是按照下图的方式: + +![](./remove_interior_replace.png) + +但是这种方式可能会影响树的高度,降低查找的效率。这里我们用另一种非常巧妙的方式。 +还记得上边提到的吗,如果你中序遍历 BST 并且输出每个节点的 key,你会发现就是一个有序的数组。 +`[1 4 12 23 29 37 41 60 71 84 90 100]`。这里我们定义两个概念,逻辑前任(predecessor)和后继(successor),请看下图: + +![](./predecessor_successor.png) + +12 在中序遍历中的逻辑前任和后继分别是 4 和 23 节点。于是我们还有一种方法来删除 12 这个节点: + +- 找到待删除节点 N(12) 的后继节点 S(23) +- 复制节点 S 到节点 N +- 从 N 的右子树中删除节点 S,并更新其删除后继节点后的右子树 + +说白了就是找到后继并且替换,这里之所以能保证这种方法是正确的,你会发现替换后依旧是保持了 BST 的性质。 +有个问题是如何找到后继节点呢?待删除节点的右子树的最小的节点不就是后继嘛,上边我们已经实现了找到最小 key 的方法了。 + + +![](./find_successor.png) + +我们开始编写代码实现,和之前的操作类似,我们还是通过辅助函数的形式来实现,这个递归函数会比较复杂,请你仔细理解: + +```py + def _bst_remove(self, subtree, key): + """删除节点并返回根节点""" + if subtree is None: + return None + elif key < subtree.key: + subtree.left = self._bst_remove(subtree.left, key) + return subtree + elif key > subtree.key: + subtree.right = self._bst_remove(subtree.right, key) + return subtree + else: # 找到了需要删除的节点 + if subtree.left is None and subtree.right is None: # 叶节点,返回 None 把其父亲指向它的指针置为 None + return None + elif subtree.left is None or subtree.right is None: # 只有一个孩子 + if subtree.left is not None: + return subtree.left # 返回它的孩子并让它的父亲指过去 + else: + return subtree.right + else: # 俩孩子,寻找后继节点替换,并从待删节点的右子树中删除后继节点 + successor_node = self._bst_min_node(subtree.right) + subtree.key, subtree.value = successor_node.key, successor_node.value + subtree.right = self._bst_remove(subtree.right, successor_node.key) + return subtree + + def remove(self, key): + assert key in self + self.size -= 1 + return self._bst_remove(self.root, key) +``` + +完整代码你可以在本章的 bst.py 找到。 +另外推荐一个可以在线演示过程的网址大家可以手动执行下看看效果: https://www.cs.usfca.edu/~galles/visualization/BST.html + +# 时间复杂度分析 + +上边介绍的操作时间复杂度和二叉树的形状有关。平均来说时间复杂度是和树的高度成正比的,树的高度 h 是 log(n), +但是最坏情况下以上操作的时间复杂度都是 O(n)。为了改善 BST 有很多变种,感兴趣请参考延伸阅读中的内容。 + +![](./bst_worstcase.png) + + +# 练习题: +- 请你实现查找 BST 最大值的函数 + + +# 延伸阅读 +- 《Data Structures and Algorithms in Python》14 章,树的概念和算法还有很多,我们这里介绍最基本的帮你打个基础 +- 了解红黑树。普通二叉查找树有个很大的问题就是难以保证树的平衡,极端情况下某些节点可能会非常深,导致查找复杂度大幅退化。而平衡二叉树就是为了解决这个问题。请搜索对应资料了解下。 +- 了解 mysql 索引使用的 B-Tree 结构(多路平衡查找树),这个是后端面试数据库的常考点。想想为什么?当元素非常多的时候,二叉树的深度会很深,导致多次磁盘查找。[从B树、B+树、B*树谈到R 树](https://blog.csdn.net/v_JULY_v/article/details/6530142) + + +# Leetcode + +验证是否是合法二叉搜索树 [validate-binary-search-tree](https://leetcode.com/problems/validate-binary-search-tree/ diff --git a/docs/17/bst.png 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zlNA&?pC7xCJ~0Sb_Omb^2CXjsvA%%;a_ylZsNr(Vu6qhF>nLzY$N@9u;d)jU24F8o zfTag?sf6LT$R#4+5y%g~LRs>kwgLm_zC?ut9^g1r&&KmgOMx0Wu-Mv?-cu{432eYQ f2pHHf{b!$;9eMQowIvTgS8{r~`njxgN@xNAN_jeX literal 0 HcmV?d00001 diff --git a/docs/17/bst.py b/docs/17/bst.py new file mode 100644 index 0000000..e1ceef3 --- /dev/null +++ b/docs/17/bst.py @@ -0,0 +1,155 @@ +# -*- coding: utf-8 -*- + + +class BSTNode(object): + def __init__(self, key, value, left=None, right=None): + self.key, self.value, self.left, self.right = key, value, left, right + + +class BST(object): + def __init__(self, root=None): + self.root = root + + @classmethod + def build_from(cls, node_list): + cls.size = 0 + key_to_node_dict = {} + for node_dict in node_list: + key = node_dict['key'] + key_to_node_dict[key] = BSTNode(key, value=key) # 这里值暂时用 和 key一样的 + + for node_dict in node_list: + key = node_dict['key'] + node = key_to_node_dict[key] + if node_dict['is_root']: + root = node + node.left = key_to_node_dict.get(node_dict['left']) + node.right = key_to_node_dict.get(node_dict['right']) + cls.size += 1 + return cls(root) + + def _bst_search(self, subtree, key): + if subtree is None: # 没找到 + return None + elif key < subtree.key: + return self._bst_search(subtree.left, key) + elif key > subtree.key: + return self._bst_search(subtree.right, key) + else: + return subtree + + def __contains__(self, key): + """实现 in 操作符""" + return self._bst_search(self.root, key) is not None + + def get(self, key, default=None): + node = self._bst_search(self.root, key) + if node is None: + return default + else: + return node.value + + def _bst_min_node(self, subtree): + if subtree is None: + return None + elif subtree.left is None: # 找到左子树的头 + return subtree + else: + return self._bst_min_node(subtree.left) + + def bst_min(self): + node = self._bst_min_node(self.root) + return node.value if node else None + + def _bst_insert(self, subtree, key, value): + """ 插入并且返回根节点 + + :param subtree: + :param key: + :param value: + """ + if subtree is None: # 插入的节点一定是根节点,包括 root 为空的情况 + subtree = BSTNode(key, value) + elif key < subtree.key: + subtree.left = self._bst_insert(subtree.left, key, value) + elif key > subtree.key: + subtree.right = self._bst_insert(subtree.right, key, value) + return subtree + + def add(self, key, value): + node = self._bst_search(self.root, key) + if node is not None: # 更新已经存在的 key + node.value = value + return False + else: + self.root = self._bst_insert(self.root, key, value) + self.size += 1 + return True + + def _bst_remove(self, subtree, key): + """删除节点并返回根节点""" + if subtree is None: + return None + elif key < subtree.key: + subtree.left = self._bst_remove(subtree.left, key) + return subtree + elif key > subtree.key: + subtree.right = self._bst_remove(subtree.right, key) + return subtree + else: # 找到了需要删除的节点 + if subtree.left is None and subtree.right is None: # 叶节点,返回 None 把其父亲指向它的指针置为 None + return None + elif subtree.left is None or subtree.right is None: # 只有一个孩子 + if subtree.left is not None: + return subtree.left # 返回它的孩子并让它的父亲指过去 + else: + return subtree.right + else: # 俩孩子,寻找后继节点替换,并删除其右子树的后继节点,同时更新其右子树 + successor_node = self._bst_min_node(subtree.right) + subtree.key, subtree.value = successor_node.key, successor_node.value + subtree.right = self._bst_remove(subtree.right, successor_node.key) + return subtree + + def remove(self, key): + assert key in self + self.size -= 1 + return self._bst_remove(self.root, key) + + +NODE_LIST = [ + {'key': 60, 'left': 12, 'right': 90, 'is_root': True}, + {'key': 12, 'left': 4, 'right': 41, 'is_root': False}, + {'key': 4, 'left': 1, 'right': None, 'is_root': False}, + {'key': 1, 'left': None, 'right': None, 'is_root': False}, + {'key': 41, 'left': 29, 'right': None, 'is_root': False}, + {'key': 29, 'left': 23, 'right': 37, 'is_root': False}, + {'key': 23, 'left': None, 'right': None, 'is_root': False}, + {'key': 37, 'left': None, 'right': None, 'is_root': False}, + {'key': 90, 'left': 71, 'right': 100, 'is_root': False}, + {'key': 71, 'left': None, 'right': 84, 'is_root': False}, + {'key': 100, 'left': None, 'right': None, 'is_root': False}, + {'key': 84, 'left': None, 'right': None, 'is_root': False}, +] + + +def test_bst_tree(): + bst = BST.build_from(NODE_LIST) + for node_dict in NODE_LIST: + key = node_dict['key'] + assert bst.get(key) == key + assert bst.size == len(NODE_LIST) + assert bst.get(-1) is None # 单例的 None 我们用 is 来比较 + + assert bst.bst_min() == 1 + + bst.add(0, 0) + assert bst.bst_min() == 0 + + bst.remove(12) + assert bst.get(12) is None + + bst.remove(1) + assert bst.get(1) is None + + bst.remove(29) + assert bst.get(29) is None diff --git a/docs/17/bst_insert.png b/docs/17/bst_insert.png new file mode 100644 index 0000000000000000000000000000000000000000..5d69b37c3930e1cc2364461efb4f2406aa8d3015 GIT binary patch literal 66789 zcmeFXV{~QT@-7@(9jCivn;qM>ZQC|Fwy`^Q$F@2h+qP}ro!|N6obitFez{-o{@rSusSouW%qBAczv;!ipduV1*zcpaC#Yz&m<~e`7&Fun8@MgybcJ zgb3sv?MyAKO+Y}@Ly|q9mC@&fWb 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zMz2!z93uE%IW8YtC^fGI*7M=*y!om5)SSu8b?E_7BQ|qujNW5Yd6V&{FG~)C@UtoJ zWdDoF?0~Bp!bMLX?E7>E*e`Wu6kv~o z!@uDF|5U`g8JX9-G>jA zFWdXqGXEX_;cSiY1C!#8vR*p>zgLri-<1RZW?}K)U;3{l7t{Qg)hjEv&bckk@CfiS MyljFl*K>&cKf&_A2LJ#7 literal 0 HcmV?d00001 diff --git a/docs/19/builtins.md b/docs/19/builtins.md new file mode 100644 index 0000000..40d838b --- /dev/null +++ b/docs/19/builtins.md @@ -0,0 +1,23 @@ +# Python 常用内置算法和数据结构 +相信到这里大家对常用的数据结构和算法及其实现都比较熟悉了。 +之前在每章的数据结构和算法中涉及到的章节我都会提到对应的 python 内置模块,一般如果内置的可以满足需求,我们优先使用内置模块, +因为在性能和容错性方面内置模块要好于我们自己实现(比如有些是 c 实现的)。本章我们不会再对每个模块的原理详细说明,仅列举出一些常见模块供大家参考, +如果有需要最好的学习方式就是参考 Python 的官方文档。很多高级的数据结构我们也可以通过 google 搜索现成的库拿来直接用。 + +- 常用内置数据类型:list, tuple, dict, set, frozenset +- collections +- heapq +- bisect + +下边我列了一个常用的表格,如果有遗漏可以在 issue 中提出。确保你了解这些数据结构和算法的使用以及时间、空间复杂度。 + +| 数据结构/算法 | 语言内置 | 内置库 | +|----------------|---------------------------------|---------------------------------------------------------------| +| 线性结构 | list(列表)/tuple(元祖) | array(数组,不常用)/collections.namedtuple | +| 链式结构 | | collections.deque(双端队列) | +| 字典结构 | dict(字典) | collections.Counter(计数器)/OrderedDict(有序字典)/defaultdict | +| 集合结构 | set(集合)/frozenset(不可变集合) | | +| 排序算法 | sorted | | +| 二分算法 | | bisect模块 | +| 堆算法 | | heapq模块 | +| 缓存算法 | | functools.lru_cache(Least Recent Used, python3) | diff --git a/docs/20/interview.md b/docs/20/interview.md new file mode 100644 index 0000000..5958458 --- /dev/null +++ b/docs/20/interview.md @@ -0,0 +1,43 @@ +# 注意事项 + +- 电子简历尽量用 pdf 格式,方便跨平台打开。doc 等格式在不同的电脑上打开会有排版问题,很多后端技术面试官可能使用的是 mac 或者 linux。 +- 提前复习回顾重点知识,防止卡在基础上。比如 mac 下著名的 brew 工具作者面试 google 就因为没写出来反转二叉树被拒,后来去了苹果😂.(这就只能看人品和运气和眼缘了,如果没见到二面面试官或者 hr,大概率是挂了)。(树、链表、哈希表、二分、快排、TCP/UDP、HTTP、数据库ACID、索引优化等常考点)。 +- 白板编程,练习在纸上手写代码。虽然很多求职者都很抵触手写代码,但是白板编程确实是一种比较好的区分方式。你的思考过程、编码习惯、编码规范等都能看出来。 +- 如果被问到工程里不会使用但是比较刁钻的算法题,建议你和面试官沟通的时候问问这个算法或者题目在开发中有哪些实际使用场景,看看对方怎么说😎。少数公司喜欢考一些算法竞赛题,这种对于没有ACM,信息学竞赛背景的开发者来说比较吃力。大部分业务开发岗位应该只会考察基础算法题 +- 面试的时候准备充分,简历要与招聘方需求对等,甚至可以针对不同公司准备不同的简历内容。笔者每次面试都会带上白纸、笔、简历、电脑等,即使面试没过,至少也让面试官感觉我是有诚意的,给对方留下好印象。 +- 加分项:github、个人技术博客、开源项目、技术论坛帐号等,让面试官有更多渠道了解你,有时候仅仅根据几十分钟的面试来评判面试者是有失偏颇的。(比如面试者临场发挥不好;面试官个人偏好;会的都不问,问的都不会等) + + +# 白板编程 +其实我个人是反对出纯算法题目的,尤其是有些比较刁钻的直接出算法竞赛题,这对与很多做工程的同学来说是比较吃亏的。没事的时候可以去 LeetCode 之类的网站刷刷基础题。 +一般来说 web 业务开发者掌握常见的编程语言内置算法和数据结构就够用了。 + +- 练习手写常见的算法,比如快排,二分,归并等,记住常见排序算法时间复杂度 +- 逻辑正确是前提 +- 有图示描述思路最好,如果时间紧代码没写出来,可以直接描述自己的思路。 +- 字不要写太大,尽量工整。每行代码之间留有一定的空隙,方便你修改(甚至笔者之前会带上铅笔和橡皮手写代码) +- 如果实在写不出来可以和面试官交流,很多时候如果给不出最优方案尽量想一个次优方案,别上来就说不会 +- 想不起来的函数名写伪代码,一般面试官不会强制说让你记住每个 api 的名字 +- 如果有多余的时间(一般不会有)注意一些边界条件,防御性编程、代码风格、单元测试等东西,想好一些异常情况(空值、边界值等)的测试用例 + +# 手写代码注意事项 + +这里我就直接引用《剑指offer》里内容,大家写代码的时候可以多加注意,对于应对算法面试,如果准备时间比较多,推荐看下这本书,并且刷一下 +leetcode 上的基础题目练练手感。 + +- 规范性:书写清晰、布局清晰、命令合理 +- 完整性:完成基本功能,考虑边界条件,做好错误处理 +- 鲁棒性:防御式编程,处理无效输入 + + +# 结语 +这套教程列举的算法很有限,包括图算法、贪心,动态规划,分布式,机器学习算法等很多没有涉及到,因为它们确实需要读者更深入的理论基础,而且这套教程的目的也不是针对算法竞赛。 +不过了解了本教程涉及到的大部分算法是可以应付绝大多数的业务开发的。如果读者对算法有兴趣,本教程引用的几本参考书都可以去深入学习。希望本教程能对你学习算法、养成良好的思维方式和编码习惯等有所帮助。 + +# 延伸阅读 +目前市面上有一些专门针对算法面试的书供大家参考,如果你正在准备算法面试,我强烈建议你看看下面的参考资料学习解题技巧: + +- [那些年,我们一起跪过的算法题[视频]](https://zhuanlan.zhihu.com/p/35175401) +- [《程序员面试金典(第5版)》](https://book.douban.com/subject/25753386/) +- [《剑指Offer》](https://book.douban.com/subject/25910559/) +- [python leetCode](https://github.com/HuberTRoy/leetCodek) diff --git a/mkdocs.yml b/mkdocs.yml index 94ca6f8..163cc9c 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -7,29 +7,30 @@ markdown_extensions: - mdx_math: enable_dollar_delimiter: True #for use of inline $..$ -pages: +nav: - 课程简介: 'index.md' - - 00_课程简介之笨方法学算法: '00_课程简介之笨方法学算法/why_and_how_to_learn.md' - - 01_抽象数据类型和面向对象编程: '01_抽象数据类型和面向对象编程/ADT_OOP.md' - - 02_数组和列表: '02_数组和列表/array_and_list.md' - - 03_链表: '03_链表/linked_list.md' - - 04_队列: '04_队列/queue.md' - - 05_栈: '05_栈/stack.md' - - 06_算法分析: '06_算法分析/big_o.md' - - 07_哈希表: '07_哈希表/hashtable.md' - - 08_字典: '08_字典/dict.md' - - 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zmyd{E?6EbM(!Eol?u4|w~O6yeZUync0$kLyuEO%vj+skpYtl|8h z$NV#2@Z=9i6k}Tjb>AM6&2#SQDzgSYfoO9s3FCz1E}7Xi^OqYCz&zw|5NGqVUxhP1 z6zQktcwT)LGCl{q3PN0c_v#*$Pegu_PG|rpgLF7BS@Hx9&p0>zesDv(SCoQ7tSH!| z>NM}vde~f`Nm3&R943i^s92k(AU&axIb5U!Bzo2>brW2cY;s!joR2K|XwkHWD^TPU z(=pD9q~rK#7WZ^^JPe;8pZ5C-nu%a!c$4Juu<4xacp;rQwr!>viD%{<>DU4=RtZ00 zyt}YgiOpm`#%-o3ntTFm6LuQ;N%ki`7D7p07T8Rx^|)a0<#sFo^icC03$7DyY*n%s zG^pkPU-!rM=9S@7W3>H$azO4z3&!A!DnWYW6l+T~83d9-QN*8~r=Sj(vso;70yyDa z7AE!&T4wK>%&A7<(eB3MIYegcZN_*fJMY#bc1bz zC<$&xBdTfb=(Y{q8WDgzjlmunIH%DIr#QAAP+dT}{piLdREamvDdy??ufEJjE*&hr zDg_3ly|5by8V;{z&^C)3M8k{t>KCn%Wk%$6+zjVT0D^hJ|}wyrO-!;xCN z?Fqu|DTUtTm{b$1@cjp23{7zfE#8AWb7lD2n3gV6b3C7rSP|kLqL?kuq4Q+QR8vqR zz_#=&vVuZ|Q-t48q)25NLeI=E9AGV?GZFQKb=ebZN*3V%s%VHwbmGnBOS4{n>_N= self.maxsize: + raise Exception('full') + self._elements[self._count] = value + self._count += 1 + self._siftup(self._count-1) # 维持堆的特性 + + def _siftup(self, ndx): + if ndx > 0: + parent = int((ndx-1)/2) + if self._elements[ndx] > self._elements[parent]: # 如果插入的值大于 parent,一直交换 + self._elements[ndx], self._elements[parent] = self._elements[parent], self._elements[ndx] + self._siftup(parent) # 递归 + + def extract(self): + if self._count <= 0: + raise Exception('empty') + value = self._elements[0] # 保存 root 值 + self._count -= 1 + self._elements[0] = self._elements[self._count] # 最右下的节点放到root后siftDown + self._siftdown(0) # 维持堆特性 + return value + + def _siftdown(self, ndx): + left = 2 * ndx + 1 + right = 2 * ndx + 2 + # determine which node contains the larger value + largest = ndx + if (left < self._count and # 有左孩子 + self._elements[left] >= self._elements[largest] and + self._elements[left] >= self._elements[right]): # 原书这个地方没写实际上找的未必是largest + largest = left + elif right < self._count and self._elements[right] >= self._elements[largest]: + largest = right + if largest != ndx: + self._elements[ndx], self._elements[largest] = self._elements[largest], self._elements[ndx] + self._siftdown(largest) + + +def test_maxheap(): + import random + n = 5 + h = MaxHeap(n) + for i in range(n): + h.add(i) + for i in reversed(range(n)): + assert i == h.extract() +``` + +# 实现堆排序 +上边我们实现了最大堆,每次我们都能 extract 一个最大的元素了,于是一个倒序排序函数就能很容易写出来了: + +```py +def heapsort_reverse(array): + length = len(array) + maxheap = MaxHeap(length) + for i in array: + maxheap.add(i) + res = [] + for i in range(length): + res.append(maxheap.extract()) + return res + + +def test_heapsort_reverse(): + import random + l = list(range(10)) + random.shuffle(l) + assert heapsort_reverse(l) == sorted(l, reverse=True) +``` + +# Python 里的 heapq 模块 +python 其实自带了 heapq 模块,用来实现堆的相关操作,原理是类似的。请你阅读相关文档并使用内置的 heapq 模块完成堆排序。 +一般我们刷题或者写业务代码的时候,使用这个内置的 heapq 模块就够用了,内置的实现了是最小堆。 + + +# Top K 问题 +面试题中有这样一类问题,让求出大量数据中的top k 个元素,比如一亿个数字中最大的100个数字。 +对于这种问题有很多种解法,比如直接排序、mapreduce、trie 树、分治法等,当然如果内存够用直接排序是最简单的。 +如果内存不够用呢? 这里我们提一下使用固定大小的堆来解决这个问题的方式。 + +一开始的思路可能是,既然求最大的 k 个数,是不是应该维护一个包含 k 个元素的最大堆呢? +稍微尝试下你会发现走不通。我们先用数组的前面 k 个元素建立最大堆,然后对剩下的元素进行比对,但是最大堆只能每次获取堆顶 +最大的一个元素,如果我们取下一个大于堆顶的值和堆顶替换,你会发现堆底部的小数一直不会被换掉。如果下一个元素小于堆顶 +就替换也不对,这样可能最大的元素就被我们丢掉了。 + +相反我们用最小堆呢? +先迭代前 k 个元素建立一个最小堆,之后的元素如果小于堆顶最小值,跳过,否则替换堆顶元素并重新调整堆。你会发现最小堆里 +慢慢就被替换成了最大的那些值,并且最后堆顶是最大的 topk 个值中的最小值。 +(比如1000个数找10个,最后堆里剩余的是 [990, 991, 992, 996, 994, 993, 997, 998, 999, 995],第一个 990 最小) + +按照这个思路很容易写出来代码: + +```py +import heapq + + +class TopK: + """获取大量元素 topk 大个元素,固定内存 + 思路: + 1. 先放入元素前 k 个建立一个最小堆 + 2. 迭代剩余元素: + 如果当前元素小于堆顶元素,跳过该元素(肯定不是前 k 大) + 否则替换堆顶元素为当前元素,并重新调整堆 + """ + + def __init__(self, iterable, k): + self.minheap = [] + self.capacity = k + self.iterable = iterable + + def push(self, val): + if len(self.minheap) >= self.capacity: + min_val = self.minheap[0] + if val < min_val: # 当然你可以直接 if val > min_val操作,这里我只是显示指出跳过这个元素 + pass + else: + heapq.heapreplace(self.minheap, val) # 返回并且pop堆顶最小值,推入新的 val 值并调整堆 + else: + heapq.heappush(self.minheap, val) # 前面 k 个元素直接放入minheap + + def get_topk(self): + for val in self.iterable: + self.push(val) + return self.minheap + + +def test(): + import random + i = list(range(1000)) # 这里可以是一个可迭代元素,节省内存 + random.shuffle(i) + _ = TopK(i, 10) + print(_.get_topk()) # [990, 991, 992, 996, 994, 993, 997, 998, 999, 995] + + +if __name__ == '__main__': + test() +``` + + +# 练习题 + +- 这里我用最大堆实现了一个 heapsort_reverse 函数,请你实现一个正序排序的函数。似乎不止一种方式 +- 请你实现一个最小堆,你需要修改那些代码呢? +- 我们实现的堆排序是 inplace 的吗,如果不是,你能改成 inplace 的吗? +- 堆排序的时间复杂度是多少? siftup 和 siftdown 的时间复杂度是多少?(小提示:考虑树的高度,它决定了操作次数) +- 请你思考 Top K 问题的时间复杂度是多少? + + +# 延伸阅读 +- 《算法导论》第 6 章 Heapsort +- 《Data Structures and Algorithms in Python》 13.5 节 Heapsort +- 阅读 Python heapq 模块的文档 + +# Leetcode + +合并 k 个有序链表 https://leetcode.com/problems/merge-k-sorted-lists/description/ diff --git a/docs/15/heap_and_heapsort.py b/docs/15/heap_and_heapsort.py new file mode 100644 index 0000000..252008d --- /dev/null +++ b/docs/15/heap_and_heapsort.py @@ -0,0 +1,140 @@ +# -*- coding:utf-8 -*- + +# 第二章拷贝的 Array 代码 + + +class Array(object): + + def __init__(self, size=32): + self._size = size + self._items = [None] * size + + def __getitem__(self, index): + return self._items[index] + + def __setitem__(self, index, value): + self._items[index] = value + + def __len__(self): + return self._size + + def clear(self, value=None): + for i in range(len(self._items)): + self._items[i] = value + + def __iter__(self): + for item in self._items: + yield item + +##################################################### +# heap 实现 +##################################################### + + +class MaxHeap(object): + """ + Heaps: + 完全二叉树,最大堆的非叶子节点的值都比孩子大,最小堆的非叶子结点的值都比孩子小 + Heap包含两个属性,order property 和 shape property(a complete binary tree),在插入 + 一个新节点的时候,始终要保持这两个属性 + 插入操作:保持堆属性和完全二叉树属性, sift-up 操作维持堆属性 + extract操作:只获取根节点数据,并把树最底层最右节点copy到根节点后,sift-down操作维持堆属性 + + 用数组实现heap,从根节点开始,从上往下从左到右给每个节点编号,则根据完全二叉树的 + 性质,给定一个节点i, 其父亲和孩子节点的编号分别是: + parent = (i-1) // 2 + left = 2 * i + 1 + rgiht = 2 * i + 2 + 使用数组实现堆一方面效率更高,节省树节点的内存占用,一方面还可以避免复杂的指针操作,减少 + 调试难度。 + + """ + + def __init__(self, maxsize=None): + self.maxsize = maxsize + self._elements = Array(maxsize) + self._count = 0 + + def __len__(self): + return self._count + + def add(self, value): + if self._count >= self.maxsize: + raise Exception('full') + self._elements[self._count] = value + self._count += 1 + self._siftup(self._count-1) # 维持堆的特性 + + def _siftup(self, ndx): + if ndx > 0: + parent = int((ndx-1)/2) + if self._elements[ndx] > self._elements[parent]: # 如果插入的值大于 parent,一直交换 + self._elements[ndx], self._elements[parent] = self._elements[parent], self._elements[ndx] + self._siftup(parent) # 递归 + + def extract(self): + if self._count <= 0: + raise Exception('empty') + value = self._elements[0] # 保存 root 值 + self._count -= 1 + self._elements[0] = self._elements[self._count] # 最右下的节点放到root后siftDown + self._siftdown(0) # 维持堆特性 + return value + + def _siftdown(self, ndx): + left = 2 * ndx + 1 + right = 2 * ndx + 2 + # determine which node contains the larger value + largest = ndx + if (left < self._count and # 有左孩子 + self._elements[left] >= self._elements[largest] and + self._elements[left] >= self._elements[right]): # 原书这个地方没写实际上找的未必是largest + largest = left + elif right < self._count and self._elements[right] >= self._elements[largest]: + largest = right + if largest != ndx: + self._elements[ndx], self._elements[largest] = self._elements[largest], self._elements[ndx] + self._siftdown(largest) + + +def test_maxheap(): + import random + n = 5 + h = MaxHeap(n) + for i in range(n): + h.add(i) + for i in reversed(range(n)): + assert i == h.extract() + + +def heapsort_reverse(array): + length = len(array) + maxheap = MaxHeap(length) + for i in array: + maxheap.add(i) + res = [] + for i in range(length): + res.append(maxheap.extract()) + return res + + +def test_heapsort_reverse(): + import random + l = list(range(10)) + random.shuffle(l) + assert heapsort_reverse(l) == sorted(l, reverse=True) + + +def heapsort_use_heapq(iterable): + from heapq import heappush, heappop + items = 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zmyd{E?6EbM(!Eol?u4|w~O6yeZUync0$kLyuEO%vj+skpYtl|8h z$NV#2@Z=9i6k}Tjb>AM6&2#SQDzgSYfoO9s3FCz1E}7Xi^OqYCz&zw|5NGqVUxhP1 z6zQktcwT)LGCl{q3PN0c_v#*$Pegu_PG|rpgLF7BS@Hx9&p0>zesDv(SCoQ7tSH!| z>NM}vde~f`Nm3&R943i^s92k(AU&axIb5U!Bzo2>brW2cY;s!joR2K|XwkHWD^TPU z(=pD9q~rK#7WZ^^JPe;8pZ5C-nu%a!c$4Juu<4xacp;rQwr!>viD%{<>DU4=RtZ00 zyt}YgiOpm`#%-o3ntTFm6LuQ;N%ki`7D7p07T8Rx^|)a0<#sFo^icC03$7DyY*n%s zG^pkPU-!rM=9S@7W3>H$azO4z3&!A!DnWYW6l+T~83d9-QN*8~r=Sj(vso;70yyDa z7AE!&T4wK>%&A7<(eB3MIYegcZN_*fJMY#bc1bz zC<$&xBdTfb=(Y{q8WDgzjlmunIH%DIr#QAAP+dT}{piLdREamvDdy??ufEJjE*&hr zDg_3ly|5by8V;{z&^C)3M8k{t>KCn%Wk%$6+zjVT0D^hJ|}wyrO-!;xCN z?Fqu|DTUtTm{b$1@cjp23{7zfE#8AWb7lD2n3gV6b3C7rSP|kLqL?kuq4Q+QR8vqR zz_#=&vVuZ|Q-t48q)25NLeI=E9AGV?GZFQKb=ebZN*3V%s%VHwbmGnBOS4{n>_N= self.maxsize: - raise Exception('full') - self._elements[self._count] = value - self._count += 1 - self._siftup(self._count-1) # 维持堆的特性 - - def _siftup(self, ndx): - if ndx > 0: - parent = int((ndx-1)/2) - if self._elements[ndx] > self._elements[parent]: # 如果插入的值大于 parent,一直交换 - self._elements[ndx], self._elements[parent] = self._elements[parent], self._elements[ndx] - self._siftup(parent) # 递归 - - def extract(self): - if self._count <= 0: - raise Exception('empty') - value = self._elements[0] # 保存 root 值 - self._count -= 1 - self._elements[0] = self._elements[self._count] # 最右下的节点放到root后siftDown - self._siftdown(0) # 维持堆特性 - return value - - def _siftdown(self, ndx): - left = 2 * ndx + 1 - right = 2 * ndx + 2 - # determine which node contains the larger value - largest = ndx - if (left < self._count and # 有左孩子 - self._elements[left] >= self._elements[largest] and - self._elements[left] >= self._elements[right]): # 原书这个地方没写实际上找的未必是largest - largest = left - elif right < self._count and self._elements[right] >= self._elements[largest]: - largest = right - if largest != ndx: - self._elements[ndx], self._elements[largest] = self._elements[largest], self._elements[ndx] - self._siftdown(largest) - - -def test_maxheap(): - import random - n = 5 - h = MaxHeap(n) - for i in range(n): - h.add(i) - for i in reversed(range(n)): - assert i == h.extract() -``` - -# 实现堆排序 -上边我们实现了最大堆,每次我们都能 extract 一个最大的元素了,于是一个倒序排序函数就能很容易写出来了: - -```py -def heapsort_reverse(array): - length = len(array) - maxheap = MaxHeap(length) - for i in array: - maxheap.add(i) - res = [] - for i in range(length): - res.append(maxheap.extract()) - return res - - -def test_heapsort_reverse(): - import random - l = list(range(10)) - random.shuffle(l) - assert heapsort_reverse(l) == sorted(l, reverse=True) -``` - -# Python 里的 heapq 模块 -python 其实自带了 heapq 模块,用来实现堆的相关操作,原理是类似的。请你阅读相关文档并使用内置的 heapq 模块完成堆排序。 -一般我们刷题或者写业务代码的时候,使用这个内置的 heapq 模块就够用了,内置的实现了是最小堆。 - - -# Top K 问题 -面试题中有这样一类问题,让求出大量数据中的top k 个元素,比如一亿个数字中最大的100个数字。 -对于这种问题有很多种解法,比如直接排序、mapreduce、trie 树、分治法等,当然如果内存够用直接排序是最简单的。 -如果内存不够用呢? 这里我们提一下使用固定大小的堆来解决这个问题的方式。 - -一开始的思路可能是,既然求最大的 k 个数,是不是应该维护一个包含 k 个元素的最大堆呢? -稍微尝试下你会发现走不通。我们先用数组的前面 k 个元素建立最大堆,然后对剩下的元素进行比对,但是最大堆只能每次获取堆顶 -最大的一个元素,如果我们取下一个大于堆顶的值和堆顶替换,你会发现堆底部的小数一直不会被换掉。如果下一个元素小于堆顶 -就替换也不对,这样可能最大的元素就被我们丢掉了。 - -相反我们用最小堆呢? -先迭代前 k 个元素建立一个最小堆,之后的元素如果小于堆顶最小值,跳过,否则替换堆顶元素并重新调整堆。你会发现最小堆里 -慢慢就被替换成了最大的那些值,并且最后堆顶是最大的 topk 个值中的最小值。 -(比如1000个数找10个,最后堆里剩余的是 [990, 991, 992, 996, 994, 993, 997, 998, 999, 995],第一个 990 最小) - -按照这个思路很容易写出来代码: - -```py -import heapq - - -class TopK: - """获取大量元素 topk 大个元素,固定内存 - 思路: - 1. 先放入元素前 k 个建立一个最小堆 - 2. 迭代剩余元素: - 如果当前元素小于堆顶元素,跳过该元素(肯定不是前 k 大) - 否则替换堆顶元素为当前元素,并重新调整堆 - """ - - def __init__(self, iterable, k): - self.minheap = [] - self.capacity = k - self.iterable = iterable - - def push(self, val): - if len(self.minheap) >= self.capacity: - min_val = self.minheap[0] - if val < min_val: # 当然你可以直接 if val > min_val操作,这里我只是显示指出跳过这个元素 - pass - else: - heapq.heapreplace(self.minheap, val) # 返回并且pop堆顶最小值,推入新的 val 值并调整堆 - else: - heapq.heappush(self.minheap, val) # 前面 k 个元素直接放入minheap - - def get_topk(self): - for val in self.iterable: - self.push(val) - return self.minheap - - -def test(): - import random - i = list(range(1000)) # 这里可以是一个可迭代元素,节省内存 - random.shuffle(i) - _ = TopK(i, 10) - print(_.get_topk()) # [990, 991, 992, 996, 994, 993, 997, 998, 999, 995] - - -if __name__ == '__main__': - test() -``` - - -# 练习题 - -- 这里我用最大堆实现了一个 heapsort_reverse 函数,请你实现一个正序排序的函数。似乎不止一种方式 -- 请你实现一个最小堆,你需要修改那些代码呢? -- 我们实现的堆排序是 inplace 的吗,如果不是,你能改成 inplace 的吗? -- 堆排序的时间复杂度是多少? siftup 和 siftdown 的时间复杂度是多少?(小提示:考虑树的高度,它决定了操作次数) -- 请你思考 Top K 问题的时间复杂度是多少? - - -# 延伸阅读 -- 《算法导论》第 6 章 Heapsort -- 《Data Structures and Algorithms in Python》 13.5 节 Heapsort -- 阅读 Python heapq 模块的文档 - -# Leetcode - -合并 k 个有序链表 https://leetcode.com/problems/merge-k-sorted-lists/description/ diff --git "a/docs/15_\345\240\206\344\270\216\345\240\206\346\216\222\345\272\217/heap_and_heapsort.py" "b/docs/15_\345\240\206\344\270\216\345\240\206\346\216\222\345\272\217/heap_and_heapsort.py" deleted file mode 100644 index 252008d..0000000 --- "a/docs/15_\345\240\206\344\270\216\345\240\206\346\216\222\345\272\217/heap_and_heapsort.py" +++ /dev/null @@ -1,140 +0,0 @@ -# -*- coding:utf-8 -*- - -# 第二章拷贝的 Array 代码 - - -class Array(object): - - def __init__(self, size=32): - self._size = size - self._items = [None] * size - - def __getitem__(self, index): - return self._items[index] - - def __setitem__(self, index, value): - self._items[index] = value - - def __len__(self): - return self._size - - def clear(self, value=None): - for i in range(len(self._items)): - self._items[i] = value - - def __iter__(self): - for item in self._items: - yield item - -##################################################### -# heap 实现 -##################################################### - - -class MaxHeap(object): - """ - Heaps: - 完全二叉树,最大堆的非叶子节点的值都比孩子大,最小堆的非叶子结点的值都比孩子小 - Heap包含两个属性,order property 和 shape property(a complete binary tree),在插入 - 一个新节点的时候,始终要保持这两个属性 - 插入操作:保持堆属性和完全二叉树属性, sift-up 操作维持堆属性 - extract操作:只获取根节点数据,并把树最底层最右节点copy到根节点后,sift-down操作维持堆属性 - - 用数组实现heap,从根节点开始,从上往下从左到右给每个节点编号,则根据完全二叉树的 - 性质,给定一个节点i, 其父亲和孩子节点的编号分别是: - parent = (i-1) // 2 - left = 2 * i + 1 - rgiht = 2 * i + 2 - 使用数组实现堆一方面效率更高,节省树节点的内存占用,一方面还可以避免复杂的指针操作,减少 - 调试难度。 - - """ - - def __init__(self, maxsize=None): - self.maxsize = maxsize - self._elements = Array(maxsize) - self._count = 0 - - def __len__(self): - return self._count - - def add(self, value): - if self._count >= self.maxsize: - raise Exception('full') - self._elements[self._count] = value - self._count += 1 - self._siftup(self._count-1) # 维持堆的特性 - - def _siftup(self, ndx): - if ndx > 0: - parent = int((ndx-1)/2) - if self._elements[ndx] > self._elements[parent]: # 如果插入的值大于 parent,一直交换 - self._elements[ndx], self._elements[parent] = self._elements[parent], self._elements[ndx] - self._siftup(parent) # 递归 - - def extract(self): - if self._count <= 0: - raise Exception('empty') - value = self._elements[0] # 保存 root 值 - self._count -= 1 - self._elements[0] = self._elements[self._count] # 最右下的节点放到root后siftDown - self._siftdown(0) # 维持堆特性 - return value - - def _siftdown(self, ndx): - left = 2 * ndx + 1 - right = 2 * ndx + 2 - # determine which node contains the larger value - largest = ndx - if (left < self._count and # 有左孩子 - self._elements[left] >= self._elements[largest] and - self._elements[left] >= self._elements[right]): # 原书这个地方没写实际上找的未必是largest - largest = left - elif right < self._count and self._elements[right] >= self._elements[largest]: - largest = right - if largest != ndx: - self._elements[ndx], self._elements[largest] = self._elements[largest], self._elements[ndx] - self._siftdown(largest) - - -def test_maxheap(): - import random - n = 5 - h = MaxHeap(n) - for i in range(n): - h.add(i) - for i in reversed(range(n)): - assert i == h.extract() - - -def heapsort_reverse(array): - length = len(array) - maxheap = MaxHeap(length) - for i in array: - maxheap.add(i) - res = [] - for i in range(length): - 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value): - self._items[index] = value - - def __len__(self): - return self._size - - def clear(self, value=None): - for i in range(len(self._items)): - self._items[i] = value - - def __iter__(self): - for item in self._items: - yield item - - -class FullError(Exception): - pass - - -class ArrayQueue(object): - def __init__(self, maxsize): - self.maxsize = maxsize - self.array = Array(maxsize) - self.head = 0 - self.tail = 0 - - def push(self, value): - if len(self) >= self.maxsize: - raise FullError('queue full') - self.array[self.head % self.maxsize] = value - self.head += 1 - - def pop(self): - value = self.array[self.tail % self.maxsize] - self.tail += 1 - return value - - def __len__(self): - return self.head - self.tail - - -def test_queue(): - import pytest # pip install pytest - size = 5 - q = ArrayQueue(size) - for i in range(size): - q.push(i) - - with pytest.raises(FullError) as excinfo: # 我们来测试是否真的抛出了异常 - q.push(size) - assert 'full' in str(excinfo.value) - - assert len(q) == 5 - - assert q.pop() == 0 - assert q.pop() == 1 - - q.push(5) - - assert len(q) == 4 - - assert q.pop() == 2 - assert q.pop() == 3 - assert q.pop() == 4 - assert q.pop() == 5 - - assert len(q) == 0 diff --git "a/docs/04_\351\230\237\345\210\227/deque.py" "b/docs/04_\351\230\237\345\210\227/deque.py" deleted file mode 100644 index 0cc58f2..0000000 --- "a/docs/04_\351\230\237\345\210\227/deque.py" +++ /dev/null @@ -1 +0,0 @@ -# 留给读者练习,如果实在想不出,在第 5 章栈里 stack.py 会有实现 diff --git "a/docs/04_\351\230\237\345\210\227/queue.md" "b/docs/04_\351\230\237\345\210\227/queue.md" deleted file mode 100644 index bae29fb..0000000 --- "a/docs/04_\351\230\237\345\210\227/queue.md" +++ /dev/null @@ -1,88 +0,0 @@ -# 队列和栈 - -前面讲了线性和链式结构,如果你顺利掌握了,下边的队列和栈就小菜一碟了。因为我们会用前两章讲到的东西来实现队列和栈。 -之所以放到一起讲是因为这两个东西很类似,队列是先进先出结构(FIFO, first in first out), -栈是后进先出结构(LIFO, last in first out)。 - -生活中的数据结构: - -- 队列。没错就是咱平常排队,第一个来的第一个走 - -本章我们详细讲讲常用的队列 - -# 队列 Queue - -这里卖个关子,如果你熟悉了上两节讲的内容,这里你会选取哪个数据结构作为队列的底层存储? -还记得第一章讲的如何实现 ADT 吗?我视频了说了三个注意事项: - -- 1.如何选用恰当的数据结构作为存储? -- 2.选取的数据结构能否满足 ADT 的功能需求 -- 3.实现效率如何? - -我们先来看看 list 可以不?对照这个三个需求,看看能否满足: - -- 1.我们选择了 list -- 2.看起来队列需要从头删除,向尾部增加元素,也就是 list.pop(0) 和 list.append(element) -- 3.嗯,貌似 list.pop(0) 会导致所有其后所有元素向前移动一个位置,O(n)复杂度。append 平均倒是O(1),但是如果内存不够还要重新分配内存。 - -你看,使用了 list 的话频繁 pop(0) 是非常低效的。(当然list 实现还有另一种方式就是插入用 list.insert(0, item),删除用list.pop()) - -脑子再转转, 我们第二章实现了 链表 LinkedList,看看能否满足要求: - -- 1.这里选择 LinkedList -- 2.删除头元素 LinkedList.popleft(),追加 append(element)。都可以满足 -- 3.哇欧,这两个操作都是 O(1) 的,完美。 - -好, 就用 LinkedList 了,我们开始实现,具体看视频。这次实现我们还将演示自定义异常和测试异常。 - - -# 用数组实现队列 - -难道用数组就不能实现队列了吗?其实还是可以的。只不过数组是预先分配固定内存的,所以如果你知道了队列的最大长度,也是 -可以用数组来实现的。 - -想象一下,队列就俩操作,进进出出,一进一出,pop 和 push 操作。 -似乎只要两个下标 head, tail 就可以了。 当我们 push 的时候赋值并且前移 head,pop 的时候前移 tail 就可以了。你可以在纸上 -模拟下试试。列队的长度就是 head-pop,这个长度必须不能大于初始化的最大程度。 - -如果 head 先到了数组末尾咋办?重头来呗,只要我们保证 tail 不会超过 head 就行。 - -head = 0,1,2,3,4 ... 0,1,2,3,4 ... - -重头再来,循环往复,仿佛一个轮回。。。。 -怎么重头来呢?看上边数组的规律你如果还想不起来用取模,估计小学数学是体育老师教的。 - -```py -maxsize = 5 -for i in range(100): - print(i % maxsize) -``` - -![](./array_queue.png) - -我们来实现一个空间有限的循环队列。ArrayQueue,它的实现很简单,但是缺点是需要预先知道队列的长度来分配内存。 - - -# 双端队列 Double ended Queue -看了视频相信你已经会实现队列了,你可能还听过双端队列。上边讲到的队列 队头出,尾尾进,我们如果想头部和尾巴都能进能出呢? -这就是双端队列了,如果你用过 collections.deque 模块,就是这个东西。他能高效在两头操作。 - -假如让你实现你能想起来嘛? -似乎我们需要一个能 append() appendleft() popleft() pop() 都是 O(1) 的数据结构。 - -上边我们实现 队列的 LinkedList 可以吗?貌似就差一个 pop() 最后边的元素无法实现了。 -对,我们还有双端链表。它有这几个方法: - -- append -- appendleft -- headnode() -- tailnode() -- remove(node) # O(1) - -啊哈,似乎删除头尾都可以啦,而且都是 O(1) 的,完美。 -交给你一个艰巨的任务,实现双端队列 Deque() ADT。你可以参考前几章的任何代码,挑战一下这个任务,别忘记写单元测试呦。当然如果没想出来也没关系,后边我们实现栈的时候还会用到它,那里我们会实现这个代码。 - - -# 思考题 -- 你能用 python 的 deque 来实现 queue ADT 吗? -- 哪些经典算法里用到了队列呢? diff --git "a/docs/04_\351\230\237\345\210\227/queue.py" "b/docs/04_\351\230\237\345\210\227/queue.py" deleted file mode 100644 index 4ca6fed..0000000 --- "a/docs/04_\351\230\237\345\210\227/queue.py" +++ /dev/null @@ -1,182 +0,0 @@ -# -*- coding: utf-8 -*- - -from collections import deque - -# NOTE:注意这里是第三章 linked_list.py 里的内容,为了使文件自包含,我直接拷贝过来的 - - -class Node(object): - def __init__(self, value=None, next=None): # 这里我们 root 节点默认都是 None,所以都给了默认值 - self.value = value - self.next = next - - def __str__(self): - """方便你打出来调试,复杂的代码可能需要断点调试""" - return ''.format(self.value, self.next) - - __repr__ = __str__ - - -class LinkedList(object): - """ 链接表 ADT - [root] -> [node0] -> [node1] -> [node2] - """ - - def __init__(self, maxsize=None): - """ - :param maxsize: int or None, 如果是 None,无限扩充 - """ - self.maxsize = maxsize - self.root = Node() # 默认 root 节点指向 None - self.tailnode = None - self.length = 0 - - def __len__(self): - return self.length - - def append(self, value): # O(1) - if self.maxsize is not None and len(self) >= self.maxsize: - raise Exception('LinkedList is Full') - node = Node(value) # 构造节点 - tailnode = self.tailnode - if tailnode is None: # 还没有 append 过,length = 0, 追加到 root 后 - self.root.next = node - else: # 否则追加到最后一个节点的后边,并更新最后一个节点是 append 的节点 - tailnode.next = node - self.tailnode = node - self.length += 1 - - def appendleft(self, value): - headnode = self.root.next - node = Node(value) - self.root.next = node - node.next = headnode - self.length += 1 - - def __iter__(self): - for node in self.iter_node(): - yield node.value - - def iter_node(self): - """遍历 从 head 节点到 tail 节点""" - curnode = self.root.next - while curnode is not self.tailnode: # 从第一个节点开始遍历 - yield curnode - curnode = curnode.next # 移动到下一个节点 - yield curnode - - def remove(self, value): # O(n) - """ 删除包含值的一个节点,将其前一个节点的 next 指向被查询节点的下一个即可 - - :param value: - """ - prevnode = self.root # - curnode = self.root.next - for curnode in self.iter_node(): - if curnode.value == value: - prevnode.next = curnode.next - del curnode - self.length -= 1 - return 1 # 表明删除成功 - else: - prevnode = curnode - return -1 # 表明删除失败 - - def find(self, value): # O(n) - """ 查找一个节点,返回序号,从 0 开始 - - :param value: - """ - index = 0 - for node in self.iter_node(): # 我们定义了 __iter__,这里就可以用 for 遍历它了 - if node.value == value: - return index - index += 1 - return -1 # 没找到 - - def popleft(self): # O(1) - """ 删除第一个链表节点 - """ - if self.root.next is None: - raise Exception('pop from empty LinkedList') - headnode = self.root.next - self.root.next = headnode.next - self.length -= 1 - value = headnode.value - del headnode - return value - - def clear(self): - for node in self.iter_node(): - del node - self.root.next = None - self.length = 0 - -###################################################### -# 下边是 Queue 实现 -###################################################### - - -class EmptyError(Exception): - """自定义异常""" - pass - - -class Queue(object): - def __init__(self, maxsize=None): - self.maxsize = maxsize - self._item_link_list = LinkedList() - - def __len__(self): - return len(self._item_link_list) - - def push(self, value): # O(1) - """ 队尾添加元素 """ - return self._item_link_list.append(value) - - def pop(self): - """队列头部删除元素""" - if len(self) <= 0: - raise EmptyError('empty queue') - return self._item_link_list.popleft() - - -def test_queue(): - q = Queue() - q.push(0) - q.push(1) - q.push(2) - - assert len(q) == 3 - - assert q.pop() == 0 - assert q.pop() == 1 - assert q.pop() == 2 - - import pytest # pip install pytest - with pytest.raises(EmptyError) as excinfo: # 我们来测试是否真的抛出了异常 - q.pop() # 继续调用会抛出异常 - assert 'empty queue' == str(excinfo.value) - - -class MyQueue: - """ - 使用 collections.deque 可以迅速实现一个队列 - """ - def __init__(self): - self.items = deque() - - def append(self, val): - return self.items.append(val) - - def pop(self): - return self.items.popleft() - - def __len__(self): - return len(self.items) - - def empty(self): - return len(self.items) == 0 - - def front(self): - return self.items[0] diff --git "a/docs/05_\346\240\210/stack.md" "b/docs/05_\346\240\210/stack.md" deleted file mode 100644 index e220d8a..0000000 --- "a/docs/05_\346\240\210/stack.md" +++ /dev/null @@ -1,63 +0,0 @@ -# 栈 - -栈这个词实际上在计算机科学里使用很多,除了数据结构外,还有内存里的栈区 (和堆对应),熟悉 C 系语言的话应该不会陌生。 -上一章我们讲到了先进先出 queue,其实用 python 的内置类型 collections.deque 或者我们自己实现的 LinkedList 来实现它都很简单。 -本章我们讲讲 后进先出的栈。 - -生活中的数据结构: - -- 栈。好比在桶里头放盘子,先放的盘子放在了底下,后来的盘子放在上边。你要拿的时候,也是先拿最上边的。 - -栈其实也很简单,因为基础操作就俩,一个 push 和一个 pop,咦,咋和队列一样的? -确实方法名字一样,但是得到的结果可是不同的。 - - -# 栈 ADT - -上一章我介绍了我们怎样选取恰到的数据结构来实现新的 ADT?你能想到这里我们应该使用之前提到的哪个数据结构来实现吗? -你的大脑可能开始高(gui)速(su)旋转了,上几章学过的 array, list, deque, LinkedList, CircularDoubleLinkedList, queue -等在大脑里呼啸而过,这个时候可能已经一脸愁容了,到底该选啥? - -还用问嘛,当然是时间复杂度最小的啦,大部分情况下空间都是够用的。 -其实你会发现栈比队列还简单,因为它只在顶上操作(想象装着盘子的桶),如果有一种数据结构能方便在尾部增减元素不就满足需求了吗。 -这个时候如果你忘记了,可以翻翻前几章,看看哪个数据结构符合要求。 - -想一下,似乎 CircularDoubleLinkedList 循环双端队列是满足的,因为增删最后一个元素都是 O(1)。 -不过看了下示例代码,似乎没有 pop() 方法,对,因为我已经把实现 deque 作为思考题了。😂 -如果之前你没写出来也没关系,这里我们会再实现它。 - - -视频里我们将借助 CircularDoubleLinkedList 实现 双端队列 Deque ,并且用 Deque 实现 Stack。 - - -# Stack over flow 什么鬼? -嗯,stackoverflow 不是一个程序员问答网站吗?没错。 -函数的临时变量是存储在栈区的,如果你不幸写了一个没有出口的递归函数,就会这个错。不信你试试: - - -```py -def infinite_fib(n): - return infinite_fib(n-1) + infinite_fib(n-2) -infinite_fib(10) -``` - -一大段输出之后就会出现异常: RecursionError: maximum recursion depth exceeded。 -后边会讲到递归,递归是初学者比较难理解的概念,在树的遍历等地方还会看到它。 - - -# 数据结构头脑风暴法 - -当我们不知道使用什么数据结构来解决问题的时候,《程序员面试金典》这本书的第六章提到了一种方式叫做『数据结构头脑风暴法』。 -这种笨方法就是快速过一遍数据结构的列表,然后逐一尝试各种数据结构看看哪个最适合。 - -在你实现一个更高级的数据结构的时候,如果脑子没有思路,不妨尝试下这个方法,迅速过一遍你所知道的数据结构,看看哪种最适合。(从每个操作的时间复杂度和空间复杂度分析寻找最优解) - -# 思考题 -- 上一章我们用数组实现了队列,其实也能用数组来实现栈,你能自己用数组来实现一个栈的 ADT 吗? -- 实际上借助 python 内置的 list/collections.deque 结构就很容易实现一个栈,请你尝试实现,本章我们全部使用自己编写的数据结构而没用到 python 内置的数据结构。 -- 这里我们自己实现了 Deque,你能用 python 内置的 collections.deque 实现栈吗?有轮子能直接用的话看起来就简单多了,这里我们为了学习数据结构的实现就避免了直接使用内置结构 -- 哪些经典算法里使用到了栈呢? - -# Leetcode 练习 - -https://leetcode.com/problems/implement-queue-using-stacks/ diff --git "a/docs/05_\346\240\210/stack.py" "b/docs/05_\346\240\210/stack.py" deleted file mode 100644 index 8cd8970..0000000 --- "a/docs/05_\346\240\210/stack.py" +++ /dev/null @@ -1,188 +0,0 @@ -# -*- coding: utf-8 -*- - -# NOTE: 这里拷贝的 double_link_list.py 里的代码 - -from collections import deque - - -class Node(object): - - def __init__(self, value=None, prev=None, next=None): - self.value, self.prev, self.next = value, prev, next - - -class CircularDoubleLinkedList(object): - """循环双端链表 ADT - 多了个循环其实就是把 root 的 prev 指向 tail 节点,串起来 - """ - - def __init__(self, maxsize=None): - self.maxsize = maxsize - node = Node() - node.next, node.prev = node, node - self.root = node - self.length = 0 - - def __len__(self): - return self.length - - def headnode(self): - return self.root.next - - def tailnode(self): - return self.root.prev - - def append(self, value): # O(1), 你发现一般不用 for 循环的就是 O(1),有限个步骤 - if self.maxsize is not None and len(self) >= self.maxsize: - raise Exception('LinkedList is Full') - node = Node(value=value) - tailnode = self.tailnode() or self.root - - tailnode.next = node - node.prev = tailnode - node.next = self.root - self.root.prev = node - self.length += 1 - - def appendleft(self, value): - if self.maxsize is not None and len(self) >= self.maxsize: - raise Exception('LinkedList is Full') - node = Node(value=value) - if self.root.next is self.root: # empty - node.next = self.root - node.prev = self.root - self.root.next = node - self.root.prev = node - else: - node.prev = self.root - headnode = self.root.next - node.next = headnode - headnode.prev = node - self.root.next = node - self.length += 1 - - def remove(self, node): # O(1),传入node 而不是 value 我们就能实现 O(1) 删除 - """remove - :param node # 在 lru_cache 里实际上根据key 保存了整个node: - """ - if node is self.root: - return - else: # - node.prev.next = node.next - node.next.prev = node.prev - self.length -= 1 - return node - - def iter_node(self): - if self.root.next is self.root: - return - curnode = self.root.next - while curnode.next is not self.root: - yield curnode - curnode = curnode.next - yield curnode - - def __iter__(self): - for node in self.iter_node(): - yield node.value - - def iter_node_reverse(self): - """相比单链表独有的反序遍历""" - if self.root.prev is self.root: - return - curnode = self.root.prev - while curnode.prev is not self.root: - yield curnode - curnode = curnode.prev - yield curnode - - -############################################################ -# 分割线,下边是本章 内容实现 -############################################################ - - -class Deque(CircularDoubleLinkedList): # 注意这里我们用到了继承,嗯,貌似我说过不会用啥 OOP 特性的,抱歉 - - def pop(self): - """删除尾节点""" - if len(self) == 0: - raise Exception('empty') - tailnode = self.tailnode() - value = tailnode.value - self.remove(tailnode) - return value - - def popleft(self): - if len(self) == 0: - raise Exception('empty') - headnode = self.headnode() - value = headnode.value - self.remove(headnode) - return value - - -def test_deque(): - dq = Deque() - dq.append(1) - - dq.append(2) - assert list(dq) == [1, 2] - - dq.appendleft(0) - assert list(dq) == [0, 1, 2] - - dq.pop() - assert list(dq) == [0, 1] - - dq.popleft() - assert list(dq) == [1] - - dq.pop() - assert len(dq) == 0 - - -class Stack(object): - def __init__(self): - self.deque = Deque() # 你可以很容易替换为 python 内置的 collections.deque - - def push(self, value): - self.deque.append(value) - - def pop(self): - return self.deque.pop() - - -class Stack2(object): - - def __init__(self): - self._deque = deque() - - def push(self, value): - return self._deque.append(value) - - def pop(self): - return self._deque.pop() - - def empty(self): - return len(self._deque) == 0 - - -def test_stack(): - s = Stack() - s.push(0) - s.push(1) - s.push(2) - - assert s.pop() == 2 - assert s.pop() == 1 - assert s.pop() == 0 - - import pytest # pip install pytest - with pytest.raises(Exception) as excinfo: # 我们来测试是否真的抛出了异常 - s.pop() - assert 'empty' in str(excinfo.value) - - -if __name__ == '__main__': - test_stack() diff --git "a/docs/06_\347\256\227\346\263\225\345\210\206\346\236\220/big_o.md" "b/docs/06_\347\256\227\346\263\225\345\210\206\346\236\220/big_o.md" deleted file mode 100644 index 5f26aed..0000000 --- "a/docs/06_\347\256\227\346\263\225\345\210\206\346\236\220/big_o.md" +++ /dev/null @@ -1,122 +0,0 @@ -# 算法复杂度分析 -前面我们说了很多次时间复杂度是 O(1), O(n) 啥的,并没有仔细讲解这个 O 符号究竟是什么。 -你可以大概理解为操作的次数和数据个数的比例关系。比如 O(1) 就是有限次数操作,O(n) 就是操作正比于你的元素个数。 -这一章我们用更严谨的方式来定义它。 - - -# 大 O 表示法 -我们从一个计算矩阵的例子来引入,这里我参考了 [《Data Structures and Algorithms in Python》]( -https://book.douban.com/subject/10607365/) 中给的一个例子: - -考虑计算一个 n * n 矩阵所有元素的和(如果你不知道矩阵,就理解为一个二维数组): - -$$ - \begin{bmatrix} - 0 & 1 & 2 \\ - 3 & 4 & 5 \\ - 6 & 7 & 8 \\ - \end{bmatrix} -$$ - -这里列举两种方式: - -```py -# version1 -total_sum = 0 -for i in range(n): - row_sum[i] = 0 - for j in range(n): - row_sum[i] = row_sum[i] + matrix[i, j] - total_sum = total_sum + matrix[i, j] - -# version2 -total_sum = 0 -for i in range(n): - row_sum[i] = 0 - for j in range(n): - row_sum[i] = row_sum[i] + matrix[i, j] - total_sum = total_sum + row_sum[i] # 注意这里和上边的不同 -``` - -v1 版本的关键操作在 j 循环里,两步加法操作,由于嵌套在第一个循环里,操作步骤是 $ (2n) * n = 2n^2 $。 - -v2 版本的 total_sum 只有 n 次操作,它的操作次数是 $ n + n*n = n^2 + n $。 - - -这里你可能还感觉不到它们有多大差别,因为计算机执行的太快了,但是当 n 增长特别快的时候,总的操作次数差距就很明显了: - -n | $ 2n^2 $ | $ n^2 +n $ | --------|----------------|----------------| -10 | 200 | 110 | -100 | 20,000 | 10,100 | -1000 | 2,000,000 | 1,001,000 | -10000 | 200,000,000 | 100,010,000 | -100000 | 20,000,000,000 | 10,000,100,000 | - -通常我们不太关注每个算法具体执行了多少次,而更关心随着输入规模 n 的增加,算法运行时间将以什么速度增加。为此计算机科学家定义了一个符号, -用来表示在最糟糕的情况下算法的运行时间,大 O 符号,在数学上称之为渐进上界(《算法导论》)。 - -# 如何计算时间复杂度 -上边我们列举了两个版本的计算矩阵和的代码,你看到了两个公式: - -- v1: $ 2n*n = 2n^2 $ -- v2: $ n + n*n = n + n^2 $ - -当 n 非常大的时候,$ n^2 $ 的数值这里将占主导,我们可以忽略 n 的影响 - -- v1: $ 2n*n = 2n^2 $ -- v2: $ n + n*n = n + n^2 \leq 2n^2 $ - -这里我们可以认为两个算法的时间复杂度均为 $ O(n^2) $ - -# 常用时间复杂度 -这里我们列举一些常用的时间复杂度,按照增长速度排序,日常我们的业务代码中最常用的是指数之前的复杂度,指数和阶乘的增长速度非常快, -当输入比较大的时候用在业务代码里是不可接受的。 - -O | 名称 | 举例 | -----------|--------------|--------------------| -1 | 常量时间 | 一次赋值 | -$\log n$ | 对数时间 | 折半查找 | -$n$ | 线性时间 | 线性查找 | -n$\log n$ | 对数线性时间 | 快速排序 | -$n^2$ | 平方 | 两重循环 | -$n^3$ | 立方 | 三重循环 | -$2^n$ | 指数 | 递归求斐波那契数列 | -$n!$ | 阶乘 | 旅行商问题 | - - -# 空间复杂度 -相比时间复杂度,空间复杂度讨论比较少。因为用户老爷等不及,况且现在存储越来越白菜价了,更多时候我们为了提升响应速度宁可多 使用点空间。 -空间复杂度相对好算一些,就是每个元素的空间占用乘以总的元素数,有些算法需要额外的空间存储,有些可以本地解决。 -如果能本地搞定的我们成为 in place 的,原地操作,比如交换一个 数组中的某两个位置的元素。但是有些操作可能就需要申请额外的空间 -来完成算法了,后边我们介绍排序算法的时候会讲到。 - - -# 常见复杂度增长趋势图 -为了让你有个直观的感觉,我们来看看一些经典的时间复杂度和对应的增长趋势图,不同函数在输入规模增长的时候很快就会有巨大的增长差异 - -![函数增长趋势图](./function_growth.png) - - -# 时间换空间,空间换时间 -有一些时候时间和空间两者不可兼得,我们会牺牲其中之一来换取另一个。 - -空间换时间:比如典型的就是 python 中的集合(后面会讲到它的实现原理),虽然它比较浪费空间,但是却能用 O(1) -的时间复杂度来判重。 - -时间换空间:当我们空间不够用,典型的就是缓存失效算法,我们不可能缓存下无限容量的数据,就会使用一些缓存淘汰算法来保证空间可用。 - - -# 思考题 -- 回头看看前几章我们讲到的数据结构,以及每个操作的时间复杂度,你能理解了吗? -- 二分查找是针对有序元素的一种经典的查找算法,你知道的它的时间复杂度吗?你能简单证明下吗。 -- 斐波那契数列你肯定很熟悉,它的公式是 F(n) = F(n-1) + F(n-2),你知道计算一个斐波那契数 F(n) - 的时间复杂度吗?你会用数学公式证明吗? -- 你能指出时间和空间权衡的例子吗?往往很多高效的数据结构能同时兼顾时间和空间复杂度,但是有时候我们却得做出一定的权衡 - - -# 参考资料 -如果你对数学感兴趣,建议你阅读《算法导论》『函数的增长』这一节 和《Data Structures and Algorithms in Python》第4章。 - - -(本章我用了 [MathJax](https://www.zybuluo.com/codeep/note/163962) 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zpF(c@hv-vVoQ$ujC@=hHJf8mh$CT;52ALw}7IzYx^G{PsnDNtX^t)+o(*M#}5x<^^ za&m#-L2Uf2q*K`R?h<_Pn+($yufl5+L(O(PTCAnjiw0WP7-@a$^RsEi0=cEOwXqX2 zMVgcB3+buiPm4TZ#-AcNL-5b-t(QVZ3N#HW{!BLK@wuGin}n(@oK)r3Y{J+P^m+h2nc7 zo)!1BG$(Ghh1a{UhDudAjZ0xf+?q`JFYuEzRy0yM$!m)ev2*h(%v=67kzZ=iVZej9 zcVN5QR+0q`Zid#^qgB)I5-&e?5UEHOgSv-mc6rZRU)Z#^SGPt1B{6y_ z>unThqd*%4+9+^GQ9$5CduVhR*24ka{H2WNJF0zh5t`-|gnp3&V8_I_u@iVgVj~2O kJF2;DjcKF6e~beE4=xrKQ^VLf8~^|S07*qoM6N<$f`q4e)c^nh diff --git "a/docs/07_\345\223\210\345\270\214\350\241\250/hashtable.md" "b/docs/07_\345\223\210\345\270\214\350\241\250/hashtable.md" deleted file mode 100644 index efccb63..0000000 --- "a/docs/07_\345\223\210\345\270\214\350\241\250/hashtable.md" +++ /dev/null @@ -1,181 +0,0 @@ -# 哈希表 -不知道你有没有好奇过为什么 Python 里的 dict 和 set 查找速度这么快呢,用了什么黑魔法吗? -经常听别人说哈希表(也叫做散列表),究竟什么是哈希表呢?这一章我们来介绍哈希表,后续章节我们会看到 Python 中的字典和集合是如何实现的。 - -# 哈希表的工作过程 -前面我们已经讲到了数组和链表,数组能通过下标 O(1) 访问,但是删除一个中间元素却要移动其他元素,时间 O(n)。 -循环双端链表倒是可以在知道一个节点的情况下迅速删除它,但是吧查找又成了 O(n)。 - -难道就没有一种方法可以快速定位和删除元素吗?似乎想要快速找到一个元素除了知道下标之外别无他法,于是乎聪明的计算机科学家又想到了一种方法。 -能不能给每个元素一种『逻辑下标』,然后直接找到它呢,哈希表就是这种实现。它通过一个哈希函数来计算一个元素应该放在数组哪个位置,当然对于一个 -特定的元素,哈希函数每次计算的下标必须要一样才可以,而且范围不能超过给定的数组长度。 - -我们还是以书中的例子说明,假如我们有一个数组 T,包含 M=13 个元素,我们可以定义一个简单的哈希函数 h - -``` -h(key) = key % M -``` - -这里取模运算使得 h(key) 的结果不会超过数组的长度下标。我们来分别插入以下元素: - -765, 431, 96, 142, 579, 226, 903, 388 - -先来计算下它们应用哈希函数后的结果: - -``` -M = 13 -h(765) = 765 % M = 11 -h(431) = 431 % M = 2 -h(96) = 96 % M = 5 -h(142) = 142 % M = 12 -h(579) = 579 % M = 7 -h(226) = 226 % M = 5 -h(903) = 903 % M = 6 -h(388) = 388 % M = 11 -``` -下边我画个图演示整个插入过程(纯手工绘制,原谅我字写得不太优雅): - -![](./insert_hash.png) - - -# 哈希冲突 (collision) -这里到插入 226 这个元素的时候,不幸地发现 h(226) = h(96) = 5,不同的 key 通过我们的哈希函数计算后得到的下标一样, -这种情况成为哈希冲突。怎么办呢?聪明的计算机科学家又想到了办法,其实一种直观的想法是如果冲突了我能不能让数组中 -对应的槽变成一个链式结构呢?这就是其中一种解决方法,叫做 **链接法(chaining)**。如果我们用链接法来处理冲突,后边的插入是这样的: - -![](./insert_hash_chaining.png) - -这样就用链表解决了冲突问题,但是如果哈希函数选不好的话,可能就导致冲突太多一个链变得太长,这样查找就不再是 O(1) 的了。 -还有一种叫做开放寻址法(open addressing),它的基本思想是当一个槽被占用的时候,采用一种方式来寻找下一个可用的槽。 -(这里槽指的是数组中的一个位置),根据找下一个槽的方式不同,分为: - -- 线性探查(linear probing): 当一个槽被占用,找下一个可用的槽。 $ h(k, i) = (h^\prime(k) + i) \% m, i = 0,1,...,m-1 $ -- 二次探查(quadratic probing): 当一个槽被占用,以二次方作为偏移量。 $ h(k, i) = (h^\prime(k) + c_1 + c_2i^2) \% m , i=0,1,...,m-1 $ -- 双重散列(double hashing): 重新计算 hash 结果。 $ h(k,i) = (h_1(k) + ih_2(k)) \% m $ - -我们选一个简单的二次探查函数 $ h(k, i) = (home + i^2) \% m $,它的意思是如果 -遇到了冲突,我们就在原始计算的位置不断加上 i 的平方。我写了段代码来模拟整个计算下标的过程: - -```py -inserted_index_set = set() -M = 13 - -def h(key, M=13): - return key % M - -to_insert = [765, 431, 96, 142, 579, 226, 903, 388] -for number in to_insert: - index = h(number) - first_index = index - i = 1 - while index in inserted_index_set: # 如果计算发现已经占用,继续计算得到下一个可用槽的位置 - print('\th({number}) = {number} % M = {index} collision'.format(number=number, index=index)) - index = (first_index + i*i) % M # 根据二次方探查的公式重新计算下一个需要插入的位置 - i += 1 - else: - print('h({number}) = {number} % M = {index}'.format(number=number, index=index)) - inserted_index_set.add(index) -``` -这段代码输出的结果如下: - -``` -h(765) = 765 % M = 11 -h(431) = 431 % M = 2 -h(96) = 96 % M = 5 -h(142) = 142 % M = 12 -h(579) = 579 % M = 7 - h(226) = 226 % M = 5 collision -h(226) = 226 % M = 6 - h(903) = 903 % M = 6 collision - h(903) = 903 % M = 7 collision -h(903) = 903 % M = 10 - h(388) = 388 % M = 11 collision - h(388) = 388 % M = 12 collision - h(388) = 388 % M = 2 collision - h(388) = 388 % M = 7 collision -h(388) = 388 % M = 1 -``` - -遇到冲突之后会重新计算,每个待插入元素最终的下标就是: - -![](quadratic_hash.png) - -![](quadratic_result.png) - - -# Cpython 如何解决哈希冲突 -如果你对 cpython 解释器的实现感兴趣,可以参考下这个文件 [dictobject.c](https://github.com/python/cpython/blob/master/Objects/dictobject.c#L165)。 -不同 cpython 版本实现的探查方式是不同的,后边我们自己实现 HashTable ADT 的时候会模仿这个探查方式来解决冲突。 - - -``` -The first half of collision resolution is to visit table indices via this -recurrence: - - j = ((5*j) + 1) mod 2**i - -For any initial j in range(2**i), repeating that 2**i times generates each -int in range(2**i) exactly once (see any text on random-number generation for -proof). By itself, this doesn't help much: like linear probing (setting -j += 1, or j -= 1, on each loop trip), it scans the table entries in a fixed -order. This would be bad, except that's not the only thing we do, and it's -actually *good* in the common cases where hash keys are consecutive. In an -example that's really too small to make this entirely clear, for a table of -size 2**3 the order of indices is: - - 0 -> 1 -> 6 -> 7 -> 4 -> 5 -> 2 -> 3 -> 0 [and here it's repeating] -``` - -# 哈希函数 -到这里你应该明白哈希表插入的工作原理了,不过有个重要的问题之前没提到,就是 hash 函数怎么选? -当然是散列得到的冲突越来越小就好啦,也就是说每个 key 都能尽量被等可能地散列到 m 个槽中的任何一个,并且与其他 key 被散列到哪个槽位无关。 -如果你感兴趣,可以阅读后边提到的一些参考资料。视频里我们使用二次探查函数,它相比线性探查得到的结果冲突会更少。 - - -# 装载因子(load factor) -如果继续往我们的哈希表里塞东西会发生什么?空间不够用。这里我们定义一个负载因子的概念(load factor),其实很简单,就是已经使用的槽数比哈希表大小。 -比如我们上边的例子插入了 8 个元素,哈希表总大小是 13, 它的 load factor 就是 $ 8/13 \approx 0.62 $。当我们继续往哈希表插入数据的时候,很快就不够用了。 -通常当负载因子开始超过 0.8 的时候,就要新开辟空间并且重新进行散列了。 - - -# 重哈希(Rehashing) -当负载因子超过 0.8 的时候,需要进行 rehashing 操作了。步骤就是重新开辟一块新的空间,开多大呢?感兴趣的话可以看下 cpython 的 dictobject.c 文件然后搜索 -GROWTH_RATE 这个关键字,你会发现不同版本的 cpython 使用了不同的策略。python3.3 的策略是扩大为已经使用的槽数目的两倍。开辟了新空间以后,会把原来哈希表里 -不为空槽的数据重新插入到新的哈希表里,插入方式和之前一样。这就是 rehashing 操作。 - -# HashTable ADT -实践是检验真理的唯一标准,这里我们来实现一个简化版的哈希表 ADT,主要是为了让你更好地了解它的工作原理,有了它,后边实现起 dict 和 set 来就小菜一碟了。 -这里我们使用到了定长数组,还记得我们在数组和列表章节里实现的 Array 吧,这里要用上了。 - -解决冲突我们使用二次探查法,模拟 cpython 二次探查函数的实现。我们来实现三个哈希表最常用的基本操作,这实际上也是使用字典的时候最常用的操作。 - -- add(key, value) -- get(key, default) -- remove(key) - -```py -class Slot(object): - """定义一个 hash 表 数组的槽 - 注意,一个槽有三种状态,看你能否想明白 - 1.从未使用 HashMap.UNUSED。此槽没有被使用和冲突过,查找时只要找到 UNUSED 就不用再继续探查了 - 2.使用过但是 remove 了,此时是 HashMap.EMPTY,该探查点后边的元素扔可能是有key - 3.槽正在使用 Slot 节点 - """ - def __init__(self, key, value): - self.key, self.value = key, value - -class HashTable(object): - pass -``` - -具体的实现和代码编写在视频里讲解。这个代码可不太好实现,稍不留神就会有错,我们还是通过编写单元测试验证代码的正确性。 - -# 思考题 -- 请你分析下哈希表插入和删除元素的平均时间复杂度是多少?我们都实现代码了,相信这个问题你可以回答上来 -- Slot 在二次探查法里为什么不能直接删除?为什么我们要给它定义几个状态? - -# 延伸阅读 -- 《Data Structures and Algorithms in Python》11 章 Hash Tables -- 《算法导论》第三版 11 章散列表,了解几种哈希冲突的解决方式,以及为什么我们选择二次探查而不是线性探查法? -- 介绍 c 解释器如何实现的 python dict对象:[Python dictionary implementation](http://www.laurentluce.com/posts/python-dictionary-implementation/) -- [Python hash function implement](https://stackoverflow.com/questions/2070276/where-can-i-find-source-or-algorithm-of-pythons-hash-function) diff --git "a/docs/07_\345\223\210\345\270\214\350\241\250/hashtable.py" "b/docs/07_\345\223\210\345\270\214\350\241\250/hashtable.py" deleted file mode 100644 index f1df5e2..0000000 --- "a/docs/07_\345\223\210\345\270\214\350\241\250/hashtable.py" +++ /dev/null @@ -1,176 +0,0 @@ -# -*- coding: utf-8 -*- - -# 从数组和列表章复制的代码 - - -class Array(object): - - def __init__(self, size=32, init=None): - self._size = size - self._items = [init] * size - - def __getitem__(self, index): - return self._items[index] - - def __setitem__(self, index, value): - self._items[index] = value - - def __len__(self): - return self._size - - def clear(self, value=None): - for i in range(len(self._items)): - self._items[i] = value - - def __iter__(self): - for item in self._items: - yield item - - -class Slot(object): - """定义一个 hash 表数组的槽(slot 这里指的就是数组的一个位置) - hash table 就是一个 数组,每个数组的元素(也叫slot槽)是一个对象,对象包含两个属性 key 和 value。 - - 注意,一个槽有三种状态,看你能否想明白。相比链接法解决冲突,探查法删除一个 key 的操作稍微复杂。 - 1.从未使用 HashMap.UNUSED。此槽没有被使用和冲突过,查找时只要找到 UNUSED 就不用再继续探查了 - 2.使用过但是 remove 了,此时是 HashMap.EMPTY,该探查点后边的元素仍然可能是有key的,需要继续查找 - 3.槽正在使用 Slot 节点 - """ - - def __init__(self, key, value): - self.key, self.value = key, value - - -class HashTable(object): - - UNUSED = None # 没被使用过 - EMPTY = Slot(None, None) # 使用却被删除过 - - def __init__(self): - self._table = Array(8, init=HashTable.UNUSED) # 保持 2*i 次方 - self.length = 0 - - @property - def _load_factor(self): - # load_factor 超过 0.8 重新分配 - return self.length / float(len(self._table)) - - def __len__(self): - return self.length - - def _hash(self, key): - return abs(hash(key)) % len(self._table) - - def _find_key(self, key): - """ - 解释一个 slot 为 UNUSED 和 EMPTY 的区别 - 因为使用的是二次探查的方式,假如有两个元素 A,B 冲突了,首先A hash 得到是 slot 下标5,A 放到了第5个槽,之后插入 B 因为冲突了,所以继续根据二次探查方式放到了 slot8。 - 然后删除 A,槽 5 被置为 EMPTY。然后我去查找 B,第一次 hash 得到的是 槽5,但是这个时候我还是需要第二次计算 hash 才能找到 B。但是如果槽是 UNUSED 我就不用继续找了,我认为 B 就是不存在的元素。这个就是 UNUSED 和 EMPTY 的区别。 - """ - origin_index = index = self._hash(key) # origin_index 判断是否又走到了起点,如果查找一圈了都找不到则无此元素 - _len = len(self._table) - while self._table[index] is not HashTable.UNUSED: - if self._table[index] is HashTable.EMPTY: # 注意如果是 EMPTY,继续寻找下一个槽 - index = (index * 5 + 1) % _len - if index == origin_index: - break - continue - if self._table[index].key == key: # 找到了key - return index - else: - index = (index * 5 + 1) % _len # 没有找到继续找下一个位置 - if index == origin_index: - break - - return None - - def _find_slot_for_insert(self, key): - index = self._hash(key) - _len = len(self._table) - while not self._slot_can_insert(index): # 直到找到一个可以用的槽 - index = (index * 5 + 1) % _len - return index - - def _slot_can_insert(self, index): - return (self._table[index] is HashTable.EMPTY or self._table[index] is HashTable.UNUSED) - - def __contains__(self, key): # in operator,实现之后可以使用 in 操作符判断 - index = self._find_key(key) - return index is not None - - def add(self, key, value): - if key in self: # update - index = self._find_key(key) - self._table[index].value = value - return False - else: - index = self._find_slot_for_insert(key) - self._table[index] = Slot(key, value) - self.length += 1 - if self._load_factor >= 0.8: - self._rehash() - return True - - def _rehash(self): - old_table = self._table - newsize = len(self._table) * 2 - self._table = Array(newsize, HashTable.UNUSED) - - self.length = 0 - - for slot in old_table: - if slot is not HashTable.UNUSED and slot is not HashTable.EMPTY: - index = self._find_slot_for_insert(slot.key) - self._table[index] = slot - self.length += 1 - - def get(self, key, default=None): - index = self._find_key(key) - if index is None: - return default - else: - return self._table[index].value - - def remove(self, key): - index = self._find_key(key) - if index is None: - raise KeyError() - value = self._table[index].value - self.length -= 1 - self._table[index] = HashTable.EMPTY - return value - - def __iter__(self): - for slot in self._table: - if slot not in (HashTable.EMPTY, HashTable.UNUSED): - yield slot.key - - -def test_hash_table(): - h = HashTable() - h.add('a', 0) - h.add('b', 1) - h.add('c', 2) - assert len(h) == 3 - assert h.get('a') == 0 - assert h.get('b') == 1 - assert h.get('hehe') is None - - h.remove('a') - assert h.get('a') is None - assert sorted(list(h)) == ['b', 'c'] - - n = 50 - for i in range(n): - h.add(i, i) - - for i in range(n): - assert h.get(i) == i - - -if __name__ == '__main__': - print( - 'beg', - test_hash_table(), - 'end', - ) diff --git "a/docs/07_\345\223\210\345\270\214\350\241\250/insert_hash.png" "b/docs/07_\345\223\210\345\270\214\350\241\250/insert_hash.png" deleted file mode 100644 index 72a851600364e23c9dfa257b393ffd621a00cf3a..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 167188 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