We can solve a problem using one or more algorithms. It's essential to learn how to compare the performance of different algorithms and select the best one for a specific task.

- Regardless of the system or its settings on which the algorithm is executing.

- Demonstrate a direct relationship with the quantity of inputs.

Graphical representation of Big Oh:

Graphical representation of Big Omega:

Graphical representation of Big Theta:

The best-case scenario refers to the situation where an algorithm performs optimally, achieving the lowest possible time or space complexity. It represents the most favorable conditions under which an algorithm operates.

- Occurs when the input data is structured in such a way that the algorithm can exploit its strengths fully.

- Often used to analyze the lower bound of an algorithm's performance.

The worst-case scenario refers to the situation where an algorithm performs at its poorest, achieving the highest possible time or space complexity. It represents the most unfavorable conditions under which an algorithm operates.

- Represents the maximum time or space required by an algorithm to solve a problem.

- Occurs when the input data is structured in such a way that the algorithm encounters the most challenging conditions.

- Often used to analyze the upper bound of an algorithm's performance.

The average-case scenario refers to the expected performance of an algorithm over all possible inputs, typically calculated as the arithmetic mean of the time or space complexity.

- Represents the typical performance of an algorithm across a range of input data.

- Takes into account the distribution of inputs and their likelihood of occurrence.

- Provides a more realistic measure of an algorithm's performance compared to the best-case or worst-case scenarios.

The memory space that a code utilizes as it is being run is often referred to as space complexity. Additionally, space complexity depends on the machine, therefore rather than using the typical memory units like MB, GB, etc., we will express space complexity using the Big O notation.

2. `Linear Space Complexity (O(n))`: Algorithms that store each element of the input array in a separate variable or data structure have O(n) space complexity.

3. `Quadratic Space Complexity (O(n^2))`: Algorithms that create a two-dimensional array or matrix with dimensions based on the input size have O(n^2) space complexity.

To analyze space complexity:

- Identify the variables, data structures, and recursive calls used by the algorithm.

Consider each line takes one unit of time to run. So, to simply iterate over an array to print all elements it will take `O(n)` time, where n is the size of array.

Code:

arr = [1,2,3,4] #1

for x in arr: #2

print(x) #3

Also, as the code dosen't take any additional space except the input arr its Space Complexity is O(1) constant.

Linear search is a simple algorithm for finding an element in an array by sequentially checking each element until a match is found or the end of the array is reached. Here's an example of calculating the time and space complexity of linear search:

target = 5

print(linear_search(arr, target))

Therefore, the time complexity of linear search is `O(n)`.

The space complexity of linear search is `O(1)` since it only uses a constant amount of additional space for variables regardless of the input size.

Binary search is an efficient algorithm for finding an element in a sorted array by repeatedly dividing the search interval in half. Here's an example of calculating the time and space complexity of binary search:

arr = [1, 3, 5, 7, 9]

target = 5

print(binary_search(arr, target))

Therefore, the time complexity of binary search is `O(log n)`.

The space complexity of binary search is `O(1)` since it only uses a constant amount of additional space for variables regardless of the input size.

Let's consider an example of a function that generates Fibonacci numbers up to a given index and stores them in a list. In this case, the space complexity will not be constant because the size of the list grows with the Fibonacci sequence.

fib_sequence = fibonacci_sequence(n)

print(fib_sequence)

The while loop iterates until the length of the Fibonacci sequence list reaches n, so it takes `O(n)` iterations in the `worst case`.Inside the loop, each operation takes constant time (O(1)).