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‎README.md

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## Changelog
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- August, 2025: Overhaul of CP-Algorithms [donation system](https://github.com/sponsors/cp-algorithms). Please consider supporting us, so that we can grow!
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- August, 2025: Launched a [Discord server](https://discord.gg/HZ5AecN3KX)!
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- October, 2024: Welcome new maintainers: [jxu](https://github.com/jxu), [mhayter](https://github.com/mhayter) and [kostero](https://github.com/kostero)!
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- October, 15, 2024: GitHub pages based mirror is now served at [https://gh.cp-algorithms.com/](https://gh.cp-algorithms.com/), and an auxiliary competitive programming library is available at [https://lib.cp-algorithms.com/](https://lib.cp-algorithms.com/).
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- July 16, 2024: Major overhaul of the [Finding strongly connected components / Building condensation graph](https://cp-algorithms.com/graph/strongly-connected-components.html) article.
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### New articles
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- (19 August 2025) [Minimum Enclosing Circle](https://cp-algorithms.com/geometry/enclosing-circle.html)
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- (21 May 2025) [Simulated Annealing](https://cp-algorithms.com/num_methods/simulated_annealing.html)
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- (12 July 2024) [Manhattan distance](https://cp-algorithms.com/geometry/manhattan-distance.html)
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- (8 June 2024) [Knapsack Problem](https://cp-algorithms.com/dynamic_programming/knapsack.html)

‎mkdocs.yml

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copyright: Text is available under the <a href="https://github.com/cp-algorithms/cp-algorithms/blob/main/LICENSE">Creative Commons Attribution Share Alike 4.0 International</a> License<br/>Copyright &copy; 2014 - 2025 by <a href="https://github.com/cp-algorithms/cp-algorithms/graphs/contributors">cp-algorithms contributors</a>
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extra_javascript:
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- javascript/config.js
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- javascript/donation-banner.js
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- https://cdnjs.cloudflare.com/polyfill/v3/polyfill.min.js?features=es6
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- https://unpkg.com/mathjax@3/es5/tex-mml-chtml.js
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extra_css:
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- rss
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extra:
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social:
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- icon: fontawesome/brands/github
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link: https://github.com/cp-algorithms/cp-algorithms
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- icon: fontawesome/brands/discord
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link: https://discord.gg/HZ5AecN3KX
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- icon: fontawesome/solid/circle-dollar-to-slot
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link: https://github.com/sponsors/cp-algorithms
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analytics:
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provider: google
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property: G-7FLS2HCYHH

‎src/algebra/binary-exp.md

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a_{41} & a_ {42} & a_ {43} & a_ {44}
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\end{pmatrix}$$
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that, when multiplied by a vector with the old coordinates and an unit gives a new vector with the new coordinates and an unit:
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that, when multiplied by a vector with the old coordinates and a unit gives a new vector with the new coordinates and a unit:
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$$\begin{pmatrix} x & y & z & 1 \end{pmatrix} \cdot
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\begin{pmatrix}

‎src/algebra/chinese-remainder-theorem.md

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a & \equiv 2 \pmod{6}
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\end{align}\right.$$
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It is pretty simple to determine is a system has a solution.
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It is pretty simple to determine if a system has a solution.
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And if it has one, we can use the original algorithm to solve a slightly modified system of congruences.
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A single congruence $a \equiv a_i \pmod{m_i}$ is equivalent to the system of congruences $a \equiv a_i \pmod{p_j^{n_j}}$ where $p_1^{n_1} p_2^{n_2}\cdots p_k^{n_k}$ is the prime factorization of $m_i$.

‎src/algebra/discrete-log.md

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Instead of a `map`, we can also use a hash table (`unordered_map` in C++) which has the average time complexity $O(1)$ for inserting and searching.
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Problems often ask for the minimum $x$ which satisfies the solution.
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It is possible to get all answers and take the minimum, or reduce the first found answer using [Euler's theorem](phi-function.md#toc-tgt-2), but we can be smart about the order in which we calculate values and ensure the first answer we find is the minimum.
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It is possible to get all answers and take the minimum, or reduce the first found answer using [Euler's theorem](phi-function.md#application), but we can be smart about the order in which we calculate values and ensure the first answer we find is the minimum.
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```{.cpp file=discrete_log}
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// Returns minimum x for which a ^ x % m = b % m, a and m are coprime.

‎src/algebra/linear-diophantine-equation.md

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If $c$ is divisible by $g = \gcd(a, b)$, then the given Diophantine equation has a solution, otherwise it does not have any solution. The proof is straight-forward: a linear combination of two numbers is divisible by their common divisor.
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Now supposed that $c$ is divisible by $g$, then we have:
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Now suppose that $c$ is divisible by $g$, then we have:
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$$a \cdot x_g \cdot \frac{c}{g} + b \cdot y_g \cdot \frac{c}{g} = c$$
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‎src/algebra/polynomial.md

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$$A^{-1}(x) \equiv \frac{1}{A(x)} \equiv \frac{A(-x)}{A(x)A(-x)} \equiv \frac{A(-x)}{T(x^2)} \pmod{x^k}$$
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Note that $T(x)$ can be computed with a single multiplication, after which we're only interested in the first half of coefficients of its inverse series. This effectively reduces the initial problem of computing $A^{-1} \pmod{x^k}$ to computing $T^{-1} \pmod{x^{\lfloor k / 2 \rfloor}}$.
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Note that $T(x)$ can be computed with a single multiplication, after which we're only interested in the first half of coefficients of its inverse series. This effectively reduces the initial problem of computing $A^{-1} \pmod{x^k}$ to computing $T^{-1} \pmod{x^{\lceil k / 2 \rceil}}$.
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The complexity of this method can be estimated as
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‎src/combinatorics/burnside.md

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* [SPOJ - Sorting Machine](https://www.spoj.com/problems/SRTMACH/)
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* [Project Euler - Pizza Toppings](https://projecteuler.net/problem=281)
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* [ICPC 2011 SERCP - Alphabet Soup](https://basecamp.eolymp.com/tr/problems/3064)
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* [GCPC 2017 - Buildings](https://basecamp.eolymp.com/en/problems/11615)
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* [GCPC 2017 - Buildings](https://basecamp.eolymp.com/en/problems/11615)

‎src/data_structures/stack_queue_modification.md

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## Stack modification
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We want to modify the stack data structure in such a way, that it possible to find the smallest element in the stack in $O(1)$ time, while maintaining the same asymptotic behavior for adding and removing elements from the stack.
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We want to modify the stack data structure in such a way, that it is possible to find the smallest element in the stack in $O(1)$ time, while maintaining the same asymptotic behavior for adding and removing elements from the stack.
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Quick reminder, on a stack we only add and remove elements on one end.
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To do this, we will not only store the elements in the stack, but we will store them in pairs: the element itself and the minimum in the stack starting from this element and below.

‎src/geometry/basic-geometry.md

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To see the next important property we should take a look at the set of points $\mathbf r$ for which $\mathbf r\cdot \mathbf a = C$ for some fixed constant $C$.
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You can see that this set of points is exactly the set of points for which the projection onto $\mathbf a$ is the point $C \cdot \dfrac{\mathbf a}{|\mathbf a|}$ and they form a hyperplane orthogonal to $\mathbf a$.
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You can see that this set of points is exactly the set of points for which the projection onto $\mathbf a$ is the point $C \cdot \dfrac{\mathbf a}{|\mathbf a| ^ 2}$ and they form a hyperplane orthogonal to $\mathbf a$.
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You can see the vector $\mathbf a$ alongside with several such vectors having same dot product with it in 2D on the picture below:
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<div style="text-align: center;">

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