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Commite3a6aa3

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Merge branch 'main' into cpp-tips
2 parentsedabef5 +91672f0 commite3a6aa3

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‎.github/workflows/build.yml‎

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-name:Set up Python
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uses:actions/setup-python@v5.2.0
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with:
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python-version:'3.8'
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python-version:'3.11'
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-name:Install mkdocs-material
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run:|
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scripts/install-mkdocs.sh

‎mkdocs.yml‎

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icon:
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repo:fontawesome/brands/github
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features:
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-navigation.tabs
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-toc.integrate
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-search.suggest
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-content.code.copy
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repo_url:https://github.com/cp-algorithms/cp-algorithms
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repo_name:cp-algorithms/cp-algorithms
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edit_uri:edit/main/src/
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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 -2024 by <a href="https://github.com/cp-algorithms/cp-algorithms/graphs/contributors">cp-algorithms contributors</a>
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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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-https://cdnjs.cloudflare.com/polyfill/v3/polyfill.min.js?features=es6
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permalink:true
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plugins:
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-toggle-sidebar:
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toggle_button:all
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-mkdocs-simple-hooks:
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hooks:
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on_env:"hooks:on_env"
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-search
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-tags:
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tags_file:tags.md
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-tags
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-literate-nav:
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nav_file:navigation.md
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-git-revision-date-localized:

‎scripts/install-mkdocs.sh‎

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pip install \
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"mkdocs-material>=9.0.2" \
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mkdocs-toggle-sidebar-plugin \
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mkdocs-macros-plugin \
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mkdocs-literate-nav \
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mkdocs-git-authors-plugin \

‎src/algebra/bit-manipulation.md‎

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With the new knowledge in hand we can come up with the following algorithm:
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- Find the highest power of $2$ that is lesser than or equal to the given number. Let this number be $x$.
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- Calculate the number of set bits from $1$ to $2^x - 1$ by using theformua $x \cdot 2^{x-1}$.
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- Calculate the number of set bits from $1$ to $2^x - 1$ by using theformula $x \cdot 2^{x-1}$.
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- Count the no. of set bits in the most significant bit from $2^x$ to $n$ and add it.
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- Subtract $2^x$ from $n$ and repeat the above steps using the new $n$.
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‎src/algebra/factorization.md‎

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Here is a visualization of such a sequence $\{x_i \bmod p\}$ with $n = 2206637$, $p = 317$, $x_0 = 2$ and $f(x) = x^2 + 1$.
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From the form of the sequence you can see very clearly why the algorithm is called Pollard's $\rho$ algorithm.
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<center>![Pollard's rho visualization](pollard_rho.png)</center>
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<divstyle="text-align:center;">
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<imgsrc="pollard_rho.png"alt="Pollard's rho visualization">
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</div>
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Yet, there is still an open question.
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How can we exploit the properties of the sequence $\{x_i \bmod p\}$ to our advantage without even knowing the number $p$ itself?

‎src/algebra/fft.md‎

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$$A(x) = A_0(x^2) + x A_1(x^2).$$
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The polynomials $A_0$ and $A_1$are only half asmuch coefficients as the polynomial $A$.
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The polynomials $A_0$ and $A_1$have only half asmany coefficients as the polynomial $A$.
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If we can compute the $\text{DFT}(A)$ in linear time using $\text{DFT}(A_0)$ and $\text{DFT}(A_1)$, then we get the recurrence $T_{\text{DFT}}(n) = 2 T_{\text{DFT}}\left(\frac{n}{2}\right) + O(n)$ for the time complexity, which results in $T_{\text{DFT}}(n) = O(n \log n)$ by the**master theorem**.
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Let's learn how we can accomplish that.

‎src/algebra/phi-function.md‎

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## Euler totient function from $1$ to $n$ in $O(n \log\log{n})$ { #etf_1_to_n data-toc-label="Euler totient function from 1 to n in <script type=\"math/tex\">O(n log log n)</script>" }
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If we needall allthe totient of all numbers between $1$ and $n$, then factorizing all $n$ numbers is not efficient.
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If we need the totient of all numbers between $1$ and $n$, then factorizing all $n$ numbers is not efficient.
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We can use the same idea as the [Sieve of Eratosthenes](sieve-of-eratosthenes.md).
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It is still based on the property shown above, but instead of updating the temporary result for each prime factor for each number, we find all prime numbers and for each one update the temporary results of all numbers that are divisible by that prime number.
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‎src/algebra/sieve-of-eratosthenes.md‎

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In the following image you can see a visualization of the algorithm for computing all prime numbers in the range $[1; 16]$. It can be seen, that quite often we mark numbers as composite multiple times.
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<center>![Sieve of Eratosthenes](sieve_eratosthenes.png)</center>
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<divstyle="text-align:center;">
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<imgsrc="sieve_eratosthenes.png"alt="Sieve of Eratosthenes">
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</div>
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The idea behind is this:
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A number is prime, if none of the smaller prime numbers divides it.

‎src/combinatorics/burnside.md‎

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return sum / s.size();
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}
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```
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## Practice Problems
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* [CSES - Counting Necklaces](https://cses.fi/problemset/task/2209)
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* [CSES - Counting Grids](https://cses.fi/problemset/task/2210)
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* [Codeforces - Buildings](https://codeforces.com/gym/101873/problem/B)
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* [CS Academy - Cube Coloring](https://csacademy.com/contest/beta-round-8/task/cube-coloring/)

‎src/data_structures/fenwick.md‎

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The following image shows a possible interpretation of the Fenwick treeas tree.
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The nodes of the tree show the ranges they cover.
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<center>![Binary Indexed Tree](binary_indexed_tree.png)</center>
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<divstyle="text-align: center;">
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<imgsrc="binary_indexed_tree.png"alt="Binary Indexed Tree">
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</div>
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## Implementation
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