add BinaryGCD.md

tatyam-prime · May 31, 2024 · 9151361 · 9151361
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diff --git a/.verify-helper/docs/static/notebook.html b/.verify-helper/docs/static/notebook.html
@@ -37,6 +37,7 @@ <h1 id="title">ICPC Notebook</h1>
     </ol>
     <li><a class="section" href="#math">math</a></li>
     <ol>
+    <li><a href="#BinaryGCD.hpp">BinaryGCD.hpp</a></li>
     </ol>
     <li><a class="section" href="#modint">modint</a></li>
     <ol>
@@ -223,6 +224,26 @@ <h2 id="FastSet.hpp">FastSet.hpp</h2>
 
     <h2 class="section" id="math">math</h2>
 
+    <div class="filename">
+        <h2 id="BinaryGCD.hpp">BinaryGCD.hpp</h2>
+        md5: f3ab31
+
+    </div>
+    <div class="content content-hpp">
+    <pre><code>u64 <span class="token function">ctz</span><span class="token punctuation">(</span>u64 x<span class="token punctuation">)</span> <span class="token punctuation">{</span> <span class="token keyword">return</span> <span class="token function">countr_zero</span><span class="token punctuation">(</span>x<span class="token punctuation">)</span><span class="token punctuation">;</span> <span class="token punctuation">}</span>
+u64 <span class="token function">binary_gcd</span><span class="token punctuation">(</span>u64 x<span class="token punctuation">,</span> u64 y<span class="token punctuation">)</span> <span class="token punctuation">{</span>
+   <span class="token keyword">if</span><span class="token punctuation">(</span><span class="token operator">!</span>x <span class="token operator">||</span> <span class="token operator">!</span>y<span class="token punctuation">)</span> <span class="token keyword">return</span> x <span class="token operator">|</span> y<span class="token punctuation">;</span>
+   u64 n <span class="token operator">=</span> <span class="token function">ctz</span><span class="token punctuation">(</span>x<span class="token punctuation">)</span><span class="token punctuation">,</span> m <span class="token operator">=</span> <span class="token function">ctz</span><span class="token punctuation">(</span>y<span class="token punctuation">)</span><span class="token punctuation">;</span>
+   x <span class="token operator">>>=</span> n<span class="token punctuation">,</span> y <span class="token operator">>>=</span> m<span class="token punctuation">;</span>
+   <span class="token keyword">while</span><span class="token punctuation">(</span>x <span class="token operator">!=</span> y<span class="token punctuation">)</span> <span class="token punctuation">{</span>
+      <span class="token keyword">if</span><span class="token punctuation">(</span>x <span class="token operator">></span> y<span class="token punctuation">)</span> x <span class="token operator">=</span> <span class="token punctuation">(</span>x <span class="token operator">-</span> y<span class="token punctuation">)</span> <span class="token operator">>></span> <span class="token function">ctz</span><span class="token punctuation">(</span>x <span class="token operator">-</span> y<span class="token punctuation">)</span><span class="token punctuation">;</span>
+      <span class="token keyword">else</span> y <span class="token operator">=</span> <span class="token punctuation">(</span>y <span class="token operator">-</span> x<span class="token punctuation">)</span> <span class="token operator">>></span> <span class="token function">ctz</span><span class="token punctuation">(</span>y <span class="token operator">-</span> x<span class="token punctuation">)</span><span class="token punctuation">;</span>
+   <span class="token punctuation">}</span>
+   <span class="token keyword">return</span> x <span class="token operator">&lt;&lt;</span> <span class="token function">min</span><span class="token punctuation">(</span>n<span class="token punctuation">,</span> m<span class="token punctuation">)</span><span class="token punctuation">;</span>
+<span class="token punctuation">}</span>
+</code></pre>
+</div>
+
     <h2 class="section" id="modint">modint</h2>
 
     <div class="filename">
@@ -663,16 +684,6 @@ <h4>高度合成数</h4>
 <td>768</td>
 <td>1344</td>
 </tr>
-<tr>
-<td>factor</td>
-<td>3111</td>
-<td>3311</td>
-<td>33111</td>
-<td>421111</td>
-<td>631111</td>
-<td>5311111</td>
-<td>6321111</td>
-</tr>
 </tbody></table>
 <table>
 <thead>
@@ -684,6 +695,9 @@ <h4>高度合成数</h4>
 <th>$10^{13}$</th>
 <th>$10^{14}$</th>
 <th>$10^{15}$</th>
+<th>$10^{16}$</th>
+<th>$10^{17}$</th>
+<th>$10^{18}$</th>
 </tr>
 </thead>
 <tbody><tr>
@@ -694,37 +708,65 @@ <h4>高度合成数</h4>
 <td>10752</td>
 <td>17280</td>
 <td>26880</td>
-</tr>
-<tr>
-<td>factor</td>
-<td>53211111</td>
-<td>63221111</td>
-<td>642111111</td>
-<td>6321111111</td>
-<td>5422111111</td>
-<td>64211111111</td>
+<td>41472</td>
+<td>64512</td>
+<td>103680</td>
 </tr>
 </tbody></table>
+<h4>素数階乗</h4>
 <table>
 <thead>
 <tr>
-<th>$≤n$</th>
-<th>$10^{16}$</th>
-<th>$10^{17}$</th>
-<th>$10^{18}$</th>
+<th>$n$</th>
+<th>$2$</th>
+<th>$3$</th>
+<th>$5$</th>
+<th>$7$</th>
+<th>$11$</th>
+<th>$13$</th>
+<th>$17$</th>
+<th>$19$</th>
+<th>$23$</th>
+<th>$29$</th>
 </tr>
 </thead>
 <tbody><tr>
-<td>$d^0(x)$</td>
-<td>41472</td>
-<td>64512</td>
-<td>103680</td>
+<td>$n\#$</td>
+<td>2</td>
+<td>6</td>
+<td>30</td>
+<td>210</td>
+<td>2310</td>
+<td>30030</td>
+<td>510510</td>
+<td>9699690</td>
+<td>$2.2 \times 10^8$</td>
+<td>$6.5 \times 10^9$</td>
 </tr>
+</tbody></table>
+<h4>階乗</h4>
+<table>
+<thead>
 <tr>
-<td>factor</td>
-<td>83221111111</td>
-<td>632211111111</td>
-<td>842211111111</td>
+<th>$5!$</th>
+<th>$6!$</th>
+<th>$7!$</th>
+<th>$8!$</th>
+<th>$9!$</th>
+<th>$10!$</th>
+<th>$11!$</th>
+<th>$12!$</th>
+</tr>
+</thead>
+<tbody><tr>
+<td>120</td>
+<td>720</td>
+<td>5040</td>
+<td>40320</td>
+<td>362880</td>
+<td>3628800</td>
+<td>$4.0 \times 10^7$</td>
+<td>$4.8 \times 10^8$</td>
 </tr>
 </tbody></table>
 </div>

diff --git a/.verify-helper/docs/static/notebook.pdf b/.verify-helper/docs/static/notebook.pdf
diff --git a/build/notebook.html b/build/notebook.html
@@ -37,6 +37,7 @@ <h1 id="title">ICPC Notebook</h1>
     </ol>
     <li><a class="section" href="#math">math</a></li>
     <ol>
+    <li><a href="#BinaryGCD.hpp">BinaryGCD.hpp</a></li>
     </ol>
     <li><a class="section" href="#modint">modint</a></li>
     <ol>
@@ -223,6 +224,26 @@ <h2 id="FastSet.hpp">FastSet.hpp</h2>
 
     <h2 class="section" id="math">math</h2>
 
+    <div class="filename">
+        <h2 id="BinaryGCD.hpp">BinaryGCD.hpp</h2>
+        md5: f3ab31
+
+    </div>
+    <div class="content content-hpp">
+    <pre><code>u64 <span class="token function">ctz</span><span class="token punctuation">(</span>u64 x<span class="token punctuation">)</span> <span class="token punctuation">{</span> <span class="token keyword">return</span> <span class="token function">countr_zero</span><span class="token punctuation">(</span>x<span class="token punctuation">)</span><span class="token punctuation">;</span> <span class="token punctuation">}</span>
+u64 <span class="token function">binary_gcd</span><span class="token punctuation">(</span>u64 x<span class="token punctuation">,</span> u64 y<span class="token punctuation">)</span> <span class="token punctuation">{</span>
+   <span class="token keyword">if</span><span class="token punctuation">(</span><span class="token operator">!</span>x <span class="token operator">||</span> <span class="token operator">!</span>y<span class="token punctuation">)</span> <span class="token keyword">return</span> x <span class="token operator">|</span> y<span class="token punctuation">;</span>
+   u64 n <span class="token operator">=</span> <span class="token function">ctz</span><span class="token punctuation">(</span>x<span class="token punctuation">)</span><span class="token punctuation">,</span> m <span class="token operator">=</span> <span class="token function">ctz</span><span class="token punctuation">(</span>y<span class="token punctuation">)</span><span class="token punctuation">;</span>
+   x <span class="token operator">>>=</span> n<span class="token punctuation">,</span> y <span class="token operator">>>=</span> m<span class="token punctuation">;</span>
+   <span class="token keyword">while</span><span class="token punctuation">(</span>x <span class="token operator">!=</span> y<span class="token punctuation">)</span> <span class="token punctuation">{</span>
+      <span class="token keyword">if</span><span class="token punctuation">(</span>x <span class="token operator">></span> y<span class="token punctuation">)</span> x <span class="token operator">=</span> <span class="token punctuation">(</span>x <span class="token operator">-</span> y<span class="token punctuation">)</span> <span class="token operator">>></span> <span class="token function">ctz</span><span class="token punctuation">(</span>x <span class="token operator">-</span> y<span class="token punctuation">)</span><span class="token punctuation">;</span>
+      <span class="token keyword">else</span> y <span class="token operator">=</span> <span class="token punctuation">(</span>y <span class="token operator">-</span> x<span class="token punctuation">)</span> <span class="token operator">>></span> <span class="token function">ctz</span><span class="token punctuation">(</span>y <span class="token operator">-</span> x<span class="token punctuation">)</span><span class="token punctuation">;</span>
+   <span class="token punctuation">}</span>
+   <span class="token keyword">return</span> x <span class="token operator">&lt;&lt;</span> <span class="token function">min</span><span class="token punctuation">(</span>n<span class="token punctuation">,</span> m<span class="token punctuation">)</span><span class="token punctuation">;</span>
+<span class="token punctuation">}</span>
+</code></pre>
+</div>
+
     <h2 class="section" id="modint">modint</h2>
 
     <div class="filename">

diff --git a/docs/math/BinaryGCD.md b/docs/math/BinaryGCD.md
@@ -0,0 +1,11 @@
+---
+title: Binary GCD
+documentation_of: //src/math/BinaryGCD.hpp
+---
+
+割り算を使わない高速な GCD
+
+## 使い方
+
+- `u64 binary_gcd(u64 x, u64 y)`：$x$ と $y$ の最大公約数を計算する．
+    - $O(\log(x + y))$ 時間
diff --git a/docs/modint/BarrettReduction.md b/docs/modint/BarrettReduction.md
@@ -2,14 +2,15 @@
 title: Barrett Reduction
 documentation_of: //src/modint/BarrettReduction.hpp
 ---
+- 参考実装：[AC Library](https://github.com/atcoder/ac-library/blob/master/atcoder/internal_math.hpp)
 
 同じ mod でたくさん計算するとき，剰余算を掛け算に変換して高速化する．
 
 ## 使い方
 
 - `Barrett br(mod)`：Barrett Reduction を準備する．
     - 制約：`mod < 2^32`
-- `br.mul(u64 a, u64 b)`：`a * b % mod` を計算する．
+- `u64 br.mul(u64 a, u64 b)`：`a * b % mod` を計算する．
     - 制約：`a * b < 2^64`
 
 ## 仕組み

diff --git a/docs/modint/modint.md b/docs/modint/modint.md
@@ -2,6 +2,7 @@
 title: Modint
 documentation_of: //src/modint/modint.hpp
 ---
+- 参考実装：[AC Library (初期実装)](https://github.com/atcoder/ac-library/blob/8250de484ae0ab597391db58040a602e0dc1a419/atcoder/convolution.hpp)
 
 - タイプ速度重視の簡易 modint
 - 速度が欲しいとき：[32 bit で加減算をちゃんと書いた Modint](../extra/modint_fast.hpp)

diff --git a/notebook.pdf b/notebook.pdf
diff --git a/src/math/FastGCD.hpp → src/math/BinaryGCD.hpp b/src/math/FastGCD.hpp → src/math/BinaryGCD.hpp
diff --git a/test/math/FastGCD.test.cpp → test/math/BinaryGCD.test.cpp b/test/math/FastGCD.test.cpp → test/math/BinaryGCD.test.cpp
@@ -1,13 +1,15 @@
 #define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A"
 #include "test/template.hpp"
 using u64 = uint64_t;
-#include "src/math/FastGCD.hpp"
+#include "src/math/BinaryGCD.hpp"
 
 int main() {
    mt19937_64 rnd;
-   rep(i, 0, (ll)1e7) {
-      u64 a = rnd(), b = rnd();
-      assert(gcd(a, b) == binary_gcd(a, b));
+   rep(shift, 0, 64) {
+      rep(i, 0, (ll)1e5) {
+         u64 a = rnd() >> shift, b = rnd() >> shift;
+         assert(gcd(a, b) == binary_gcd(a, b));
+      }
    }
    puts("Hello World");
 }