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docs(leetcode/69): update README and implementation notes for Claude 4.6 extended
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‎Algorithm/BinarySearch/leetcode/69. Sqrt(x)/Claude4.6 extended/README.md‎

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## 目次
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- [概要](#overview)
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- [アルゴリズム要点 TL;DR](#tldr)
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- [図解](#figures)
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- [正しさのスケッチ](#correctness)
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- [計算量](#complexity)
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- [Python 実装](#impl)
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- [CPython 最適化ポイント](#cpython)
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- [エッジケースと検証観点](#edgecases)
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- [FAQ](#faq)
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- [Overview (概要)](#overview)
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- [Algorithm (アルゴリズム)](#algorithm)
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- [Complexity (計算量)](#complexity)
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- [Implementation (実装)](#implementation)
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- [Optimization (最適化)](#optimization)
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---
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<h2 id="overview">概要</h2>
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<h2 id="overview">Overview (概要)</h2>
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### 問題要約
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@@ -64,7 +60,7 @@ class Solution:
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---
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<h2 id="figures">図解</h2>
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### 図解
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### フローチャート
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---
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<h2 id="correctness">正しさのスケッチ</h2>
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### 正しさのスケッチ
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### ループ不変条件（Loop Invariant）
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ループの各反復前に以下が成立する：
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> `(low - 1)^2 ≤ x` かつ `(high + 1)^2 > x`
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つまり、**真の答え `floor(√x)` は常に `[low, high]` の閉区間内に存在する**。
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この不変条件により、**真の答え `floor(√x)` は常に `[low - 1, high]` の範囲に含まれる**ことが保証される。ループ終了時 (`low > high`) には `high = floor(√x)` が確定する。
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| フェーズ | 不変条件の確認 |
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| -------------- | -------------------------------------------------------------------------- |
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---
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<h2 id="complexity">計算量</h2>
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<h2 id="complexity">Complexity (計算量)</h2>
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| 観点 | 計算量 | 補足 |
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| -------------- | -------- | --------------------------------------------- |
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---
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<h2 id="impl">Python 実装</h2>
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<h2 id="implementation">Implementation (実装)</h2>
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### Python 実装
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```python
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from __future__ import annotations

‎Algorithm/BinarySearch/leetcode/69. Sqrt(x)/Claude4.6 extended/README_react.html‎

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<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.29.0/prism.min.js"></script>
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<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.29.0/components/prism-python.min.js"></script>
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<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.29.0/components/prism-typescript.min.js"></script>
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<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.29.0/components/prism-rust.min.js"></script>
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<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.29.0/plugins/line-numbers/prism-line-numbers.min.js"></script>
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<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.29.0/plugins/toolbar/prism-toolbar.min.js"></script>
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<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.29.0/plugins/copy-to-clipboard/prism-copy-to-clipboard.min.js"></script>
@@ -350,9 +351,9 @@ <h3 class="outfit font-bold text-slate-800 text-base mb-3">入出力例</h3>
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// エッジケース: x=0, x=1 は即リターン
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if (x < 2) return x;
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// 探索範囲: [1, x >> 1]
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// 探索範囲: [1, x >>> 1]
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let low: number = 1;
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let high: number = x >> 1; // x / 2（ビットシフト整数除算）
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let high: number = x >>> 1; // x / 2（符号なし右シフト整数除算）
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// Loop Invariant:
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// low - 1 の二乗は x 以下
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function TraceTable({ x }) {
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const rows = [];
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let low = 1,
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high = x >> 1;
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let iter = 0;
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while (low <= high && iter < 20) {
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iter++;
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const mid = (low + high) >> 1;
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const sq = mid * mid;
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let action = '';
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if (sq === x) {
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rows.push({ iter, low, high, mid, sq, action: 'return mid ✓' });
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break;
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} else if (sq < x) {
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action = `low = ${mid + 1}`;
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rows.push({ iter, low, high, mid, sq, action });
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low = mid + 1;
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} else {
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action = `high = ${mid - 1}`;
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rows.push({ iter, low, high, mid, sq, action });
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high = mid - 1;
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}
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}
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if (
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rows.length === 0 ||
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rows[rows.length - 1].action.startsWith('low') ||
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rows[rows.length - 1].action.startsWith('high')
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) {
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// 早期リターンケース: x < 2
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if (x < 2) {
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rows.push({
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iter: iter + 1,
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low,
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high,
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iter: 0,
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low: '-',
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high: '-',
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mid: '-',
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sq: '-',
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action: `return high=${high} ✓`,
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action: `x < 2 → return ${x} ✓`,
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});
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} else {
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let low = 1,
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high = x >>> 1;
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let iter = 0;
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// 二分探索ループ: 実装と同じ終了条件 (while low <= high)
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while (low <= high) {
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iter++;
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const mid = (low + high) >>> 1;
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const sq = mid * mid;
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let action = '';
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if (sq === x) {
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rows.push({ iter, low, high, mid, sq, action: `return mid=${mid} ✓` });
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break;
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} else if (sq < x) {
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action = `low = ${mid + 1}`;
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rows.push({ iter, low, high, mid, sq, action });
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low = mid + 1;
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} else {
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action = `high = ${mid - 1}`;
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rows.push({ iter, low, high, mid, sq, action });
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high = mid - 1;
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}
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}
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// ループ終了後: high = floor(√x)
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// 完全平方数でない場合のみ最終行を追加
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if (rows.length === 0 || !rows[rows.length - 1].action.includes('return mid')) {
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rows.push({
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iter: iter + 1,
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low,
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high,
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mid: '-',
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sq: '-',
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action: `return high=${high} ✓`,
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});
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}
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}
10021018

10031019
return (

‎Algorithm/BinarySearch/leetcode/69. Sqrt(x)/Claude4.6 extended/Sqrt(x)_rust.md‎

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## 競技プログラミング視点での分析
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- 探索空間 `[0, x]` は単調増加 → **二分探索**が最適（最大31回のイテレーション）
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- `u32::MAX = 2³¹ - 1 = 2147483647` → `mid * mid` は最大 `u64` で `~4.6 × 10¹⁸` に達するため、**`u64` へのキャスト**でオーバーフローを防止
6+
- `u32::MAX = 2³¹ - 1 = 2147483647` → `mid` の最大値は `x >> 1` で約 `2³⁰`、よって `mid * mid` は最大 `(2³⁰)² ≈ 1.15×10¹⁸` に達するため、**`u64` へのキャスト**でオーバーフローを防止
77
- スタックのみ使用・ヒープアロケーション完全ゼロ → キャッシュに最適
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- 全変数が `Copy` 型（`u32`, `u64`） → 借用・所有権の複雑性なし
99

‎Algorithm/BinarySearch/leetcode/69. Sqrt(x)/Claude4.6 extended/Sqrt(x)_typescript.md‎

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@@ -86,9 +86,9 @@ function mySqrt(x: number): number {
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let high: number = x >> 1; // == Math.floor(x / 2), ビットシフトで整数除算
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// Loop Invariant:
89-
// low の時点では low * low <= x が未確認
90-
// high の時点では high * high >= x が未確認
91-
// → 最終的に low > high になった時点で high = floor(√x)
89+
// (low - 1) * (low - 1) <= x (lowより小さい値の二乗はx以下)
90+
// (high + 1) * (high + 1) > x (highより大きい値の二乗はxより大きい)
91+
// → この不変条件により、ループ終了時に high = floor(√x) が保証される
9292
while (low <= high) {
9393
// オーバーフロー防止: (low + high) >>> 1
9494
const mid: number = (low + high) >>> 1;
@@ -107,8 +107,10 @@ function mySqrt(x: number): number {
107107
}
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109109
// ループ終了後、high = floor(√x)
110-
// 例: x=8 → mid=3(9>8) → high=2, mid=1(1<8) → low=2
111-
// low=high=2 → mid=2(4<8) → low=3 → 終了: high=2 ✓
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// 例: x=8 → 初期 low=1, high=4
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// Iter1: mid=2, sq=4<8 → low=3
112+
// Iter2: mid=3, sq=9>8 → high=2
113+
// 終了: low(3) > high(2) → return high=2 ✓
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return high;
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}
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```

‎public/Algorithm/BinarySearch/leetcode/69. Sqrt(x)/Claude4.6 extended/README_react.html‎

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<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.29.0/prism.min.js"></script>
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<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.29.0/components/prism-python.min.js"></script>
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<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.29.0/components/prism-typescript.min.js"></script>
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<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.29.0/components/prism-rust.min.js"></script>
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<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.29.0/plugins/line-numbers/prism-line-numbers.min.js"></script>
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<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.29.0/plugins/toolbar/prism-toolbar.min.js"></script>
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<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.29.0/plugins/copy-to-clipboard/prism-copy-to-clipboard.min.js"></script>
@@ -350,9 +351,9 @@ <h3 class="outfit font-bold text-slate-800 text-base mb-3">入出力例</h3>
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// エッジケース: x=0, x=1 は即リターン
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if (x < 2) return x;
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// 探索範囲: [1, x >> 1]
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// 探索範囲: [1, x >>> 1]
354355
let low: number = 1;
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let high: number = x >> 1; // x / 2（ビットシフト整数除算）
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let high: number = x >>> 1; // x / 2（符号なし右シフト整数除算）
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// Loop Invariant:
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// low - 1 の二乗は x 以下
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function TraceTable({ x }) {
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const rows = [];
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let low = 1,
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high = x >> 1;
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let iter = 0;
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while (low <= high && iter < 20) {
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iter++;
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const mid = (low + high) >> 1;
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const sq = mid * mid;
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let action = '';
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if (sq === x) {
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rows.push({ iter, low, high, mid, sq, action: 'return mid ✓' });
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break;
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} else if (sq < x) {
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action = `low = ${mid + 1}`;
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rows.push({ iter, low, high, mid, sq, action });
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low = mid + 1;
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} else {
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action = `high = ${mid - 1}`;
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rows.push({ iter, low, high, mid, sq, action });
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high = mid - 1;
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}
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}
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if (
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rows.length === 0 ||
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rows[rows.length - 1].action.startsWith('low') ||
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rows[rows.length - 1].action.startsWith('high')
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) {
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// 早期リターンケース: x < 2
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if (x < 2) {
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rows.push({
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iter: iter + 1,
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low,
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high,
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iter: 0,
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low: '-',
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high: '-',
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mid: '-',
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sq: '-',
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action: `return high=${high} ✓`,
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action: `x < 2 → return ${x} ✓`,
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});
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} else {
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let low = 1,
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high = x >>> 1;
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let iter = 0;
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// 二分探索ループ: 実装と同じ終了条件 (while low <= high)
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while (low <= high) {
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iter++;
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const mid = (low + high) >>> 1;
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const sq = mid * mid;
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let action = '';
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if (sq === x) {
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rows.push({ iter, low, high, mid, sq, action: `return mid=${mid} ✓` });
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break;
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} else if (sq < x) {
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action = `low = ${mid + 1}`;
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rows.push({ iter, low, high, mid, sq, action });
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low = mid + 1;
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} else {
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action = `high = ${mid - 1}`;
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rows.push({ iter, low, high, mid, sq, action });
1001+
high = mid - 1;
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}
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}
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// ループ終了後: high = floor(√x)
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// 完全平方数でない場合のみ最終行を追加
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if (rows.length === 0 || !rows[rows.length - 1].action.includes('return mid')) {
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rows.push({
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iter: iter + 1,
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low,
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high,
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mid: '-',
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sq: '-',
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action: `return high=${high} ✓`,
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});
1016+
}
10011017
}
10021018

10031019
return (

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