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第 1 章   アルゴリズムとの出会い
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第 1 章   アルゴリズムとの出会い
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1.1   アルゴリズムはどこにでもある
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1.2   アルゴリズムとは
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1.3   まとめ
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第 2 章   計算量の解析
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第 2 章   計算量の解析
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2.1   アルゴリズムの効率の評価
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2.2   反復と再帰
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2.3   時間計算量
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2.4   空間計算量
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2.5   まとめ
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2.6   練習問題
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第 3 章   データ構造
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第 3 章   データ構造
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3.1   データ構造の分類
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3.2   基本データ型
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3.3   数値の符号化 *
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3.4   文字エンコーディング *
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3.5   まとめ
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3.6   練習問題
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第 4 章   配列と連結リスト
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第 4 章   配列と連結リスト
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4.1   配列
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4.2   連結リスト
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4.3   リスト
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4.4   メモリとキャッシュ *
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4.5   まとめ
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4.6   練習問題
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第 5 章   スタックとキュー
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第 5 章   スタックとキュー
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5.1   スタック
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5.2   キュー
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5.3   両端キュー
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5.4   まとめ
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5.5   練習問題
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第 6 章   ハッシュテーブル
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第 6 章   ハッシュテーブル
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6.1   ハッシュテーブル
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6.2   ハッシュ衝突
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6.3   ハッシュアルゴリズム
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6.4   まとめ
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6.5   練習問題
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第 7 章   木
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第 7 章   木
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7.1   二分木
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7.2   二分木の走査
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7.3   二分木の配列表現
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7.4   二分探索木
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7.5   AVL 木 *
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7.6   まとめ
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7.7   練習問題
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第 8 章   ヒープ
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第 8 章   ヒープ
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8.1   ヒープ
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8.2   ヒープの構築
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8.3   Top-k 問題
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8.4   まとめ
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8.5   練習問題
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第 9 章   グラフ
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第 9 章   グラフ
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9.1   グラフ
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9.2   グラフの基本操作
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9.3   グラフの走査
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9.4   まとめ
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9.5   練習問題
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第 10 章   探索
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第 10 章   探索
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10.1   二分探索
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10.2   二分探索で挿入位置を探す
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10.3   二分探索で境界を探す
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10.4   ハッシュを使った最適化
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10.5   探索アルゴリズムを改めて考える
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10.5   探索アルゴリズムを改めて考える
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10.5.1   全探索
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10.5.2   データの特性に応じた探索
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10.5.3   探索方法の選択
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10.6   まとめ
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10.7   練習問題
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第 11 章   ソート
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第 11 章   ソート
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11.1   ソートアルゴリズム
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11.2   選択ソート
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11.3   バブルソート
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11.4   挿入ソート
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11.5   クイックソート
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11.6   マージソート
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11.7   ヒープソート
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11.8   バケットソート
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11.9   カウントソート
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11.10   基数ソート
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11.11   まとめ
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11.12   練習問題
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第 12 章   分割統治
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第 12 章   分割統治
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12.1   分割統治アルゴリズム
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12.2   分割統治による探索戦略
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12.3   二分木の構築問題
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12.4   ハノイの塔の問題
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12.5   まとめ
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12.6   練習問題
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第 13 章   バックトラッキング
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第 13 章   バックトラッキング
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13.1   バックトラッキングアルゴリズム
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13.2   順列列挙問題
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13.3   部分集合和問題
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13.4   n クイーン問題
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13.5   まとめ
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13.6   練習問題
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第 14 章   動的計画法
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第 14 章   動的計画法
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14.1   動的計画法への第一歩
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14.2   動的計画法の問題の特徴
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14.3   動的計画法の解き方
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14.4   0-1 ナップサック問題
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14.5   個数制限なしナップサック問題
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14.6   編集距離問題
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14.7   まとめ
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14.8   練習問題
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第 15 章   貪欲法
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<h1 id="105">10.5   探索アルゴリズムを改めて考える<a class="headerlink" href="#105" title="この見出しへのリンク"></a></h1>
<p><u>探索アルゴリズム(searching algorithm)</u>は、配列、連結リスト、木、グラフなどのデータ構造から、特定の条件を満たす一つまたは複数の要素を探すために使います。</p>
<p>探索アルゴリズムは、実装の考え方によって次の二種類に分けられます。</p>
<ul>
<li><strong>データ構造を走査して目的の要素を見つける方法</strong>です。配列、連結リスト、木、グラフの走査などが該当します。</li>
<li><strong>データの構造や既知の情報を利用して、効率よく要素を探す方法</strong>です。二分探索、ハッシュ探索、二分探索木による探索などが該当します。</li>
</ul>
<p>これらはすでに前の章で紹介した内容なので、探索アルゴリズムは私たちにとってなじみのないものではありません。この節では、より体系的な視点から探索アルゴリズムを改めて考えます。</p>
<h2 id="1051">10.5.1   全探索<a class="headerlink" href="#1051" title="この見出しへのリンク"></a></h2>
<p>全探索は、データ構造の各要素を走査して目的の要素を見つけます。</p>
<ul>
<li>「線形探索」は、配列や連結リストなどの線形データ構造に適しています。一方の端から要素を順に訪問し、目的の要素を見つけるか、見つからないままもう一方の端に達するまで続けます。</li>
<li>「幅優先探索」と「深さ優先探索」は、グラフや木を走査する二つの戦略です。幅優先探索は開始ノードから層ごとに探索し、近いノードから遠いノードへと訪問します。深さ優先探索は開始ノードから一本の経路を行き止まりまで進み、戻って別の経路を試し、データ構造全体を走査するまで続けます。</li>
</ul>
<p>全探索の長所は単純で汎用性が高く、<strong>データの前処理や追加のデータ構造が不要なこと</strong>です。</p>
<p>一方、<strong>時間計算量は <span class="arithmatex">\(O(n)\)</span></strong> です。<span class="arithmatex">\(n\)</span> は要素数なので、データが多い場合は性能が低くなります。</p>
<h2 id="1052">10.5.2   データの特性に応じた探索<a class="headerlink" href="#1052" title="この見出しへのリンク"></a></h2>
<p>データ固有の性質(整列済みであることなど)を利用して探索過程を最適化し、目的の要素をより効率よく見つける方法です。</p>
<ul>
<li>「二分探索」は、データが整列済みであることを利用して効率よく探索します。配列にのみ適用できます。</li>
<li>「ハッシュ探索」は、ハッシュテーブルを利用し、検索するデータと目的のデータをキーと値のペアとして対応付けて照会します。</li>
<li>「木探索」は、二分探索木などの特定の木構造でノードの値を比較し、候補となるノードを素早く除外して目的の要素を見つけます。</li>
</ul>
<p>これらのアルゴリズムの長所は効率が高く、<strong>時間計算量が <span class="arithmatex">\(O(\log n)\)</span>、さらには <span class="arithmatex">\(O(1)\)</span> にもなること</strong>です。</p>
<p>ただし、<strong>利用にはデータの前処理が必要となることが多いです</strong>。例えば、二分探索では配列をあらかじめソートする必要があり、ハッシュ探索と木探索では追加のデータ構造が必要です。それらの維持にも時間と領域が余分にかかります。</p>
<div class="admonition tip">
<p class="admonition-title">ヒント</p>
<p>データの特性に応じた探索アルゴリズムは、検索アルゴリズムとも呼ばれ、<strong>主に特定のデータ構造から目的の要素を素早く取得するために使います</strong>。</p>
</div>
<h2 id="1053">10.5.3   探索方法の選択<a class="headerlink" href="#1053" title="この見出しへのリンク"></a></h2>
<p>大きさ <span class="arithmatex">\(n\)</span> のデータ集合から目的の要素を探すには、線形探索、二分探索、木探索、ハッシュ探索などの方法が使えます。それぞれの仕組みを図 10-11 に示します。</p>
<p><ha-svg-figure class="animation-figure" data-book-key="ja/chapter_searching/searching_algorithm_revisited.assets/searching_algorithms.png" data-dark-src="/assets/figure-svg/figures/681449de5f69a52a120c3a35.json" data-src="/assets/figure-svg/figures/ab6c4ef64198c3d41a086d7b.json" style=";--figure-ratio:960.0/540.0"><object data="/ja/chapter_searching/searching_algorithm_revisited.assets/searching_algorithms.svg" data-theme="light" tabindex="-1" type="image/svg+xml">さまざまな探索戦略</object><object data="/ja/chapter_searching/searching_algorithm_revisited.assets/searching_algorithms.dark.svg" data-theme="dark" tabindex="-1" type="image/svg+xml">さまざまな探索戦略</object></ha-svg-figure></p>
<p align="center"> 図 10-11   さまざまな探索戦略 </p>
<p>これらの方法の操作効率と特徴を表 10-1 に示します。</p>
<p align="center"> 表 10-1   探索アルゴリズムの効率の比較 </p>
<div class="center-table">
<table>
<thead>
<tr>
<th></th>
<th>線形探索</th>
<th>二分探索</th>
<th>木探索</th>
<th>ハッシュ探索</th>
</tr>
</thead>
<tbody>
<tr>
<td>要素の探索</td>
<td><span class="arithmatex">\(O(n)\)</span></td>
<td><span class="arithmatex">\(O(\log n)\)</span></td>
<td><span class="arithmatex">\(O(\log n)\)</span></td>
<td><span class="arithmatex">\(O(1)\)</span></td>
</tr>
<tr>
<td>要素の挿入</td>
<td><span class="arithmatex">\(O(1)\)</span></td>
<td><span class="arithmatex">\(O(n)\)</span></td>
<td><span class="arithmatex">\(O(\log n)\)</span></td>
<td><span class="arithmatex">\(O(1)\)</span></td>
</tr>
<tr>
<td>要素の削除</td>
<td><span class="arithmatex">\(O(n)\)</span></td>
<td><span class="arithmatex">\(O(n)\)</span></td>
<td><span class="arithmatex">\(O(\log n)\)</span></td>
<td><span class="arithmatex">\(O(1)\)</span></td>
</tr>
<tr>
<td>追加領域</td>
<td><span class="arithmatex">\(O(1)\)</span></td>
<td><span class="arithmatex">\(O(1)\)</span></td>
<td><span class="arithmatex">\(O(n)\)</span></td>
<td><span class="arithmatex">\(O(n)\)</span></td>
</tr>
<tr>
<td>データの前処理</td>
<td>/</td>
<td>ソート <span class="arithmatex">\(O(n \log n)\)</span></td>
<td>木の構築 <span class="arithmatex">\(O(n \log n)\)</span></td>
<td>ハッシュテーブルの構築 <span class="arithmatex">\(O(n)\)</span></td>
</tr>
<tr>
<td>データの整列状態</td>
<td>未整列</td>
<td>整列済み</td>
<td>整列済み</td>
<td>未整列</td>
</tr>
</tbody>
</table>
</div>
<p>探索アルゴリズムの選択は、規模、必要な探索性能、データの照会と更新の頻度などにも左右されます。</p>
<p><strong>線形探索</strong></p>
<ul>
<li>汎用性が高く、データの前処理が不要です。照会が一回だけなら、ほかの三つの方法では前処理の時間が線形探索の時間より長くなります。</li>
<li>データが少ない場合に適しています。この場合、時間計算量が効率に与える影響は小さくなります。</li>
<li>データを追加で管理する必要がないため、更新頻度が高い場合に適しています。</li>
</ul>
<p><strong>二分探索</strong></p>
<ul>
<li>データが多い場合に適しており、効率が安定しています。最悪時間計算量は <span class="arithmatex">\(O(\log n)\)</span> です。</li>
<li>配列の格納には連続したメモリ領域が必要なので、データが多すぎる場合には適しません。</li>
<li>整列済み配列の維持にかかる負担が大きいため、データを頻繁に追加・削除する場合には適しません。</li>
</ul>
<p><strong>ハッシュ探索</strong></p>
<ul>
<li>平均時間計算量が <span class="arithmatex">\(O(1)\)</span> なので、高い照会性能が求められる場合に適しています。</li>
<li>ハッシュテーブルはデータの順序を維持できないため、整列済みデータや範囲検索が必要な場合には適しません。</li>
<li>ハッシュ関数とハッシュ衝突への対処方法に大きく依存し、性能が悪化するリスクが比較的高くなります。</li>
<li>衝突をできるだけ減らして照会性能を高く保つために追加領域が必要なので、データが多すぎる場合には適しません。</li>
</ul>
<p><strong>木探索</strong></p>
<ul>
<li>木のノードはメモリ内に分散して格納されるため、膨大な量のデータにも適しています。</li>
<li>データの順序の維持や範囲検索が必要な場合に適しています。</li>
<li>ノードの追加・削除を続けるうちに二分探索木が偏り、時間計算量が <span class="arithmatex">\(O(n)\)</span> に悪化する可能性があります。</li>
<li>AVL 木や赤黒木を使えば、各操作を安定して <span class="arithmatex">\(O(\log n)\)</span> で実行できますが、平衡を維持するための追加の処理が必要になります。</li>
</ul>
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