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第 8 章 &nbsp; ヒープ
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第 9 章 &nbsp; グラフ
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第 10 章 &nbsp; 探索
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第 11 章 &nbsp; ソート
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12.1 &nbsp; 分割統治アルゴリズム
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第 13 章 &nbsp; バックトラッキング
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第 14 章 &nbsp; 動的プログラミング
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<h1 id="105">10.5 &nbsp; 探索アルゴリズムの再検討<a class="headerlink" href="#105" title="Permanent link">&para;</a></h1>
<p><u>探索アルゴリズム（検索アルゴリズム）</u>は、配列、連結リスト、木、グラフなどのデータ構造内で特定の基準を満たす1つ以上の要素を取得するために使用されます。</p>
<p>探索アルゴリズムは、そのアプローチに基づいて以下の2つのカテゴリに分けることができます。</p>
<ul>
<li><strong>データ構造を横断することで目標要素を特定する</strong>：配列、連結リスト、木、グラフの横断など。</li>
<li><strong>データの組織構造や既存のデータを使用して効率的な要素探索を実現する</strong>：二分探索、ハッシュ探索、二分探索木探索など。</li>
</ul>
<p>これらのトピックは前の章で紹介されたため、私たちには馴染みのないものではありません。このセクションでは、より体系的な観点から探索アルゴリズムを再検討します。</p>
<h2 id="1051">10.5.1 &nbsp; 総当たり探索<a class="headerlink" href="#1051" title="Permanent link">&para;</a></h2>
<p>総当たり探索は、データ構造のすべての要素を横断することで目標要素を特定します。</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。データ構造の一端から開始し、目標要素が見つかるか、目標要素を見つけることなく他端に到達するまで、各要素に一つずつアクセスします。</li>
<li>「幅優先探索」と「深さ優先探索」は、グラフと木の2つの横断戦略です。幅優先探索は初期ノードから開始し、層ごと（左から右へ）に探索し、近くから遠くのノードにアクセスします。深さ優先探索は初期ノードから開始し、パスの終端（上から下へ）まで追跡し、その後バックトラックして他のパスを試し、データ構造全体が横断されるまで続行します。</li>
</ul>
<p>総当たり探索の利点は、その単純さと汎用性であり、**データの前処理や追加のデータ構造の助けが不要**です。</p>
<p>ただし、**このタイプのアルゴリズムの時間計算量は<span class="arithmatex">\(O(n)\)</span>**で、<span class="arithmatex">\(n\)</span>は要素数であるため、大規模なデータセットでは性能が悪くなります。</p>
<h2 id="1052">10.5.2 &nbsp; 適応的探索<a class="headerlink" href="#1052" title="Permanent link">&para;</a></h2>
<p>適応的探索は、データの固有の性質（順序など）を使用して探索プロセスを最適化し、それにより目標要素をより効率的に特定します。</p>
<ul>
<li>「二分探索」はデータの整列性を使用して効率的な探索を実現し、配列にのみ適用可能です。</li>
<li>「ハッシュ探索」はハッシュテーブルを使用して探索データと目標データの間にキーと値のマッピングを確立し、それによりクエリ操作を実装します。</li>
<li>特定の木構造（二分探索木など）での「木探索」は、ノード値の比較に基づいてノードを迅速に除外し、それにより目標要素を特定します。</li>
</ul>
<p>これらのアルゴリズムの利点は高効率であり、<strong>時間計算量が<span class="arithmatex">\(O(\log n)\)</span>または<span class="arithmatex">\(O(1)\)</span>にまで達します</strong>。</p>
<p>ただし、<strong>これらのアルゴリズムを使用するには、多くの場合データの前処理が必要です</strong>。例えば、二分探索では事前に配列をソートする必要があり、ハッシュ探索と木探索の両方で追加のデータ構造の助けが必要です。これらの構造を維持することも、時間と空間の面でより多くのオーバーヘッドが必要です。</p>
<div class="admonition tip">
<p class="admonition-title">Tip</p>
<p>適応的探索アルゴリズムは、多くの場合探索アルゴリズムと呼ばれ、<strong>主に特定のデータ構造内で目標要素を迅速に取得するために使用されます</strong>。</p>
</div>
<h2 id="1053">10.5.3 &nbsp; 探索方法の選択<a class="headerlink" href="#1053" title="Permanent link">&para;</a></h2>
<p>サイズ<span class="arithmatex">\(n\)</span>のデータセットが与えられた場合、線形探索、二分探索、木探索、ハッシュ探索、またはその他の方法を使用して目標要素を取得できます。これらの方法の動作原理を下図に示します。</p>
<p><a class="glightbox" href="../searching_algorithm_revisited.assets/searching_algorithms.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Various search strategies" class="animation-figure" src="../searching_algorithm_revisited.assets/searching_algorithms.png" /></a></p>
<p align="center"> 図 10-11 &nbsp; Various search strategies </p>
<p>前述の方法の特性と操作効率を以下の表に示します。</p>
<p align="center"> 表 10-1 &nbsp; 探索アルゴリズム効率の比較 </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>追加空間</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>データ前処理</td>
<td>/</td>
<td>ソート <span class="arithmatex">\(O(n \log n)\)</span></td>
<td>木構築 <span class="arithmatex">\(O(n \log n)\)</span></td>
<td>ハッシュテーブル構築 <span class="arithmatex">\(O(n)\)</span></td>
</tr>
<tr>
<td>データ順序性</td>
<td>無順序</td>
<td>順序</td>
<td>順序</td>
<td>無順序</td>
</tr>
</tbody>
</table>
</div>
<p>探索アルゴリズムの選択は、データ量、探索性能要件、データクエリと更新の頻度などにも依存します。</p>
<p><strong>線形探索</strong></p>
<ul>
<li>汎用性が良く、データ前処理操作が不要です。データを一度だけクエリする必要がある場合、他の3つの方法のデータ前処理時間は線形探索の時間よりも長くなります。</li>
<li>小容量のデータに適しており、時間計算量が効率に与える影響は小さいです。</li>
<li>データ更新が非常に頻繁なシナリオに適しています。この方法はデータの追加メンテナンスを必要としないためです。</li>
</ul>
<p><strong>二分探索</strong></p>
<ul>
<li>より大きなデータ量に適しており、安定した性能と最悪ケースの時間計算量<span class="arithmatex">\(O(\log n)\)</span>を持ちます。</li>
<li>ただし、データ量が大きすぎることはできません。配列の保存には連続したメモリ空間が必要だからです。</li>
<li>頻繁な追加と削除があるシナリオには適していません。順序付き配列の維持に多くのオーバーヘッドが発生するためです。</li>
</ul>
<p><strong>ハッシュ探索</strong></p>
<ul>
<li>高速クエリ性能が不可欠なシナリオに適しており、平均時間計算量は<span class="arithmatex">\(O(1)\)</span>です。</li>
<li>順序付きデータや範囲探索が必要なシナリオには適していません。ハッシュテーブルはデータの順序性を維持できないためです。</li>
<li>ハッシュ関数とハッシュ衝突処理戦略への依存度が高く、性能劣化のリスクが大きいです。</li>
<li>過度に大容量のデータには適していません。ハッシュテーブルは衝突を最小化し、良好なクエリ性能を提供するために追加の空間が必要だからです。</li>
</ul>
<p><strong>木探索</strong></p>
<ul>
<li>大容量データに適しています。木ノードはメモリ内に分散して保存されるためです。</li>
<li>順序付きデータの維持や範囲探索に適しています。</li>
<li>ノードの継続的な追加と削除により、二分探索木は偏る可能性があり、時間計算量が<span class="arithmatex">\(O(n)\)</span>に劣化する可能性があります。</li>
<li>AVL木や赤黒木を使用する場合、操作は<span class="arithmatex">\(O(\log n)\)</span>効率で安定して実行できますが、木のバランスを維持する操作により追加のオーバーヘッドが追加されます。</li>
</ul>
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