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第 11 章 &nbsp; 排序
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<h1 id="105">10.5 &nbsp; 重識搜尋演算法<a class="headerlink" href="#105" title="Permanent link">&para;</a></h1>
<p><u>搜尋演算法（searching algorithm）</u>用於在資料結構（例如陣列、鏈結串列、樹或圖）中搜索一個或一組滿足特定條件的元素。</p>
<p>搜尋演算法可根據實現思路分為以下兩類。</p>
<ul>
<li><strong>透過走訪資料結構來定位目標元素</strong>，例如陣列、鏈結串列、樹和圖的走訪等。</li>
<li><strong>利用資料組織結構或資料包含的先驗資訊，實現高效元素查詢</strong>，例如二分搜尋、雜湊查詢和二元搜尋樹查詢等。</li>
</ul>
<p>不難發現，這些知識點都已在前面的章節中介紹過，因此搜尋演算法對於我們來說並不陌生。在本節中，我們將從更加系統的視角切入，重新審視搜尋演算法。</p>
<h2 id="1051">10.5.1 &nbsp; 暴力搜尋<a class="headerlink" href="#1051" title="Permanent link">&para;</a></h2>
<p>暴力搜尋透過走訪資料結構的每個元素來定位目標元素。</p>
<ul>
<li>“線性搜尋”適用於陣列和鏈結串列等線性資料結構。它從資料結構的一端開始，逐個訪問元素，直到找到目標元素或到達另一端仍沒有找到目標元素為止。</li>
<li>“廣度優先搜尋”和“深度優先搜尋”是圖和樹的兩種走訪策略。廣度優先搜尋從初始節點開始逐層搜尋，由近及遠地訪問各個節點。深度優先搜尋從初始節點開始，沿著一條路徑走到頭，再回溯並嘗試其他路徑，直到走訪完整個資料結構。</li>
</ul>
<p>暴力搜尋的優點是簡單且通用性好，<strong>無須對資料做預處理和藉助額外的資料結構</strong>。</p>
<p>然而，<strong>此類演算法的時間複雜度為 <span class="arithmatex">\(O(n)\)</span></strong> ，其中 <span class="arithmatex">\(n\)</span> 為元素數量，因此在資料量較大的情況下效能較差。</p>
<h2 id="1052">10.5.2 &nbsp; 自適應搜尋<a class="headerlink" href="#1052" title="Permanent link">&para;</a></h2>
<p>自適應搜尋利用資料的特有屬性（例如有序性）來最佳化搜尋過程，從而更高效地定位目標元素。</p>
<ul>
<li>“二分搜尋”利用資料的有序性實現高效查詢，僅適用於陣列。</li>
<li>“雜湊查詢”利用雜湊表將搜尋資料和目標資料建立為鍵值對對映，從而實現查詢操作。</li>
<li>“樹查詢”在特定的樹結構（例如二元搜尋樹）中，基於比較節點值來快速排除節點，從而定位目標元素。</li>
</ul>
<p>此類演算法的優點是效率高，<strong>時間複雜度可達到 <span class="arithmatex">\(O(\log n)\)</span> 甚至 <span class="arithmatex">\(O(1)\)</span></strong> 。</p>
<p>然而，<strong>使用這些演算法往往需要對資料進行預處理</strong>。例如，二分搜尋需要預先對陣列進行排序，雜湊查詢和樹查詢都需要藉助額外的資料結構，維護這些資料結構也需要額外的時間和空間開銷。</p>
<div class="admonition tip">
<p class="admonition-title">Tip</p>
<p>自適應搜尋演算法常被稱為查詢演算法，<strong>主要用於在特定資料結構中快速檢索目標元素</strong>。</p>
</div>
<h2 id="1053">10.5.3 &nbsp; 搜尋方法選取<a class="headerlink" href="#1053" title="Permanent link">&para;</a></h2>
<p>給定大小為 <span class="arithmatex">\(n\)</span> 的一組資料，我們可以使用線性搜尋、二分搜尋、樹查詢、雜湊查詢等多種方法從中搜索目標元素。各個方法的工作原理如圖 10-11 所示。</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="多種搜尋策略" class="animation-figure" src="../searching_algorithm_revisited.assets/searching_algorithms.png" /></a></p>
<p align="center"> 圖 10-11 &nbsp; 多種搜尋策略 </p>
<p>上述幾種方法的操作效率與特性如表 10-1 所示。</p>
<p align="center"> 表 10-1 &nbsp; 查詢演算法效率對比 </p>
<div class="center-table">
<table>
<thead>
<tr>
<th></th>
<th>線性搜尋</th>
<th>二分搜尋</th>
<th>樹查詢</th>
<th>雜湊查詢</th>
</tr>
</thead>
<tbody>
<tr>
<td>查詢元素</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>插入元素</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>刪除元素</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>額外空間</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>資料預處理</td>
<td>/</td>
<td>排序 <span class="arithmatex">\(O(n \log n)\)</span></td>
<td>建樹 <span class="arithmatex">\(O(n \log n)\)</span></td>
<td>建雜湊表 <span class="arithmatex">\(O(n)\)</span></td>
</tr>
<tr>
<td>資料是否有序</td>
<td>無序</td>
<td>有序</td>
<td>有序</td>
<td>無序</td>
</tr>
</tbody>
</table>
</div>
<p>搜尋演算法的選擇還取決於資料體量、搜尋效能要求、資料查詢與更新頻率等。</p>
<p><strong>線性搜尋</strong></p>
<ul>
<li>通用性較好，無須任何資料預處理操作。假如我們僅需查詢一次資料，那麼其他三種方法的資料預處理的時間比線性搜尋的時間還要更長。</li>
<li>適用於體量較小的資料，此情況下時間複雜度對效率影響較小。</li>
<li>適用於資料更新頻率較高的場景，因為該方法不需要對資料進行任何額外維護。</li>
</ul>
<p><strong>二分搜尋</strong></p>
<ul>
<li>適用於大資料量的情況，效率表現穩定，最差時間複雜度為 <span class="arithmatex">\(O(\log n)\)</span> 。</li>
<li>資料量不能過大，因為儲存陣列需要連續的記憶體空間。</li>
<li>不適用於高頻增刪資料的場景，因為維護有序陣列的開銷較大。</li>
</ul>
<p><strong>雜湊查詢</strong></p>
<ul>
<li>適合對查詢效能要求很高的場景，平均時間複雜度為 <span class="arithmatex">\(O(1)\)</span> 。</li>
<li>不適合需要有序資料或範圍查詢的場景，因為雜湊表無法維護資料的有序性。</li>
<li>對雜湊函式和雜湊衝突處理策略的依賴性較高，具有較大的效能劣化風險。</li>
<li>不適合資料量過大的情況，因為雜湊表需要額外空間來最大程度地減少衝突，從而提供良好的查詢效能。</li>
</ul>
<p><strong>樹查詢</strong></p>
<ul>
<li>適用於海量資料，因為樹節點在記憶體中是分散儲存的。</li>
<li>適合需要維護有序資料或範圍查詢的場景。</li>
<li>在持續增刪節點的過程中，二元搜尋樹可能產生傾斜，時間複雜度劣化至 <span class="arithmatex">\(O(n)\)</span> 。</li>
<li>若使用 AVL 樹或紅黑樹，則各項操作可在 <span class="arithmatex">\(O(\log n)\)</span> 效率下穩定執行，但維護樹平衡的操作會增加額外的開銷。</li>
</ul>
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