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第 8 章 &nbsp; 堆積
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第 9 章 &nbsp; 圖
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第 10 章 &nbsp; 搜尋
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10.2 &nbsp; 二分搜尋插入點
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第 11 章 &nbsp; 排序
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11.8 &nbsp; 桶排序
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第 12 章 &nbsp; 分治
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12.3 &nbsp; 構建樹問題
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第 13 章 &nbsp; 回溯
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<h1 id="121">12.1 &nbsp; 分治演算法<a class="headerlink" href="#121" title="Permanent link">&para;</a></h1>
<p><u>分治（divide and conquer）</u>，全稱分而治之，是一種非常重要且常見的演算法策略。分治通常基於遞迴實現，包括“分”和“治”兩個步驟。</p>
<ol>
<li><strong>分（劃分階段）</strong>：遞迴地將原問題分解為兩個或多個子問題，直至到達最小子問題時終止。</li>
<li><strong>治（合併階段）</strong>：從已知解的最小子問題開始，從底至頂地將子問題的解進行合併，從而構建出原問題的解。</li>
</ol>
<p>如圖 12-1 所示，“合併排序”是分治策略的典型應用之一。</p>
<ol>
<li><strong>分</strong>：遞迴地將原陣列（原問題）劃分為兩個子陣列（子問題），直到子陣列只剩一個元素（最小子問題）。</li>
<li><strong>治</strong>：從底至頂地將有序的子陣列（子問題的解）進行合併，從而得到有序的原陣列（原問題的解）。</li>
</ol>
<p><a class="glightbox" href="../divide_and_conquer.assets/divide_and_conquer_merge_sort.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="合併排序的分治策略" class="animation-figure" src="../divide_and_conquer.assets/divide_and_conquer_merge_sort.png" /></a></p>
<p align="center"> 圖 12-1 &nbsp; 合併排序的分治策略 </p>
<h2 id="1211">12.1.1 &nbsp; 如何判斷分治問題<a class="headerlink" href="#1211" title="Permanent link">&para;</a></h2>
<p>一個問題是否適合使用分治解決，通常可以參考以下幾個判斷依據。</p>
<ol>
<li><strong>問題可以分解</strong>：原問題可以分解成規模更小、類似的子問題，以及能夠以相同方式遞迴地進行劃分。</li>
<li><strong>子問題是獨立的</strong>：子問題之間沒有重疊，互不依賴，可以獨立解決。</li>
<li><strong>子問題的解可以合併</strong>：原問題的解透過合併子問題的解得來。</li>
</ol>
<p>顯然，合併排序滿足以上三個判斷依據。</p>
<ol>
<li><strong>問題可以分解</strong>：遞迴地將陣列（原問題）劃分為兩個子陣列（子問題）。</li>
<li><strong>子問題是獨立的</strong>：每個子陣列都可以獨立地進行排序（子問題可以獨立進行求解）。</li>
<li><strong>子問題的解可以合併</strong>：兩個有序子陣列（子問題的解）可以合併為一個有序陣列（原問題的解）。</li>
</ol>
<h2 id="1212">12.1.2 &nbsp; 透過分治提升效率<a class="headerlink" href="#1212" title="Permanent link">&para;</a></h2>
<p><strong>分治不僅可以有效地解決演算法問題，往往還可以提升演算法效率</strong>。在排序演算法中，快速排序、合併排序、堆積排序相較於選擇、冒泡、插入排序更快，就是因為它們應用了分治策略。</p>
<p>那麼，我們不禁發問：<strong>為什麼分治可以提升演算法效率，其底層邏輯是什麼</strong>？換句話說，將大問題分解為多個子問題、解決子問題、將子問題的解合併為原問題的解，這幾步的效率為什麼比直接解決原問題的效率更高？這個問題可以從操作數量和平行計算兩方面來討論。</p>
<h3 id="1">1. &nbsp; 操作數量最佳化<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
<p>以“泡沫排序”為例，其處理一個長度為 <span class="arithmatex">\(n\)</span> 的陣列需要 <span class="arithmatex">\(O(n^2)\)</span> 時間。假設我們按照圖 12-2 所示的方式，將陣列從中點處分為兩個子陣列，則劃分需要 <span class="arithmatex">\(O(n)\)</span> 時間，排序每個子陣列需要 <span class="arithmatex">\(O((n / 2)^2)\)</span> 時間，合併兩個子陣列需要 <span class="arithmatex">\(O(n)\)</span> 時間，總體時間複雜度為：</p>
<div class="arithmatex">\[
O(n + (\frac{n}{2})^2 \times 2 + n) = O(\frac{n^2}{2} + 2n)
\]</div>
<p><a class="glightbox" href="../divide_and_conquer.assets/divide_and_conquer_bubble_sort.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="劃分陣列前後的泡沫排序" class="animation-figure" src="../divide_and_conquer.assets/divide_and_conquer_bubble_sort.png" /></a></p>
<p align="center"> 圖 12-2 &nbsp; 劃分陣列前後的泡沫排序 </p>
<p>接下來，我們計算以下不等式，其左邊和右邊分別為劃分前和劃分後的操作總數：</p>
<div class="arithmatex">\[
\begin{aligned}
n^2 &amp; &gt; \frac{n^2}{2} + 2n \newline
n^2 - \frac{n^2}{2} - 2n &amp; &gt; 0 \newline
n(n - 4) &amp; &gt; 0
\end{aligned}
\]</div>
<p><strong>這意味著當 <span class="arithmatex">\(n &gt; 4\)</span> 時，劃分後的操作數量更少，排序效率應該更高</strong>。請注意，劃分後的時間複雜度仍然是平方階 <span class="arithmatex">\(O(n^2)\)</span> ，只是複雜度中的常數項變小了。</p>
<p>進一步想，<strong>如果我們把子陣列不斷地再從中點處劃分為兩個子陣列</strong>，直至子陣列只剩一個元素時停止劃分呢？這種思路實際上就是“合併排序”，時間複雜度為 <span class="arithmatex">\(O(n \log n)\)</span> 。</p>
<p>再思考，<strong>如果我們多設定幾個劃分點</strong>，將原陣列平均劃分為 <span class="arithmatex">\(k\)</span> 個子陣列呢？這種情況與“桶排序”非常類似，它非常適合排序海量資料，理論上時間複雜度可以達到 <span class="arithmatex">\(O(n + k)\)</span> 。</p>
<h3 id="2">2. &nbsp; 平行計算最佳化<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
<p>我們知道，分治生成的子問題是相互獨立的，<strong>因此通常可以並行解決</strong>。也就是說，分治不僅可以降低演算法的時間複雜度，<strong>還有利於作業系統的並行最佳化</strong>。</p>
<p>並行最佳化在多核或多處理器的環境中尤其有效，因為系統可以同時處理多個子問題，更加充分地利用計算資源，從而顯著減少總體的執行時間。</p>
<p>比如在圖 12-3 所示的“桶排序”中，我們將海量的資料平均分配到各個桶中，則可將所有桶的排序任務分散到各個計算單元，完成後再合併結果。</p>
<p><a class="glightbox" href="../divide_and_conquer.assets/divide_and_conquer_parallel_computing.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="桶排序的平行計算" class="animation-figure" src="../divide_and_conquer.assets/divide_and_conquer_parallel_computing.png" /></a></p>
<p align="center"> 圖 12-3 &nbsp; 桶排序的平行計算 </p>
<h2 id="1213">12.1.3 &nbsp; 分治常見應用<a class="headerlink" href="#1213" title="Permanent link">&para;</a></h2>
<p>一方面，分治可以用來解決許多經典演算法問題。</p>
<ul>
<li><strong>尋找最近點對</strong>：該演算法首先將點集分成兩部分，然後分別找出兩部分中的最近點對，最後找出跨越兩部分的最近點對。</li>
<li><strong>大整數乘法</strong>：例如 Karatsuba 演算法，它將大整數乘法分解為幾個較小的整數的乘法和加法。</li>
<li><strong>矩陣乘法</strong>：例如 Strassen 演算法，它將大矩陣乘法分解為多個小矩陣的乘法和加法。</li>
<li><strong>河內塔問題</strong>：河內塔問題可以透過遞迴解決，這是典型的分治策略應用。</li>
<li><strong>求解逆序對</strong>：在一個序列中，如果前面的數字大於後面的數字，那麼這兩個數字構成一個逆序對。求解逆序對問題可以利用分治的思想，藉助合併排序進行求解。</li>
</ul>
<p>另一方面，分治在演算法和資料結構的設計中應用得非常廣泛。</p>
<ul>
<li><strong>二分搜尋</strong>：二分搜尋是將有序陣列從中點索引處分為兩部分，然後根據目標值與中間元素值比較結果，決定排除哪一半區間，並在剩餘區間執行相同的二分操作。</li>
<li><strong>合併排序</strong>：本節開頭已介紹，不再贅述。</li>
<li><strong>快速排序</strong>：快速排序是選取一個基準值，然後把陣列分為兩個子陣列，一個子陣列的元素比基準值小，另一子陣列的元素比基準值大，再對這兩部分進行相同的劃分操作，直至子陣列只剩下一個元素。</li>
<li><strong>桶排序</strong>：桶排序的基本思想是將資料分散到多個桶，然後對每個桶內的元素進行排序，最後將各個桶的元素依次取出，從而得到一個有序陣列。</li>
<li><strong>樹</strong>：例如二元搜尋樹、AVL 樹、紅黑樹、B 樹、B+ 樹等，它們的查詢、插入和刪除等操作都可以視為分治策略的應用。</li>
<li><strong>堆積</strong>：堆積是一種特殊的完全二元樹，其各種操作，如插入、刪除和堆積化，實際上都隱含了分治的思想。</li>
<li><strong>雜湊表</strong>：雖然雜湊表並不直接應用分治，但某些雜湊衝突解決方案間接應用了分治策略，例如，鏈式位址中的長鏈結串列會被轉化為紅黑樹，以提升查詢效率。</li>
</ul>
<p>可以看出，<strong>分治是一種“潤物細無聲”的演算法思想</strong>，隱含在各種演算法與資料結構之中。</p>
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