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第 12 章 &nbsp; 分治
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<h1 id="91">9.1 &nbsp; 圖<a class="headerlink" href="#91" title="Permanent link">&para;</a></h1>
<p><u>圖（graph）</u>是一種非線性資料結構，由<u>頂點（vertex）</u>和<u>邊（edge）</u>組成。我們可以將圖 <span class="arithmatex">\(G\)</span> 抽象地表示為一組頂點 <span class="arithmatex">\(V\)</span> 和一組邊 <span class="arithmatex">\(E\)</span> 的集合。以下示例展示了一個包含 5 個頂點和 7 條邊的圖。</p>
<div class="arithmatex">\[
\begin{aligned}
V &amp; = \{ 1, 2, 3, 4, 5 \} \newline
E &amp; = \{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \} \newline
G &amp; = \{ V, E \} \newline
\end{aligned}
\]</div>
<p>如果將頂點看作節點，將邊看作連線各個節點的引用（指標），我們就可以將圖看作一種從鏈結串列拓展而來的資料結構。如圖 9-1 所示，<strong>相較於線性關係（鏈結串列）和分治關係（樹），網路關係（圖）的自由度更高</strong>，因而更為複雜。</p>
<p><a class="glightbox" href="../graph.assets/linkedlist_tree_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="鏈結串列、樹、圖之間的關係" class="animation-figure" src="../graph.assets/linkedlist_tree_graph.png" /></a></p>
<p align="center"> 圖 9-1 &nbsp; 鏈結串列、樹、圖之間的關係 </p>
<h2 id="911">9.1.1 &nbsp; 圖的常見型別與術語<a class="headerlink" href="#911" title="Permanent link">&para;</a></h2>
<p>根據邊是否具有方向，可分為<u>無向圖（undirected graph）</u>和<u>有向圖（directed graph）</u>，如圖 9-2 所示。</p>
<ul>
<li>在無向圖中，邊表示兩頂點之間的“雙向”連線關係，例如微信或 QQ 中的“好友關係”。</li>
<li>在有向圖中，邊具有方向性，即 <span class="arithmatex">\(A \rightarrow B\)</span> 和 <span class="arithmatex">\(A \leftarrow B\)</span> 兩個方向的邊是相互獨立的，例如微博或抖音上的“關注”與“被關注”關係。</li>
</ul>
<p><a class="glightbox" href="../graph.assets/directed_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="有向圖與無向圖" class="animation-figure" src="../graph.assets/directed_graph.png" /></a></p>
<p align="center"> 圖 9-2 &nbsp; 有向圖與無向圖 </p>
<p>根據所有頂點是否連通，可分為<u>連通圖（connected graph）</u>和<u>非連通圖（disconnected graph）</u>，如圖 9-3 所示。</p>
<ul>
<li>對於連通圖，從某個頂點出發，可以到達其餘任意頂點。</li>
<li>對於非連通圖，從某個頂點出發，至少有一個頂點無法到達。</li>
</ul>
<p><a class="glightbox" href="../graph.assets/connected_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="連通圖與非連通圖" class="animation-figure" src="../graph.assets/connected_graph.png" /></a></p>
<p align="center"> 圖 9-3 &nbsp; 連通圖與非連通圖 </p>
<p>我們還可以為邊新增“權重”變數，從而得到如圖 9-4 所示的<u>有權圖（weighted graph）</u>。例如在《王者榮耀》等手遊中，系統會根據共同遊戲時間來計算玩家之間的“親密度”，這種親密度網路就可以用有權圖來表示。</p>
<p><a class="glightbox" href="../graph.assets/weighted_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="有權圖與無權圖" class="animation-figure" src="../graph.assets/weighted_graph.png" /></a></p>
<p align="center"> 圖 9-4 &nbsp; 有權圖與無權圖 </p>
<p>圖資料結構包含以下常用術語。</p>
<ul>
<li><u>鄰接（adjacency）</u>：當兩頂點之間存在邊相連時，稱這兩頂點“鄰接”。在圖 9-4 中，頂點 1 的鄰接頂點為頂點 2、3、5。</li>
<li><u>路徑（path）</u>：從頂點 A 到頂點 B 經過的邊構成的序列被稱為從 A 到 B 的“路徑”。在圖 9-4 中，邊序列 1-5-2-4 是頂點 1 到頂點 4 的一條路徑。</li>
<li><u>度（degree）</u>：一個頂點擁有的邊數。對於有向圖，<u>入度（in-degree）</u>表示有多少條邊指向該頂點，<u>出度（out-degree）</u>表示有多少條邊從該頂點指出。</li>
</ul>
<h2 id="912">9.1.2 &nbsp; 圖的表示<a class="headerlink" href="#912" title="Permanent link">&para;</a></h2>
<p>圖的常用表示方式包括“鄰接矩陣”和“鄰接表”。以下使用無向圖進行舉例。</p>
<h3 id="1">1. &nbsp; 鄰接矩陣<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
<p>設圖的頂點數量為 <span class="arithmatex">\(n\)</span> ，<u>鄰接矩陣（adjacency matrix）</u>使用一個 <span class="arithmatex">\(n \times n\)</span> 大小的矩陣來表示圖，每一行（列）代表一個頂點，矩陣元素代表邊，用 <span class="arithmatex">\(1\)</span> 或 <span class="arithmatex">\(0\)</span> 表示兩個頂點之間是否存在邊。</p>
<p>如圖 9-5 所示，設鄰接矩陣為 <span class="arithmatex">\(M\)</span>、頂點串列為 <span class="arithmatex">\(V\)</span> ，那麼矩陣元素 <span class="arithmatex">\(M[i, j] = 1\)</span> 表示頂點 <span class="arithmatex">\(V[i]\)</span> 到頂點 <span class="arithmatex">\(V[j]\)</span> 之間存在邊，反之 <span class="arithmatex">\(M[i, j] = 0\)</span> 表示兩頂點之間無邊。</p>
<p><a class="glightbox" href="../graph.assets/adjacency_matrix.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="圖的鄰接矩陣表示" class="animation-figure" src="../graph.assets/adjacency_matrix.png" /></a></p>
<p align="center"> 圖 9-5 &nbsp; 圖的鄰接矩陣表示 </p>
<p>鄰接矩陣具有以下特性。</p>
<ul>
<li>頂點不能與自身相連，因此鄰接矩陣主對角線元素沒有意義。</li>
<li>對於無向圖，兩個方向的邊等價，此時鄰接矩陣關於主對角線對稱。</li>
<li>將鄰接矩陣的元素從 <span class="arithmatex">\(1\)</span> 和 <span class="arithmatex">\(0\)</span> 替換為權重，則可表示有權圖。</li>
</ul>
<p>使用鄰接矩陣表示圖時，我們可以直接訪問矩陣元素以獲取邊，因此增刪查改操作的效率很高，時間複雜度均為 <span class="arithmatex">\(O(1)\)</span> 。然而，矩陣的空間複雜度為 <span class="arithmatex">\(O(n^2)\)</span> ，記憶體佔用較多。</p>
<h3 id="2">2. &nbsp; 鄰接表<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
<p><u>鄰接表（adjacency list）</u>使用 <span class="arithmatex">\(n\)</span> 個鏈結串列來表示圖，鏈結串列節點表示頂點。第 <span class="arithmatex">\(i\)</span> 個鏈結串列對應頂點 <span class="arithmatex">\(i\)</span> ，其中儲存了該頂點的所有鄰接頂點（與該頂點相連的頂點）。圖 9-6 展示了一個使用鄰接表儲存的圖的示例。</p>
<p><a class="glightbox" href="../graph.assets/adjacency_list.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="圖的鄰接表表示" class="animation-figure" src="../graph.assets/adjacency_list.png" /></a></p>
<p align="center"> 圖 9-6 &nbsp; 圖的鄰接表表示 </p>
<p>鄰接表僅儲存實際存在的邊，而邊的總數通常遠小於 <span class="arithmatex">\(n^2\)</span> ，因此它更加節省空間。然而，在鄰接表中需要透過走訪鏈結串列來查詢邊，因此其時間效率不如鄰接矩陣。</p>
<p>觀察圖 9-6 ，<strong>鄰接表結構與雜湊表中的“鏈式位址”非常相似，因此我們也可以採用類似的方法來最佳化效率</strong>。比如當鏈結串列較長時，可以將鏈結串列轉化為 AVL 樹或紅黑樹，從而將時間效率從 <span class="arithmatex">\(O(n)\)</span> 最佳化至 <span class="arithmatex">\(O(\log n)\)</span> ；還可以把鏈結串列轉換為雜湊表，從而將時間複雜度降至 <span class="arithmatex">\(O(1)\)</span> 。</p>
<h2 id="913">9.1.3 &nbsp; 圖的常見應用<a class="headerlink" href="#913" title="Permanent link">&para;</a></h2>
<p>如表 9-1 所示，許多現實系統可以用圖來建模，相應的問題也可以約化為圖計算問題。</p>
<p align="center"> 表 9-1 &nbsp; 現實生活中常見的圖 </p>
<div class="center-table">
<table>
<thead>
<tr>
<th></th>
<th>頂點</th>
<th>邊</th>
<th>圖計算問題</th>
</tr>
</thead>
<tbody>
<tr>
<td>社交網路</td>
<td>使用者</td>
<td>好友關係</td>
<td>潛在好友推薦</td>
</tr>
<tr>
<td>地鐵線路</td>
<td>站點</td>
<td>站點間的連通性</td>
<td>最短路線推薦</td>
</tr>
<tr>
<td>太陽系</td>
<td>星體</td>
<td>星體間的萬有引力作用</td>
<td>行星軌道計算</td>
</tr>
</tbody>
</table>
</div>
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