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7.2 &nbsp; 二元樹走訪
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7.3 &nbsp; 二元樹陣列表示
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7.4 &nbsp; 二元搜尋樹
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7.5 &nbsp; AVL *
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第 8 章 &nbsp; 堆積
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第 9 章 &nbsp; 圖
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第 9 章 &nbsp; 圖
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9.1 &nbsp; 圖
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9.2 &nbsp; 圖基礎操作
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9.3 &nbsp; 圖的走訪
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第 10 章 &nbsp; 搜尋
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10.1 &nbsp; 二分搜尋
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10.2 &nbsp; 二分搜尋插入點
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10.3 &nbsp; 二分搜尋邊界
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10.4 &nbsp; 雜湊最佳化策略
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10.5 &nbsp; 重識搜尋演算法
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第 11 章 &nbsp; 排序
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11.1 &nbsp; 排序演算法
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11.2 &nbsp; 選擇排序
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11.3 &nbsp; 泡沫排序
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11.4 &nbsp; 插入排序
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11.5 &nbsp; 快速排序
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11.6 &nbsp; 合併排序
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11.7 &nbsp; 堆積排序
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11.8 &nbsp; 桶排序
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11.9 &nbsp; 計數排序
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11.10 &nbsp; 基數排序
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11.11 &nbsp; 小結
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第 12 章 &nbsp; 分治
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12.1 &nbsp; 分治演算法
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12.3 &nbsp; 構建樹問題
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12.4 &nbsp; 河內塔問題
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12.5 &nbsp; 小結
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第 13 章 &nbsp; 回溯
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13.2 &nbsp; 全排列問題
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13.3 &nbsp; 子集和問題
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13.4 &nbsp; N 皇后問題
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第 14 章 &nbsp; 動態規劃
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14.1 &nbsp; 初探動態規劃
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14.2 &nbsp; DP 問題特性
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14.3 &nbsp; DP 解題思路
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14.4 &nbsp; 0-1 背包問題
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<h1 id="64">6.4 &nbsp; 小結<a class="headerlink" href="#64" title="Permanent link">&para;</a></h1>
<h3 id="1">1. &nbsp; 重點回顧<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
<ul>
<li>輸入 <code>key</code> ，雜湊表能夠在 <span class="arithmatex">\(O(1)\)</span> 時間內查詢到 <code>value</code> ，效率非常高。</li>
<li>常見的雜湊表操作包括查詢、新增鍵值對、刪除鍵值對和走訪雜湊表等。</li>
<li>雜湊函式將 <code>key</code> 對映為陣列索引，從而訪問對應桶並獲取 <code>value</code> 。</li>
<li>兩個不同的 <code>key</code> 可能在經過雜湊函式後得到相同的陣列索引，導致查詢結果出錯，這種現象被稱為雜湊衝突。</li>
<li>雜湊表容量越大，雜湊衝突的機率就越低。因此可以透過擴容雜湊表來緩解雜湊衝突。與陣列擴容類似，雜湊表擴容操作的開銷很大。</li>
<li>負載因子定義為雜湊表中元素數量除以桶數量，反映了雜湊衝突的嚴重程度，常用作觸發雜湊表擴容的條件。</li>
<li>鏈式位址透過將單個元素轉化為鏈結串列，將所有衝突元素儲存在同一個鏈結串列中。然而，鏈結串列過長會降低查詢效率，可以透過進一步將鏈結串列轉換為紅黑樹來提高效率。</li>
<li>開放定址透過多次探測來處理雜湊衝突。線性探查使用固定步長，缺點是不能刪除元素，且容易產生聚集。多次雜湊使用多個雜湊函式進行探測，相較線性探查更不易產生聚集，但多個雜湊函式增加了計算量。</li>
<li>不同程式語言採取了不同的雜湊表實現。例如，Java 的 <code>HashMap</code> 使用鏈式位址，而 Python 的 <code>Dict</code> 採用開放定址。</li>
<li>在雜湊表中，我們希望雜湊演算法具有確定性、高效率和均勻分佈的特點。在密碼學中，雜湊演算法還應該具備抗碰撞性和雪崩效應。</li>
<li>雜湊演算法通常採用大質數作為模數，以最大化地保證雜湊值均勻分佈，減少雜湊衝突。</li>
<li>常見的雜湊演算法包括 MD5、SHA-1、SHA-2 和 SHA-3 等。MD5 常用於校驗檔案完整性，SHA-2 常用於安全應用與協議。</li>
<li>程式語言通常會為資料型別提供內建雜湊演算法，用於計算雜湊表中的桶索引。通常情況下，只有不可變物件是可雜湊的。</li>
</ul>
<h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
<p><strong>Q</strong>：雜湊表的時間複雜度在什麼情況下是 <span class="arithmatex">\(O(n)\)</span> ？</p>
<p>當雜湊衝突比較嚴重時，雜湊表的時間複雜度會退化至 <span class="arithmatex">\(O(n)\)</span> 。當雜湊函式設計得比較好、容量設定比較合理、衝突比較平均時，時間複雜度是 <span class="arithmatex">\(O(1)\)</span> 。我們使用程式語言內建的雜湊表時，通常認為時間複雜度是 <span class="arithmatex">\(O(1)\)</span> 。</p>
<p><strong>Q</strong>：為什麼不使用雜湊函式 <span class="arithmatex">\(f(x) = x\)</span> 呢？這樣就不會有衝突了。</p>
<p>在 <span class="arithmatex">\(f(x) = x\)</span> 雜湊函式下，每個元素對應唯一的桶索引，這與陣列等價。然而，輸入空間通常遠大於輸出空間（陣列長度），因此雜湊函式的最後一步往往是對陣列長度取模。換句話說，雜湊表的目標是將一個較大的狀態空間對映到一個較小的空間，並提供 <span class="arithmatex">\(O(1)\)</span> 的查詢效率。</p>
<p><strong>Q</strong>：雜湊表底層實現是陣列、鏈結串列、二元樹，但為什麼效率可以比它們更高呢？</p>
<p>首先，雜湊表的時間效率變高，但空間效率變低了。雜湊表有相當一部分記憶體未使用。</p>
<p>其次，只是在特定使用場景下時間效率變高了。如果一個功能能夠在相同的時間複雜度下使用陣列或鏈結串列實現，那麼通常比雜湊表更快。這是因為雜湊函式計算需要開銷，時間複雜度的常數項更大。</p>
<p>最後，雜湊表的時間複雜度可能發生劣化。例如在鏈式位址中，我們採取在鏈結串列或紅黑樹中執行查詢操作，仍然有退化至 <span class="arithmatex">\(O(n)\)</span> 時間的風險。</p>
<p><strong>Q</strong>：多次雜湊有不能直接刪除元素的缺陷嗎？標記為已刪除的空間還能再次使用嗎？</p>
<p>多次雜湊是開放定址的一種，開放定址法都有不能直接刪除元素的缺陷，需要透過標記刪除。標記為已刪除的空間可以再次使用。當將新元素插入雜湊表，並且透過雜湊函式找到標記為已刪除的位置時，該位置可以被新元素使用。這樣做既能保持雜湊表的探測序列不變，又能保證雜湊表的空間使用率。</p>
<p><strong>Q</strong>：為什麼線上性探查中，查詢元素的時候會出現雜湊衝突呢？</p>
<p>查詢的時候透過雜湊函式找到對應的桶和鍵值對，發現 <code>key</code> 不匹配，這就代表有雜湊衝突。因此，線性探查法會根據預先設定的步長依次向下查詢，直至找到正確的鍵值對或無法找到跳出為止。</p>
<p><strong>Q</strong>：為什麼雜湊表擴容能夠緩解雜湊衝突？</p>
<p>雜湊函式的最後一步往往是對陣列長度 <span class="arithmatex">\(n\)</span> 取模（取餘），讓輸出值落在陣列索引範圍內；在擴容後，陣列長度 <span class="arithmatex">\(n\)</span> 發生變化，而 <code>key</code> 對應的索引也可能發生變化。原先落在同一個桶的多個 <code>key</code> ，在擴容後可能會被分配到多個桶中，從而實現雜湊衝突的緩解。</p>
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