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第 7 章 &nbsp; 樹
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7.1 &nbsp; 二元樹
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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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7.6 &nbsp; 小結
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第 8 章 &nbsp; 堆積
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第 8 章 &nbsp; 堆積
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8.1 &nbsp; 堆積
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8.2 &nbsp; 建堆積操作
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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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10.6 &nbsp; 小結
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第 11 章 &nbsp; 排序
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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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12.2 &nbsp; 分治搜尋策略
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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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13.1 &nbsp; 回溯演算法
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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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13.5 &nbsp; 小結
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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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<h1 id="34">3.4 &nbsp; 字元編碼 *<a class="headerlink" href="#34" title="Permanent link">&para;</a></h1>
<p>在計算機中，所有資料都是以二進位制數的形式儲存的，字元 <code>char</code> 也不例外。為了表示字元，我們需要建立一套“字符集”，規定每個字元和二進位制數之間的一一對應關係。有了字符集之後，計算機就可以透過查表完成二進位制數到字元的轉換。</p>
<h2 id="341-ascii">3.4.1 &nbsp; ASCII 字符集<a class="headerlink" href="#341-ascii" title="Permanent link">&para;</a></h2>
<p><u>ASCII 碼</u>是最早出現的字符集，其全稱為 American Standard Code for Information Interchange（美國標準資訊交換程式碼）。它使用 7 位二進位制數（一個位元組的低 7 位）表示一個字元，最多能夠表示 128 個不同的字元。如圖 3-6 所示，ASCII 碼包括英文字母的大小寫、數字 0 ~ 9、一些標點符號，以及一些控制字元（如換行符和製表符）。</p>
<p><a class="glightbox" href="../character_encoding.assets/ascii_table.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="ASCII 碼" class="animation-figure" src="../character_encoding.assets/ascii_table.png" /></a></p>
<p align="center"> 圖 3-6 &nbsp; ASCII 碼 </p>
<p>然而，<strong>ASCII 碼僅能夠表示英文</strong>。隨著計算機的全球化，誕生了一種能夠表示更多語言的 <u>EASCII</u> 字符集。它在 ASCII 的 7 位基礎上擴展到 8 位，能夠表示 256 個不同的字元。</p>
<p>在世界範圍內，陸續出現了一批適用於不同地區的 EASCII 字符集。這些字符集的前 128 個字元統一為 ASCII 碼，後 128 個字元定義不同，以適應不同語言的需求。</p>
<h2 id="342-gbk">3.4.2 &nbsp; GBK 字符集<a class="headerlink" href="#342-gbk" title="Permanent link">&para;</a></h2>
<p>後來人們發現，<strong>EASCII 碼仍然無法滿足許多語言的字元數量要求</strong>。比如漢字有近十萬個，光日常使用的就有幾千個。中國國家標準總局於 1980 年釋出了 <u>GB2312</u> 字符集，其收錄了 6763 個漢字，基本滿足了漢字的計算機處理需要。</p>
<p>然而，GB2312 無法處理部分罕見字和繁體字。<u>GBK</u> 字符集是在 GB2312 的基礎上擴展得到的，它共收錄了 21886 個漢字。在 GBK 的編碼方案中，ASCII 字元使用一個位元組表示，漢字使用兩個位元組表示。</p>
<h2 id="343-unicode">3.4.3 &nbsp; Unicode 字符集<a class="headerlink" href="#343-unicode" title="Permanent link">&para;</a></h2>
<p>隨著計算機技術的蓬勃發展，字符集與編碼標準百花齊放，而這帶來了許多問題。一方面，這些字符集一般只定義了特定語言的字元，無法在多語言環境下正常工作。另一方面，同一種語言存在多種字符集標準，如果兩臺計算機使用的是不同的編碼標準，則在資訊傳遞時就會出現亂碼。</p>
<p>那個時代的研究人員就在想：<strong>如果推出一個足夠完整的字符集，將世界範圍內的所有語言和符號都收錄其中，不就可以解決跨語言環境和亂碼問題了嗎</strong>？在這種想法的驅動下，一個大而全的字符集 Unicode 應運而生。</p>
<p><u>Unicode</u> 的中文名稱為“統一碼”，理論上能容納 100 多萬個字元。它致力於將全球範圍內的字元納入統一的字符集之中，提供一種通用的字符集來處理和顯示各種語言文字，減少因為編碼標準不同而產生的亂碼問題。</p>
<p>自 1991 年釋出以來，Unicode 不斷擴充新的語言與字元。截至 2022 年 9 月，Unicode 已經包含 149186 個字元，包括各種語言的字元、符號甚至表情符號等。在龐大的 Unicode 字符集中，常用的字元佔用 2 位元組，有些生僻的字元佔用 3 位元組甚至 4 位元組。</p>
<p>Unicode 是一種通用字符集，本質上是給每個字元分配一個編號（稱為“碼點”），<strong>但它並沒有規定在計算機中如何儲存這些字元碼點</strong>。我們不禁會問：當多種長度的 Unicode 碼點同時出現在一個文字中時，系統如何解析字元？例如給定一個長度為 2 位元組的編碼，系統如何確認它是一個 2 位元組的字元還是兩個 1 位元組的字元？</p>
<p>對於以上問題，<strong>一種直接的解決方案是將所有字元儲存為等長的編碼</strong>。如圖 3-7 所示，“Hello”中的每個字元佔用 1 位元組，“演算法”中的每個字元佔用 2 位元組。我們可以透過高位填 0 將“Hello 演算法”中的所有字元都編碼為 2 位元組長度。這樣系統就可以每隔 2 位元組解析一個字元，恢復這個短語的內容了。</p>
<p><a class="glightbox" href="../character_encoding.assets/unicode_hello_algo.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Unicode 編碼示例" class="animation-figure" src="../character_encoding.assets/unicode_hello_algo.png" /></a></p>
<p align="center"> 圖 3-7 &nbsp; Unicode 編碼示例 </p>
<p>然而 ASCII 碼已經向我們證明，編碼英文只需 1 位元組。若採用上述方案，英文文字佔用空間的大小將會是 ASCII 編碼下的兩倍，非常浪費記憶體空間。因此，我們需要一種更加高效的 Unicode 編碼方法。</p>
<h2 id="344-utf-8">3.4.4 &nbsp; UTF-8 編碼<a class="headerlink" href="#344-utf-8" title="Permanent link">&para;</a></h2>
<p>目前，UTF-8 已成為國際上使用最廣泛的 Unicode 編碼方法。<strong>它是一種可變長度的編碼</strong>，使用 1 到 4 位元組來表示一個字元，根據字元的複雜性而變。ASCII 字元只需 1 位元組，拉丁字母和希臘字母需要 2 位元組，常用的中文字元需要 3 位元組，其他的一些生僻字元需要 4 位元組。</p>
<p>UTF-8 的編碼規則並不複雜，分為以下兩種情況。</p>
<ul>
<li>對於長度為 1 位元組的字元，將最高位設定為 <span class="arithmatex">\(0\)</span> ，其餘 7 位設定為 Unicode 碼點。值得注意的是，ASCII 字元在 Unicode 字符集中佔據了前 128 個碼點。也就是說，<strong>UTF-8 編碼可以向下相容 ASCII 碼</strong>。這意味著我們可以使用 UTF-8 來解析年代久遠的 ASCII 碼文字。</li>
<li>對於長度為 <span class="arithmatex">\(n\)</span> 位元組的字元（其中 <span class="arithmatex">\(n &gt; 1\)</span>），將首個位元組的高 <span class="arithmatex">\(n\)</span> 位都設定為 <span class="arithmatex">\(1\)</span> ，第 <span class="arithmatex">\(n + 1\)</span> 位設定為 <span class="arithmatex">\(0\)</span> ；從第二個位元組開始，將每個位元組的高 2 位都設定為 <span class="arithmatex">\(10\)</span> ；其餘所有位用於填充字元的 Unicode 碼點。</li>
</ul>
<p>圖 3-8 展示了“Hello演算法”對應的 UTF-8 編碼。觀察發現，由於最高 <span class="arithmatex">\(n\)</span> 位都設定為 <span class="arithmatex">\(1\)</span> ，因此系統可以透過讀取最高位 <span class="arithmatex">\(1\)</span> 的個數來解析出字元的長度為 <span class="arithmatex">\(n\)</span> 。</p>
<p>但為什麼要將其餘所有位元組的高 2 位都設定為 <span class="arithmatex">\(10\)</span> 呢？實際上，這個 <span class="arithmatex">\(10\)</span> 能夠起到校驗符的作用。假設系統從一個錯誤的位元組開始解析文字，位元組頭部的 <span class="arithmatex">\(10\)</span> 能夠幫助系統快速判斷出異常。</p>
<p>之所以將 <span class="arithmatex">\(10\)</span> 當作校驗符，是因為在 UTF-8 編碼規則下，不可能有字元的最高兩位是 <span class="arithmatex">\(10\)</span> 。這個結論可以用反證法來證明：假設一個字元的最高兩位是 <span class="arithmatex">\(10\)</span> ，說明該字元的長度為 <span class="arithmatex">\(1\)</span> ，對應 ASCII 碼。而 ASCII 碼的最高位應該是 <span class="arithmatex">\(0\)</span> ，與假設矛盾。</p>
<p><a class="glightbox" href="../character_encoding.assets/utf-8_hello_algo.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="UTF-8 編碼示例" class="animation-figure" src="../character_encoding.assets/utf-8_hello_algo.png" /></a></p>
<p align="center"> 圖 3-8 &nbsp; UTF-8 編碼示例 </p>
<p>除了 UTF-8 之外，常見的編碼方式還包括以下兩種。</p>
<ul>
<li><strong>UTF-16 編碼</strong>：使用 2 或 4 位元組來表示一個字元。所有的 ASCII 字元和常用的非英文字元，都用 2 位元組表示；少數字符需要用到 4 位元組表示。對於 2 位元組的字元，UTF-16 編碼與 Unicode 碼點相等。</li>
<li><strong>UTF-32 編碼</strong>：每個字元都使用 4 位元組。這意味著 UTF-32 比 UTF-8 和 UTF-16 更佔用空間，特別是對於 ASCII 字元佔比較高的文字。</li>
</ul>
<p>從儲存空間佔用的角度看，使用 UTF-8 表示英文字元非常高效，因為它僅需 1 位元組；使用 UTF-16 編碼某些非英文字元（例如中文）會更加高效，因為它僅需 2 位元組，而 UTF-8 可能需要 3 位元組。</p>
<p>從相容性的角度看，UTF-8 的通用性最佳，許多工具和庫優先支持 UTF-8 。</p>
<h2 id="345">3.4.5 &nbsp; 程式語言的字元編碼<a class="headerlink" href="#345" title="Permanent link">&para;</a></h2>
<p>對於以往的大多數程式語言，程式執行中的字串都採用 UTF-16 或 UTF-32 這類等長編碼。在等長編碼下，我們可以將字串看作陣列來處理，這種做法具有以下優點。</p>
<ul>
<li><strong>隨機訪問</strong>：UTF-16 編碼的字串可以很容易地進行隨機訪問。UTF-8 是一種變長編碼，要想找到第 <span class="arithmatex">\(i\)</span> 個字元，我們需要從字串的開始處走訪到第 <span class="arithmatex">\(i\)</span> 個字元，這需要 <span class="arithmatex">\(O(n)\)</span> 的時間。</li>
<li><strong>字元計數</strong>：與隨機訪問類似，計算 UTF-16 編碼的字串的長度也是 <span class="arithmatex">\(O(1)\)</span> 的操作。但是，計算 UTF-8 編碼的字串的長度需要走訪整個字串。</li>
<li><strong>字串操作</strong>：在 UTF-16 編碼的字串上，很多字串操作（如分割、連線、插入、刪除等）更容易進行。在 UTF-8 編碼的字串上，進行這些操作通常需要額外的計算，以確保不會產生無效的 UTF-8 編碼。</li>
</ul>
<p>實際上，程式語言的字元編碼方案設計是一個很有趣的話題，涉及許多因素。</p>
<ul>
<li>Java 的 <code>String</code> 型別使用 UTF-16 編碼，每個字元佔用 2 位元組。這是因為 Java 語言設計之初，人們認為 16 位足以表示所有可能的字元。然而，這是一個不正確的判斷。後來 Unicode 規範擴展到了超過 16 位，所以 Java 中的字元現在可能由一對 16 位的值（稱為“代理對”）表示。</li>
<li>JavaScript 和 TypeScript 的字串使用 UTF-16 編碼的原因與 Java 類似。當 1995 年 Netscape 公司首次推出 JavaScript 語言時，Unicode 還處於發展早期，那時候使用 16 位的編碼就足以表示所有的 Unicode 字元了。</li>
<li>C# 使用 UTF-16 編碼，主要是因為 .NET 平臺是由 Microsoft 設計的，而 Microsoft 的很多技術（包括 Windows 作業系統）都廣泛使用 UTF-16 編碼。</li>
</ul>
<p>由於以上程式語言對字元數量的低估，它們不得不採取“代理對”的方式來表示超過 16 位長度的 Unicode 字元。這是一個不得已為之的無奈之舉。一方面，包含代理對的字串中，一個字元可能佔用 2 位元組或 4 位元組，從而喪失了等長編碼的優勢。另一方面，處理代理對需要額外增加程式碼，這提高了程式設計的複雜性和除錯難度。</p>
<p>出於以上原因，部分程式語言提出了一些不同的編碼方案。</p>
<ul>
<li>Python 中的 <code>str</code> 使用 Unicode 編碼，並採用一種靈活的字串表示，儲存的字元長度取決於字串中最大的 Unicode 碼點。若字串中全部是 ASCII 字元，則每個字元佔用 1 位元組；如果有字元超出了 ASCII 範圍，但全部在基本多語言平面（BMP）內，則每個字元佔用 2 位元組；如果有超出 BMP 的字元，則每個字元佔用 4 位元組。</li>
<li>Go 語言的 <code>string</code> 型別在內部使用 UTF-8 編碼。Go 語言還提供了 <code>rune</code> 型別，它用於表示單個 Unicode 碼點。</li>
<li>Rust 語言的 <code>str</code> 和 <code>String</code> 型別在內部使用 UTF-8 編碼。Rust 也提供了 <code>char</code> 型別，用於表示單個 Unicode 碼點。</li>
</ul>
<p>需要注意的是，以上討論的都是字串在程式語言中的儲存方式，<strong>這和字串如何在檔案中儲存或在網路中傳輸是不同的問題</strong>。在檔案儲存或網路傳輸中，我們通常會將字串編碼為 UTF-8 格式，以達到最優的相容性和空間效率。</p>
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