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krahets 10 months ago
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@ -5498,7 +5498,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
<h3 id="5-olog-n">5. &nbsp; 对数阶 <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#5-olog-n" title="Permanent link">&para;</a></h3>
<p>对数阶常见于分治算法。例如归并排序,输入长度为 <span class="arithmatex">\(n\)</span> 的数组,每轮递归将数组从中点处划分为两半,形成高度为 <span class="arithmatex">\(\log n\)</span> 的递归树,使用 <span class="arithmatex">\(O(\log n)\)</span> 栈帧空间。</p>
<p>再例如将数字转化为字符串,输入一个正整数 <span class="arithmatex">\(n\)</span> ,它的位数为 <span class="arithmatex">\(\log_{10} n + 1\)</span> ,即对应字符串长度为 <span class="arithmatex">\(\log_{10} n + 1\)</span> ,因此空间复杂度为 <span class="arithmatex">\(O(\log_{10} n + 1) = O(\log n)\)</span></p>
<p>再例如将数字转化为字符串,输入一个正整数 <span class="arithmatex">\(n\)</span> ,它的位数为 <span class="arithmatex">\(\lfloor \log_{10} n \rfloor + 1\)</span> ,即对应字符串长度为 <span class="arithmatex">\(\lfloor \log_{10} n \rfloor + 1\)</span> ,因此空间复杂度为 <span class="arithmatex">\(O(\log_{10} n + 1) = O(\log n)\)</span></p>
<h2 id="244">2.4.4 &nbsp; 权衡时间与空间<a class="headerlink" href="#244" title="Permanent link">&para;</a></h2>
<p>理想情况下,我们希望算法的时间复杂度和空间复杂度都能达到最优。然而在实际情况中,同时优化时间复杂度和空间复杂度通常非常困难。</p>
<p><strong>降低时间复杂度通常需要以提升空间复杂度为代价,反之亦然</strong>。我们将牺牲内存空间来提升算法运行速度的思路称为“以空间换时间”;反之,则称为“以时间换空间”。</p>

@ -3624,7 +3624,7 @@
<li>JavaScript 和 TypeScript 的字符串使用 UTF-16 编码的原因与 Java 类似。当 1995 年 Netscape 公司首次推出 JavaScript 语言时Unicode 还处于发展早期,那时候使用 16 位的编码就足以表示所有的 Unicode 字符了。</li>
<li>C# 使用 UTF-16 编码,主要是因为 .NET 平台是由 Microsoft 设计的,而 Microsoft 的很多技术(包括 Windows 操作系统)都广泛使用 UTF-16 编码。</li>
</ul>
<p>由于以上编程语言对字符数量的低估,它们不得不采取“代理对”的方式来表示超过 16 位长度的 Unicode 字符。这是一个不得已为之的无奈之举。一方面,包含代理对的字符串中,一个字符可能占用 2 字节或 4 字节,从而丧失了等长编码的优势。另一方面,处理代理对需要增加额外代码,这提高了编程的复杂性和调试难度。</p>
<p>由于以上编程语言对字符数量的低估,它们不得不采取“代理对”的方式来表示超过 16 位长度的 Unicode 字符。这是一个不得已为之的无奈之举。一方面,包含代理对的字符串中,一个字符可能占用 2 字节或 4 字节,从而丧失了等长编码的优势。另一方面,处理代理对需要额外增加代码,这提高了编程的复杂性和调试难度。</p>
<p>出于以上原因,部分编程语言提出了一些不同的编码方案。</p>
<ul>
<li>Python 中的 <code>str</code> 使用 Unicode 编码,并采用一种灵活的字符串表示,存储的字符长度取决于字符串中最大的 Unicode 码点。若字符串中全部是 ASCII 字符,则每个字符占用 1 字节;如果有字符超出了 ASCII 范围但全部在基本多语言平面BMP则每个字符占用 2 字节;如果有超出 BMP 的字符,则每个字符占用 4 字节。</li>

@ -3520,18 +3520,17 @@
<p><strong>逻辑结构揭示了数据元素之间的逻辑关系</strong>。在数组和链表中,数据按照一定顺序排列,体现了数据之间的线性关系;而在树中,数据从顶部向下按层次排列,表现出“祖先”与“后代”之间的派生关系;图则由节点和边构成,反映了复杂的网络关系。</p>
<p>如图 3-1 所示,逻辑结构可分为“线性”和“非线性”两大类。线性结构比较直观,指数据在逻辑关系上呈线性排列;非线性结构则相反,呈非线性排列。</p>
<ul>
<li><strong>线性数据结构</strong>:数组、链表、栈、队列、哈希表。</li>
<li><strong>线性数据结构</strong>:数组、链表、栈、队列、哈希表,元素之间是一对一的顺序关系</li>
<li><strong>非线性数据结构</strong>:树、堆、图、哈希表。</li>
</ul>
<p><a class="glightbox" href="../classification_of_data_structure.assets/classification_logic_structure.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="线性数据结构与非线性数据结构" class="animation-figure" src="../classification_of_data_structure.assets/classification_logic_structure.png" /></a></p>
<p align="center"> 图 3-1 &nbsp; 线性数据结构与非线性数据结构 </p>
<p>非线性数据结构可以进一步划分为树形结构和网状结构。</p>
<ul>
<li><strong>线性结构</strong>:数组、链表、队列、栈、哈希表,元素之间是一对一的顺序关系。</li>
<li><strong>树形结构</strong>:树、堆、哈希表,元素之间是一对多的关系。</li>
<li><strong>网状结构</strong>:图,元素之间是多对多的关系。</li>
</ul>
<p><a class="glightbox" href="../classification_of_data_structure.assets/classification_logic_structure.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="线性数据结构与非线性数据结构" class="animation-figure" src="../classification_of_data_structure.assets/classification_logic_structure.png" /></a></p>
<p align="center"> 图 3-1 &nbsp; 线性数据结构与非线性数据结构 </p>
<h2 id="312">3.1.2 &nbsp; 物理结构:连续与分散<a class="headerlink" href="#312" title="Permanent link">&para;</a></h2>
<p><strong>当算法程序运行时,正在处理的数据主要存储在内存中</strong>。图 3-2 展示了一个计算机内存条,其中每个黑色方块都包含一块内存空间。我们可以将内存想象成一个巨大的 Excel 表格,其中每个单元格都可以存储一定大小的数据。</p>
<p><strong>系统通过内存地址来访问目标位置的数据</strong>。如图 3-2 所示,计算机根据特定规则为表格中的每个单元格分配编号,确保每个内存空间都有唯一的内存地址。有了这些地址,程序便可以访问内存中的数据。</p>

@ -4604,7 +4604,7 @@
<a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="cm">/* 析构函数 */</span>
<a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="kt">void</span><span class="w"> </span><span class="nf">delLinkedListQueue</span><span class="p">(</span><span class="n">LinkedListQueue</span><span class="w"> </span><span class="o">*</span><span class="n">queue</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w"> </span><span class="c1">// 释放所有节点</span>
<a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="p">;</span>
<a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">;</span>
<a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="w"> </span><span class="n">free</span><span class="p">(</span><span class="n">tmp</span><span class="p">);</span>

@ -2471,7 +2471,7 @@
<a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="cm">/* 析构函数 */</span>
<a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="kt">void</span><span class="w"> </span><span class="nf">delLinkedListQueue</span><span class="p">(</span><span class="n">LinkedListQueue</span><span class="w"> </span><span class="o">*</span><span class="n">queue</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w"> </span><span class="c1">// 释放所有节点</span>
<a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="p">;</span>
<a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">;</span>
<a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="w"> </span><span class="n">free</span><span class="p">(</span><span class="n">tmp</span><span class="p">);</span>

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@ -221,7 +221,7 @@ body {
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