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<h1 id="2">第 2 章 &nbsp; 复杂度分析<a class="headerlink" href="#2" title="Permanent link">&para;</a></h1>
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<h1 id="22">2.2 &nbsp; 迭代与递归<a class="headerlink" href="#22" title="Permanent link">&para;</a></h1>
<p>在数据结构与算法中，重复执行某个任务是很常见的，其与算法的复杂度密切相关。而要重复执行某个任务，我们通常会选用两种基本的程序结构：迭代和递归。</p>
<h2 id="221">2.2.1 &nbsp; 迭代<a class="headerlink" href="#221" title="Permanent link">&para;</a></h2>
@@ -4916,11 +4948,73 @@
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<p>总之，<strong>选择迭代还是递归取决于特定问题的性质</strong>。在编程实践中，权衡两者的优劣并根据情境选择合适的方法是至关重要的。</p>
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<h1 id="21">2.1 &nbsp; 算法效率评估<a class="headerlink" href="#21" title="Permanent link">&para;</a></h1>
<p>在算法设计中，我们先后追求以下两个层面的目标。</p>
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@@ -3479,11 +3511,73 @@
<p>复杂度是个数学概念，对于初学者可能比较抽象，学习难度相对较高。从这个角度看，复杂度分析可能不太适合作为最先介绍的内容。然而，当我们讨论某个数据结构或算法的特点时，难以避免要分析其运行速度和空间使用情况。</p>
<p>综上所述，建议你在深入学习数据结构与算法之前，<strong>先对复杂度分析建立初步的了解，以便能够完成简单算法的复杂度分析</strong>。</p>
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<h1 id="24">2.4 &nbsp; 空间复杂度<a class="headerlink" href="#24" title="Permanent link">&para;</a></h1>
<p>「空间复杂度 space complexity」用于衡量算法占用内存空间随着数据量变大时的增长趋势。这个概念与时间复杂度非常类似，只需将“运行时间”替换为“占用内存空间”。</p>
<h2 id="241">2.4.1 &nbsp; 算法相关空间<a class="headerlink" href="#241" title="Permanent link">&para;</a></h2>
@@ -3577,8 +3609,8 @@
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">class</span> <span class="nc">Node</span><span class="p">:</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;类&quot;&quot;&quot;</span>
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">x</span> <span class="c1"># 节点值</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">next</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">Node</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 指向下一节点的引用</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">x</span> <span class="c1"># 节点值</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">next</span><span class="p">:</span> <span class="n">Node</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># 指向下一节点的引用</span>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="k">def</span> <span class="nf">function</span><span class="p">()</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;函数&quot;&quot;&quot;</span>
@@ -5366,11 +5398,73 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
<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>
<p>选择哪种思路取决于我们更看重哪个方面。在大多数情况下，时间比空间更宝贵，因此“以空间换时间”通常是更常用的策略。当然，在数据量很大的情况下，控制空间复杂度也是非常重要的。</p>
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@@ -5488,6 +5565,8 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
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<p><strong>算法效率评估</strong></p>
@@ -3487,11 +3519,73 @@
<p>在实际中，因为我们通常不知道每个方法的“常数项”复杂度是多少，所以一般无法仅凭复杂度来选择 <span class="arithmatex">\(n = 8\)</span> 之下的最优解法。但对于 <span class="arithmatex">\(n = 8^5\)</span> 就很好选了，这时增长趋势已经占主导了。</p>
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<p>运行时间可以直观且准确地反映算法的效率。如果我们想要准确预估一段代码的运行时间，应该如何操作呢？</p>
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@@ -6583,11 +6615,73 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
<p>可能由于 <span class="arithmatex">\(O\)</span> 符号过于朗朗上口，我们常常使用它来表示平均时间复杂度。但从严格意义上看，这种做法并不规范。在本书和其他资料中，若遇到类似“平均时间复杂度 <span class="arithmatex">\(O(n)\)</span>”的表述，请将其直接理解为 <span class="arithmatex">\(\Theta(n)\)</span> 。</p>
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