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chapter_computational_complexity/space_complexity/index.html

@@ -3503,7 +3503,7 @@
 <p><img alt="算法使用的相关空间" src="../space_complexity.assets/space_types.png" /></p>
 <p align="center"> Fig. 算法使用的相关空间 </p>
 
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 <div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="cm">/* 类 */</span>
@@ -3754,7 +3754,7 @@
 <li><strong>以最差输入数据为准</strong>。当 <span class="arithmatex">\(n &lt; 10\)</span> 时，空间复杂度为 <span class="arithmatex">\(O(1)\)</span> ；但当 <span class="arithmatex">\(n &gt; 10\)</span> 时，初始化的数组 <code>nums</code> 占用 <span class="arithmatex">\(O(n)\)</span> 空间；因此最差空间复杂度为 <span class="arithmatex">\(O(n)\)</span> 。</li>
 <li><strong>以算法运行过程中的峰值内存为准</strong>。例如，程序在执行最后一行之前，占用 <span class="arithmatex">\(O(1)\)</span> 空间；当初始化数组 <code>nums</code> 时，程序占用 <span class="arithmatex">\(O(n)\)</span> 空间；因此最差空间复杂度为 <span class="arithmatex">\(O(n)\)</span> 。</li>
 </ul>
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 <div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -3864,7 +3864,7 @@
 </div>
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 <p><strong>在递归函数中，需要注意统计栈帧空间</strong>。例如，函数 <code>loop()</code> 在循环中调用了 <span class="arithmatex">\(n\)</span> 次 <code>function()</code> ，每轮中的 <code>function()</code> 都返回并释放了栈帧空间，因此空间复杂度仍为 <span class="arithmatex">\(O(1)\)</span> 。而递归函数 <code>recur()</code> 在运行过程中会同时存在 <span class="arithmatex">\(n\)</span> 个未返回的 <code>recur()</code> ，从而占用 <span class="arithmatex">\(O(n)\)</span> 的栈帧空间。</p>
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 <div class="highlight"><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="kt">int</span><span class="w"> </span><span class="nf">function</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
@@ -4081,7 +4081,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
 <h3 id="o1">常数阶 <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#o1" title="Permanent link">&para;</a></h3>
 <p>常数阶常见于数量与输入数据大小 <span class="arithmatex">\(n\)</span> 无关的常量、变量、对象。</p>
 <p>需要注意的是，在循环中初始化变量或调用函数而占用的内存，在进入下一循环后就会被释放，即不会累积占用空间，空间复杂度仍为 <span class="arithmatex">\(O(1)\)</span> 。</p>
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 <div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* 常数阶 */</span>
@@ -4322,7 +4322,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
 </div>
 <h3 id="on">线性阶 <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#on" title="Permanent link">&para;</a></h3>
 <p>线性阶常见于元素数量与 <span class="arithmatex">\(n\)</span> 成正比的数组、链表、栈、队列等。</p>
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 <div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* 线性阶 */</span>
@@ -4565,7 +4565,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
 </div>
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 <p>以下递归函数会同时存在 <span class="arithmatex">\(n\)</span> 个未返回的 <code>algorithm()</code> 函数，使用 <span class="arithmatex">\(O(n)\)</span> 大小的栈帧空间。</p>
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 <div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* 线性阶（递归实现） */</span>
@@ -4689,7 +4689,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
 
 <h3 id="on2">平方阶 <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#on2" title="Permanent link">&para;</a></h3>
 <p>平方阶常见于矩阵和图，元素数量与 <span class="arithmatex">\(n\)</span> 成平方关系。</p>
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 <div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* 平方阶 */</span>
@@ -4883,7 +4883,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
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 <p>在以下递归函数中，同时存在 <span class="arithmatex">\(n\)</span> 个未返回的 <code>algorithm()</code> ，并且每个函数中都初始化了一个数组，长度分别为 <span class="arithmatex">\(n, n-1, n-2, ..., 2, 1\)</span> ，平均长度为 <span class="arithmatex">\(\frac{n}{2}\)</span> ，因此总体占用 <span class="arithmatex">\(O(n^2)\)</span> 空间。</p>
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 <div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-84-1" name="__codelineno-84-1" href="#__codelineno-84-1"></a><span class="cm">/* 平方阶（递归实现） */</span>
@@ -5025,7 +5025,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
 
 <h3 id="o2n">指数阶 <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#o2n" title="Permanent link">&para;</a></h3>
 <p>指数阶常见于二叉树。高度为 <span class="arithmatex">\(n\)</span> 的「满二叉树」的节点数量为 <span class="arithmatex">\(2^n - 1\)</span> ，占用 <span class="arithmatex">\(O(2^n)\)</span> 空间。</p>
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 <div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-96-1" name="__codelineno-96-1" href="#__codelineno-96-1"></a><span class="cm">/* 指数阶（建立满二叉树） */</span>