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<p><strong>展开完整测试非常耗费资源</strong>。随着输入数据量的变化，算法会表现出不同的效率。例如，在输入数据量较小时，算法 <code>A</code> 的运行时间比算法 <code>B</code> 更少；而输入数据量较大时，测试结果可能恰恰相反。因此，为了得到有说服力的结论，我们需要测试各种规模的输入数据，而这样需要耗费大量的计算资源。</p>
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<h2 id="212">2.1.2 理论估算<a class="headerlink" href="#212" title="Permanent link">¶</a></h2>
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<p>由于实际测试具有较大的局限性，我们可以考虑仅通过一些计算来评估算法的效率。这种估算方法被称为「渐近复杂度分析 Asymptotic Complexity Analysis」，简称为「复杂度分析」。</p>
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<p><strong>复杂度分析评估的是算法运行效率随着输入数据量增多时的增长趋势</strong>。这个定义有些拗口，我们可以将其分为三个重点来理解：</p>
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<p>复杂度分析评估的是算法执行所需的时间和空间资源。<strong>它被表示为一个函数，描述了随着输入数据大小的增加，算法所需时间（空间）的增长趋势</strong>。这个定义有些拗口，我们可以将其分为三个重点来理解：</p>
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<ol>
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<li>“算法运行效率”可分为运行时间和占用空间两部分，与之对应地，复杂度可分为「时间复杂度 Time Complexity」和「空间复杂度 Space Complexity」。</li>
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<li>“随着输入数据量增多时”意味着复杂度反映了算法运行效率与输入数据量之间的关系。</li>
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<li>“时间（空间）”分别对应「时间复杂度 Time Complexity」和「空间复杂度 Space Complexity」。</li>
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<li>“随着输入数据大小的增加”意味着复杂度反映了算法运行效率与输入数据体量之间的关系。</li>
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<li>“增长趋势”表示复杂度分析关注的是算法时间与空间的增长趋势，而非具体的运行时间或占用空间。</li>
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</ol>
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<p><strong>复杂度分析克服了实际测试方法的弊端</strong>。首先，它独立于测试环境，分析结果适用于所有运行平台。其次，它可以体现不同数据量下的算法效率，尤其是在大数据量下的算法性能。</p>
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常数阶 \(O(1)\)
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1. 常数阶 \(O(1)\)
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</a>
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线性阶 \(O(n)\)
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<a href="#2-on" class="md-nav__link">
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2. 线性阶 \(O(n)\)
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</a>
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平方阶 \(O(n^2)\)
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<a href="#3-on2" class="md-nav__link">
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3. 平方阶 \(O(n^2)\)
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指数阶 \(O(2^n)\)
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<a href="#4-o2n" class="md-nav__link">
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4. 指数阶 \(O(2^n)\)
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</a>
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<a href="#olog-n" class="md-nav__link">
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对数阶 \(O(\log n)\)
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<a href="#5-olog-n" class="md-nav__link">
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5. 对数阶 \(O(\log n)\)
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</a>
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</li>
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<ul class="md-nav__list">
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<li class="md-nav__item">
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<a href="#o1" class="md-nav__link">
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常数阶 \(O(1)\)
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<a href="#1-o1" class="md-nav__link">
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1. 常数阶 \(O(1)\)
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#on" class="md-nav__link">
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线性阶 \(O(n)\)
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<a href="#2-on" class="md-nav__link">
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2. 线性阶 \(O(n)\)
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</a>
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</li>
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<a href="#on2" class="md-nav__link">
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平方阶 \(O(n^2)\)
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<a href="#3-on2" class="md-nav__link">
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3. 平方阶 \(O(n^2)\)
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#o2n" class="md-nav__link">
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指数阶 \(O(2^n)\)
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<a href="#4-o2n" class="md-nav__link">
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4. 指数阶 \(O(2^n)\)
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#olog-n" class="md-nav__link">
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对数阶 \(O(\log n)\)
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<a href="#5-olog-n" class="md-nav__link">
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5. 对数阶 \(O(\log n)\)
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</a>
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</li>
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@ -4121,7 +4121,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
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<p class="admonition-title">Tip</p>
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<p>部分示例代码需要一些前置知识，包括数组、链表、二叉树、递归算法等。如果你遇到看不懂的地方，可以在学习完后面章节后再来复习。</p>
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</div>
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<h3 id="o1">常数阶 <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#o1" title="Permanent link">¶</a></h3>
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<h3 id="1-o1">1. 常数阶 <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#1-o1" title="Permanent link">¶</a></h3>
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<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>
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<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="tabbed-set tabbed-alternate" data-tabs="4:12"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Java</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Python</label><label for="__tabbed_4_4">Go</label><label for="__tabbed_4_5">JS</label><label for="__tabbed_4_6">TS</label><label for="__tabbed_4_7">C</label><label for="__tabbed_4_8">C#</label><label for="__tabbed_4_9">Swift</label><label for="__tabbed_4_10">Zig</label><label for="__tabbed_4_11">Dart</label><label for="__tabbed_4_12">Rust</label></div>
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@ -4431,7 +4431,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
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</div>
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</div>
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</div>
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<h3 id="on">线性阶 <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#on" title="Permanent link">¶</a></h3>
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<h3 id="2-on">2. 线性阶 <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#2-on" title="Permanent link">¶</a></h3>
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<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="tabbed-set tabbed-alternate" data-tabs="5:12"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JS</label><label for="__tabbed_5_6">TS</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label><label for="__tabbed_5_11">Dart</label><label for="__tabbed_5_12">Rust</label></div>
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||||
<div class="tabbed-content">
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@ -4798,7 +4798,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
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<p><img alt="递归函数产生的线性阶空间复杂度" src="../space_complexity.assets/space_complexity_recursive_linear.png" /></p>
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<p align="center"> 图：递归函数产生的线性阶空间复杂度 </p>
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<h3 id="on2">平方阶 <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#on2" title="Permanent link">¶</a></h3>
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<h3 id="3-on2">3. 平方阶 <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#3-on2" title="Permanent link">¶</a></h3>
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<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="tabbed-set tabbed-alternate" data-tabs="7:12"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Java</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Python</label><label for="__tabbed_7_4">Go</label><label for="__tabbed_7_5">JS</label><label for="__tabbed_7_6">TS</label><label for="__tabbed_7_7">C</label><label for="__tabbed_7_8">C#</label><label for="__tabbed_7_9">Swift</label><label for="__tabbed_7_10">Zig</label><label for="__tabbed_7_11">Dart</label><label for="__tabbed_7_12">Rust</label></div>
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<div class="tabbed-content">
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@ -5133,7 +5133,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
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<p><img alt="递归函数产生的平方阶空间复杂度" src="../space_complexity.assets/space_complexity_recursive_quadratic.png" /></p>
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<p align="center"> 图：递归函数产生的平方阶空间复杂度 </p>
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<h3 id="o2n">指数阶 <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#o2n" title="Permanent link">¶</a></h3>
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<h3 id="4-o2n">4. 指数阶 <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#4-o2n" title="Permanent link">¶</a></h3>
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<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="tabbed-set tabbed-alternate" data-tabs="9:12"><input checked="checked" id="__tabbed_9_1" name="__tabbed_9" type="radio" /><input id="__tabbed_9_2" name="__tabbed_9" type="radio" /><input id="__tabbed_9_3" name="__tabbed_9" type="radio" /><input id="__tabbed_9_4" name="__tabbed_9" type="radio" /><input id="__tabbed_9_5" name="__tabbed_9" type="radio" /><input id="__tabbed_9_6" name="__tabbed_9" type="radio" /><input id="__tabbed_9_7" name="__tabbed_9" type="radio" /><input id="__tabbed_9_8" name="__tabbed_9" type="radio" /><input id="__tabbed_9_9" name="__tabbed_9" type="radio" /><input id="__tabbed_9_10" name="__tabbed_9" type="radio" /><input id="__tabbed_9_11" name="__tabbed_9" type="radio" /><input id="__tabbed_9_12" name="__tabbed_9" type="radio" /><div class="tabbed-labels"><label for="__tabbed_9_1">Java</label><label for="__tabbed_9_2">C++</label><label for="__tabbed_9_3">Python</label><label for="__tabbed_9_4">Go</label><label for="__tabbed_9_5">JS</label><label for="__tabbed_9_6">TS</label><label for="__tabbed_9_7">C</label><label for="__tabbed_9_8">C#</label><label for="__tabbed_9_9">Swift</label><label for="__tabbed_9_10">Zig</label><label for="__tabbed_9_11">Dart</label><label for="__tabbed_9_12">Rust</label></div>
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<div class="tabbed-content">
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@ -5282,7 +5282,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
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<p><img alt="满二叉树产生的指数阶空间复杂度" src="../space_complexity.assets/space_complexity_exponential.png" /></p>
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<p align="center"> 图：满二叉树产生的指数阶空间复杂度 </p>
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<h3 id="olog-n">对数阶 <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#olog-n" title="Permanent link">¶</a></h3>
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<h3 id="5-olog-n">5. 对数阶 <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#5-olog-n" title="Permanent link">¶</a></h3>
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<p>对数阶常见于分治算法和数据类型转换等。</p>
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<p>例如“归并排序”算法，输入长度为 <span class="arithmatex">\(n\)</span> 的数组，每轮递归将数组从中点划分为两半，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>
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<p>再例如“数字转化为字符串”，输入任意正整数 <span class="arithmatex">\(n\)</span> ，它的位数为 <span class="arithmatex">\(\log_{10} n\)</span> ，即对应字符串长度为 <span class="arithmatex">\(\log_{10} n\)</span> ，因此空间复杂度为 <span class="arithmatex">\(O(\log_{10} n) = O(\log n)\)</span> 。</p>
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<ul class="md-nav__list">
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<a href="#_1" class="md-nav__link">
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第一步：统计操作数量
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<a href="#1" class="md-nav__link">
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1. 第一步：统计操作数量
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</a>
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<li class="md-nav__item">
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<a href="#_2" class="md-nav__link">
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第二步：判断渐近上界
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2. 第二步：判断渐近上界
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</a>
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<a href="#o1" class="md-nav__link">
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常数阶 \(O(1)\)
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<a href="#1-o1" class="md-nav__link">
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1. 常数阶 \(O(1)\)
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#on" class="md-nav__link">
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线性阶 \(O(n)\)
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<a href="#2-on" class="md-nav__link">
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2. 线性阶 \(O(n)\)
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</a>
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<a href="#on2" class="md-nav__link">
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平方阶 \(O(n^2)\)
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<a href="#3-on2" class="md-nav__link">
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3. 平方阶 \(O(n^2)\)
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</a>
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</li>
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<a href="#o2n" class="md-nav__link">
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指数阶 \(O(2^n)\)
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<a href="#4-o2n" class="md-nav__link">
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4. 指数阶 \(O(2^n)\)
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#olog-n" class="md-nav__link">
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对数阶 \(O(\log n)\)
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<a href="#5-olog-n" class="md-nav__link">
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5. 对数阶 \(O(\log n)\)
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</a>
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</li>
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|
||||
<li class="md-nav__item">
|
||||
<a href="#on-log-n" class="md-nav__link">
|
||||
线性对数阶 \(O(n \log n)\)
|
||||
<a href="#6-on-log-n" class="md-nav__link">
|
||||
6. 线性对数阶 \(O(n \log n)\)
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#on_1" class="md-nav__link">
|
||||
阶乘阶 \(O(n!)\)
|
||||
<a href="#7-on" class="md-nav__link">
|
||||
7. 阶乘阶 \(O(n!)\)
|
||||
</a>
|
||||
|
||||
</li>
|
||||
@ -3493,15 +3493,15 @@
|
||||
<ul class="md-nav__list">
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#_1" class="md-nav__link">
|
||||
第一步：统计操作数量
|
||||
<a href="#1" class="md-nav__link">
|
||||
1. 第一步：统计操作数量
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#_2" class="md-nav__link">
|
||||
第二步：判断渐近上界
|
||||
<a href="#2" class="md-nav__link">
|
||||
2. 第二步：判断渐近上界
|
||||
</a>
|
||||
|
||||
</li>
|
||||
@ -3520,50 +3520,50 @@
|
||||
<ul class="md-nav__list">
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#o1" class="md-nav__link">
|
||||
常数阶 \(O(1)\)
|
||||
<a href="#1-o1" class="md-nav__link">
|
||||
1. 常数阶 \(O(1)\)
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#on" class="md-nav__link">
|
||||
线性阶 \(O(n)\)
|
||||
<a href="#2-on" class="md-nav__link">
|
||||
2. 线性阶 \(O(n)\)
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#on2" class="md-nav__link">
|
||||
平方阶 \(O(n^2)\)
|
||||
<a href="#3-on2" class="md-nav__link">
|
||||
3. 平方阶 \(O(n^2)\)
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#o2n" class="md-nav__link">
|
||||
指数阶 \(O(2^n)\)
|
||||
<a href="#4-o2n" class="md-nav__link">
|
||||
4. 指数阶 \(O(2^n)\)
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#olog-n" class="md-nav__link">
|
||||
对数阶 \(O(\log n)\)
|
||||
<a href="#5-olog-n" class="md-nav__link">
|
||||
5. 对数阶 \(O(\log n)\)
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#on-log-n" class="md-nav__link">
|
||||
线性对数阶 \(O(n \log n)\)
|
||||
<a href="#6-on-log-n" class="md-nav__link">
|
||||
6. 线性对数阶 \(O(n \log n)\)
|
||||
</a>
|
||||
|
||||
</li>
|
||||
|
||||
<li class="md-nav__item">
|
||||
<a href="#on_1" class="md-nav__link">
|
||||
阶乘阶 \(O(n!)\)
|
||||
<a href="#7-on" class="md-nav__link">
|
||||
7. 阶乘阶 \(O(n!)\)
|
||||
</a>
|
||||
|
||||
</li>
|
||||
@ -4157,7 +4157,7 @@ $$</p>
|
||||
<h2 id="223">2.2.3 推算方法<a class="headerlink" href="#223" title="Permanent link">¶</a></h2>
|
||||
<p>渐近上界的数学味儿有点重，如果你感觉没有完全理解，也无需担心。因为在实际使用中，我们只需要掌握推算方法，数学意义可以逐渐领悟。</p>
|
||||
<p>根据定义，确定 <span class="arithmatex">\(f(n)\)</span> 之后，我们便可得到时间复杂度 <span class="arithmatex">\(O(f(n))\)</span> 。那么如何确定渐近上界 <span class="arithmatex">\(f(n)\)</span> 呢？总体分为两步：首先统计操作数量，然后判断渐近上界。</p>
|
||||
<h3 id="_1">第一步：统计操作数量<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
|
||||
<h3 id="1">1. 第一步：统计操作数量<a class="headerlink" href="#1" title="Permanent link">¶</a></h3>
|
||||
<p>针对代码，逐行从上到下计算即可。然而，由于上述 <span class="arithmatex">\(c \cdot f(n)\)</span> 中的常数项 <span class="arithmatex">\(c\)</span> 可以取任意大小，<strong>因此操作数量 <span class="arithmatex">\(T(n)\)</span> 中的各种系数、常数项都可以被忽略</strong>。根据此原则，可以总结出以下计数简化技巧：</p>
|
||||
<ol>
|
||||
<li><strong>忽略 <span class="arithmatex">\(T(n)\)</span> 中的常数项</strong>。因为它们都与 <span class="arithmatex">\(n\)</span> 无关，所以对时间复杂度不产生影响。</li>
|
||||
@ -4365,7 +4365,7 @@ T(n) & = n^2 + n & \text{偷懒统计 (o.O)}
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<h3 id="_2">第二步：判断渐近上界<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
|
||||
<h3 id="2">2. 第二步：判断渐近上界<a class="headerlink" href="#2" title="Permanent link">¶</a></h3>
|
||||
<p><strong>时间复杂度由多项式 <span class="arithmatex">\(T(n)\)</span> 中最高阶的项来决定</strong>。这是因为在 <span class="arithmatex">\(n\)</span> 趋于无穷大时，最高阶的项将发挥主导作用，其他项的影响都可以被忽略。</p>
|
||||
<p>以下表格展示了一些例子，其中一些夸张的值是为了强调“系数无法撼动阶数”这一结论。当 <span class="arithmatex">\(n\)</span> 趋于无穷大时，这些常数变得无足轻重。</p>
|
||||
<p align="center"> 表：多项式时间复杂度示例 </p>
|
||||
@ -4417,7 +4417,7 @@ O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!
|
||||
<p class="admonition-title">Tip</p>
|
||||
<p>部分示例代码需要一些预备知识，包括数组、递归等。如果你遇到不理解的部分，可以在学习完后面章节后再回顾。现阶段，请先专注于理解时间复杂度的含义和推算方法。</p>
|
||||
</div>
|
||||
<h3 id="o1">常数阶 <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#o1" title="Permanent link">¶</a></h3>
|
||||
<h3 id="1-o1">1. 常数阶 <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#1-o1" title="Permanent link">¶</a></h3>
|
||||
<p>常数阶的操作数量与输入数据大小 <span class="arithmatex">\(n\)</span> 无关，即不随着 <span class="arithmatex">\(n\)</span> 的变化而变化。</p>
|
||||
<p>对于以下算法，尽管操作数量 <code>size</code> 可能很大，但由于其与数据大小 <span class="arithmatex">\(n\)</span> 无关，因此时间复杂度仍为 <span class="arithmatex">\(O(1)\)</span> 。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="5:12"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JS</label><label for="__tabbed_5_6">TS</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label><label for="__tabbed_5_11">Dart</label><label for="__tabbed_5_12">Rust</label></div>
|
||||
@ -4563,7 +4563,7 @@ O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<h3 id="on">线性阶 <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#on" title="Permanent link">¶</a></h3>
|
||||
<h3 id="2-on">2. 线性阶 <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#2-on" title="Permanent link">¶</a></h3>
|
||||
<p>线性阶的操作数量相对于输入数据大小以线性级别增长。线性阶通常出现在单层循环中。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="6:12"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Java</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Python</label><label for="__tabbed_6_4">Go</label><label for="__tabbed_6_5">JS</label><label for="__tabbed_6_6">TS</label><label for="__tabbed_6_7">C</label><label for="__tabbed_6_8">C#</label><label for="__tabbed_6_9">Swift</label><label for="__tabbed_6_10">Zig</label><label for="__tabbed_6_11">Dart</label><label for="__tabbed_6_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
@ -4841,7 +4841,7 @@ O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!
|
||||
</div>
|
||||
</div>
|
||||
<p>值得注意的是，<strong>数据大小 <span class="arithmatex">\(n\)</span> 需根据输入数据的类型来具体确定</strong>。比如在第一个示例中，变量 <span class="arithmatex">\(n\)</span> 为输入数据大小；在第二个示例中，数组长度 <span class="arithmatex">\(n\)</span> 为数据大小。</p>
|
||||
<h3 id="on2">平方阶 <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#on2" title="Permanent link">¶</a></h3>
|
||||
<h3 id="3-on2">3. 平方阶 <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#3-on2" title="Permanent link">¶</a></h3>
|
||||
<p>平方阶的操作数量相对于输入数据大小以平方级别增长。平方阶通常出现在嵌套循环中，外层循环和内层循环都为 <span class="arithmatex">\(O(n)\)</span> ，因此总体为 <span class="arithmatex">\(O(n^2)\)</span> 。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="8:12"><input checked="checked" id="__tabbed_8_1" name="__tabbed_8" type="radio" /><input id="__tabbed_8_2" name="__tabbed_8" type="radio" /><input id="__tabbed_8_3" name="__tabbed_8" type="radio" /><input id="__tabbed_8_4" name="__tabbed_8" type="radio" /><input id="__tabbed_8_5" name="__tabbed_8" type="radio" /><input id="__tabbed_8_6" name="__tabbed_8" type="radio" /><input id="__tabbed_8_7" name="__tabbed_8" type="radio" /><input id="__tabbed_8_8" name="__tabbed_8" type="radio" /><input id="__tabbed_8_9" name="__tabbed_8" type="radio" /><input id="__tabbed_8_10" name="__tabbed_8" type="radio" /><input id="__tabbed_8_11" name="__tabbed_8" type="radio" /><input id="__tabbed_8_12" name="__tabbed_8" type="radio" /><div class="tabbed-labels"><label for="__tabbed_8_1">Java</label><label for="__tabbed_8_2">C++</label><label for="__tabbed_8_3">Python</label><label for="__tabbed_8_4">Go</label><label for="__tabbed_8_5">JS</label><label for="__tabbed_8_6">TS</label><label for="__tabbed_8_7">C</label><label for="__tabbed_8_8">C#</label><label for="__tabbed_8_9">Swift</label><label for="__tabbed_8_10">Zig</label><label for="__tabbed_8_11">Dart</label><label for="__tabbed_8_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
@ -5273,7 +5273,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<h3 id="o2n">指数阶 <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#o2n" title="Permanent link">¶</a></h3>
|
||||
<h3 id="4-o2n">4. 指数阶 <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#4-o2n" title="Permanent link">¶</a></h3>
|
||||
<p>生物学的“细胞分裂”是指数阶增长的典型例子：初始状态为 <span class="arithmatex">\(1\)</span> 个细胞，分裂一轮后变为 <span class="arithmatex">\(2\)</span> 个，分裂两轮后变为 <span class="arithmatex">\(4\)</span> 个，以此类推，分裂 <span class="arithmatex">\(n\)</span> 轮后有 <span class="arithmatex">\(2^n\)</span> 个细胞。</p>
|
||||
<p>以下代码模拟了细胞分裂的过程。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="10:12"><input checked="checked" id="__tabbed_10_1" name="__tabbed_10" type="radio" /><input id="__tabbed_10_2" name="__tabbed_10" type="radio" /><input id="__tabbed_10_3" name="__tabbed_10" type="radio" /><input id="__tabbed_10_4" name="__tabbed_10" type="radio" /><input id="__tabbed_10_5" name="__tabbed_10" type="radio" /><input id="__tabbed_10_6" name="__tabbed_10" type="radio" /><input id="__tabbed_10_7" name="__tabbed_10" type="radio" /><input id="__tabbed_10_8" name="__tabbed_10" type="radio" /><input id="__tabbed_10_9" name="__tabbed_10" type="radio" /><input id="__tabbed_10_10" name="__tabbed_10" type="radio" /><input id="__tabbed_10_11" name="__tabbed_10" type="radio" /><input id="__tabbed_10_12" name="__tabbed_10" type="radio" /><div class="tabbed-labels"><label for="__tabbed_10_1">Java</label><label for="__tabbed_10_2">C++</label><label for="__tabbed_10_3">Python</label><label for="__tabbed_10_4">Go</label><label for="__tabbed_10_5">JS</label><label for="__tabbed_10_6">TS</label><label for="__tabbed_10_7">C</label><label for="__tabbed_10_8">C#</label><label for="__tabbed_10_9">Swift</label><label for="__tabbed_10_10">Zig</label><label for="__tabbed_10_11">Dart</label><label for="__tabbed_10_12">Rust</label></div>
|
||||
@ -5592,7 +5592,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
</div>
|
||||
</div>
|
||||
<p>指数阶增长非常迅速，在穷举法（暴力搜索、回溯等）中比较常见。对于数据规模较大的问题，指数阶是不可接受的，通常需要使用「动态规划」或「贪心」等算法来解决。</p>
|
||||
<h3 id="olog-n">对数阶 <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#olog-n" title="Permanent link">¶</a></h3>
|
||||
<h3 id="5-olog-n">5. 对数阶 <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#5-olog-n" title="Permanent link">¶</a></h3>
|
||||
<p>与指数阶相反，对数阶反映了“每轮缩减到一半”的情况。设输入数据大小为 <span class="arithmatex">\(n\)</span> ，由于每轮缩减到一半，因此循环次数是 <span class="arithmatex">\(\log_2 n\)</span> ，即 <span class="arithmatex">\(2^n\)</span> 的反函数。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="12:12"><input checked="checked" id="__tabbed_12_1" name="__tabbed_12" type="radio" /><input id="__tabbed_12_2" name="__tabbed_12" type="radio" /><input id="__tabbed_12_3" name="__tabbed_12" type="radio" /><input id="__tabbed_12_4" name="__tabbed_12" type="radio" /><input id="__tabbed_12_5" name="__tabbed_12" type="radio" /><input id="__tabbed_12_6" name="__tabbed_12" type="radio" /><input id="__tabbed_12_7" name="__tabbed_12" type="radio" /><input id="__tabbed_12_8" name="__tabbed_12" type="radio" /><input id="__tabbed_12_9" name="__tabbed_12" type="radio" /><input id="__tabbed_12_10" name="__tabbed_12" type="radio" /><input id="__tabbed_12_11" name="__tabbed_12" type="radio" /><input id="__tabbed_12_12" name="__tabbed_12" type="radio" /><div class="tabbed-labels"><label for="__tabbed_12_1">Java</label><label for="__tabbed_12_2">C++</label><label for="__tabbed_12_3">Python</label><label for="__tabbed_12_4">Go</label><label for="__tabbed_12_5">JS</label><label for="__tabbed_12_6">TS</label><label for="__tabbed_12_7">C</label><label for="__tabbed_12_8">C#</label><label for="__tabbed_12_9">Swift</label><label for="__tabbed_12_10">Zig</label><label for="__tabbed_12_11">Dart</label><label for="__tabbed_12_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
@ -5857,7 +5857,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
</div>
|
||||
</div>
|
||||
<p>对数阶常出现于基于「分治」的算法中，体现了“一分为多”和“化繁为简”的算法思想。它增长缓慢，是理想的时间复杂度，仅次于常数阶。</p>
|
||||
<h3 id="on-log-n">线性对数阶 <span class="arithmatex">\(O(n \log n)\)</span><a class="headerlink" href="#on-log-n" title="Permanent link">¶</a></h3>
|
||||
<h3 id="6-on-log-n">6. 线性对数阶 <span class="arithmatex">\(O(n \log n)\)</span><a class="headerlink" href="#6-on-log-n" title="Permanent link">¶</a></h3>
|
||||
<p>线性对数阶常出现于嵌套循环中，两层循环的时间复杂度分别为 <span class="arithmatex">\(O(\log n)\)</span> 和 <span class="arithmatex">\(O(n)\)</span> 。</p>
|
||||
<p>主流排序算法的时间复杂度通常为 <span class="arithmatex">\(O(n \log n)\)</span> ，例如快速排序、归并排序、堆排序等。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="14:12"><input checked="checked" id="__tabbed_14_1" name="__tabbed_14" type="radio" /><input id="__tabbed_14_2" name="__tabbed_14" type="radio" /><input id="__tabbed_14_3" name="__tabbed_14" type="radio" /><input id="__tabbed_14_4" name="__tabbed_14" type="radio" /><input id="__tabbed_14_5" name="__tabbed_14" type="radio" /><input id="__tabbed_14_6" name="__tabbed_14" type="radio" /><input id="__tabbed_14_7" name="__tabbed_14" type="radio" /><input id="__tabbed_14_8" name="__tabbed_14" type="radio" /><input id="__tabbed_14_9" name="__tabbed_14" type="radio" /><input id="__tabbed_14_10" name="__tabbed_14" type="radio" /><input id="__tabbed_14_11" name="__tabbed_14" type="radio" /><input id="__tabbed_14_12" name="__tabbed_14" type="radio" /><div class="tabbed-labels"><label for="__tabbed_14_1">Java</label><label for="__tabbed_14_2">C++</label><label for="__tabbed_14_3">Python</label><label for="__tabbed_14_4">Go</label><label for="__tabbed_14_5">JS</label><label for="__tabbed_14_6">TS</label><label for="__tabbed_14_7">C</label><label for="__tabbed_14_8">C#</label><label for="__tabbed_14_9">Swift</label><label for="__tabbed_14_10">Zig</label><label for="__tabbed_14_11">Dart</label><label for="__tabbed_14_12">Rust</label></div>
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@ -6025,7 +6025,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
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<p><img alt="线性对数阶的时间复杂度" src="../time_complexity.assets/time_complexity_logarithmic_linear.png" /></p>
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<p align="center"> 图：线性对数阶的时间复杂度 </p>
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<h3 id="on_1">阶乘阶 <span class="arithmatex">\(O(n!)\)</span><a class="headerlink" href="#on_1" title="Permanent link">¶</a></h3>
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<h3 id="7-on">7. 阶乘阶 <span class="arithmatex">\(O(n!)\)</span><a class="headerlink" href="#7-on" title="Permanent link">¶</a></h3>
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||||
<p>阶乘阶对应数学上的“全排列”问题。给定 <span class="arithmatex">\(n\)</span> 个互不重复的元素，求其所有可能的排列方案，方案数量为：</p>
|
||||
<div class="arithmatex">\[
|
||||
n! = n \times (n - 1) \times (n - 2) \times \cdots \times 2 \times 1
|
||||
|
||||
Reference in New Issue
Block a user