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2.2.1. 统计算法运行时间
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2.2.1. 统计时间增长趋势
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2.2.2. 统计时间增长趋势
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2.2.2. 函数渐近上界
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2.2.3. 函数渐近上界
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2.2.3. 推算方法
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2.2.4. 推算方法
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2.2.5. 常见类型
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2.2.4. 常见类型
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2.2.6. 最差、最佳、平均时间复杂度
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2.2.5. 最差、最佳、平均时间复杂度
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2.2.1. 统计算法运行时间
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2.2.1. 统计时间增长趋势
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2.2.2. 统计时间增长趋势
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2.2.2. 函数渐近上界
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2.2.3. 函数渐近上界
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2.2.3. 推算方法
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2.2.4. 推算方法
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2.2.4. 常见类型
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2.2.6. 最差、最佳、平均时间复杂度
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<a href="#225" class="md-nav__link">
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2.2.5. 最差、最佳、平均时间复杂度
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</a>
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<h1 id="22">2.2. 时间复杂度<a class="headerlink" href="#22" title="Permanent link">¶</a></h1>
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<h2 id="221">2.2.1. 统计算法运行时间<a class="headerlink" href="#221" title="Permanent link">¶</a></h2>
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<p>运行时间可以直观且准确地反映算法的效率。然而，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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<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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<ol>
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<li><strong>确定运行平台</strong>，包括硬件配置、编程语言、系统环境等，这些因素都会影响代码的运行效率。</li>
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<li><strong>评估各种计算操作所需的运行时间</strong>，例如加法操作 <code>+</code> 需要 1 ns，乘法操作 <code>*</code> 需要 10 ns，打印操作需要 5 ns 等。</li>
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<li><strong>统计代码中所有的计算操作</strong>，并将所有操作的执行时间求和，从而得到运行时间。</li>
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</ol>
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<p>例如以下代码，输入数据大小为 <span class="arithmatex">\(n\)</span> ，根据以上方法，可以得到算法运行时间为 <span class="arithmatex">\(6n + 12\)</span> ns 。</p>
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<p>例如以下代码，输入数据大小为 <span class="arithmatex">\(n\)</span> 。根据以上方法，可以得到算法运行时间为 <span class="arithmatex">\(6n + 12\)</span> ns 。</p>
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<div class="arithmatex">\[
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1 + 1 + 10 + (1 + 5) \times n = 6n + 12
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\]</div>
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@@ -3764,20 +3749,24 @@
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a>
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<div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="c1">// 在某运行平台下</span>
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<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="k">fn</span> <span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 1 ns</span>
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<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 1 ns</span>
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<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 10 ns</span>
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<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w"> </span><span class="c1">// 循环 n 次</span>
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<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 1 ns ，每轮都要执行 i++</span>
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<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w"> </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{}"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// 5 ns</span>
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<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="p">}</span>
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</code></pre></div>
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</div>
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</div>
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</div>
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<p>然而实际上，<strong>统计算法的运行时间既不合理也不现实</strong>。首先，我们不希望预估时间和运行平台绑定，因为算法需要在各种不同的平台上运行。其次，我们很难获知每种操作的运行时间，这给预估过程带来了极大的难度。</p>
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<h2 id="222">2.2.2. 统计时间增长趋势<a class="headerlink" href="#222" title="Permanent link">¶</a></h2>
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<p>但实际上，<strong>统计算法的运行时间既不合理也不现实</strong>。首先，我们不希望预估时间和运行平台绑定，因为算法需要在各种不同的平台上运行。其次，我们很难获知每种操作的运行时间，这给预估过程带来了极大的难度。</p>
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<h2 id="221">2.2.1. 统计时间增长趋势<a class="headerlink" href="#221" title="Permanent link">¶</a></h2>
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<p>「时间复杂度分析」采取了一种不同的方法，其统计的不是算法运行时间，<strong>而是算法运行时间随着数据量变大时的增长趋势</strong>。</p>
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<p>“时间增长趋势”这个概念较为抽象，我们通过一个例子来加以理解。假设输入数据大小为 <span class="arithmatex">\(n\)</span> ，给定三个算法 <code>A</code> , <code>B</code> , <code>C</code> 。</p>
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<ul>
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<li>算法 <code>A</code> 只有 <span class="arithmatex">\(1\)</span> 个打印操作，算法运行时间不随着 <span class="arithmatex">\(n\)</span> 增大而增长。我们称此算法的时间复杂度为「常数阶」。</li>
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<li>算法 <code>B</code> 中的打印操作需要循环 <span class="arithmatex">\(n\)</span> 次，算法运行时间随着 <span class="arithmatex">\(n\)</span> 增大呈线性增长。此算法的时间复杂度被称为「线性阶」。</li>
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<li>算法 <code>C</code> 中的打印操作需要循环 <span class="arithmatex">\(1000000\)</span> 次，但运行时间仍与输入数据大小 <span class="arithmatex">\(n\)</span> 无关。因此 <code>C</code> 的时间复杂度和 <code>A</code> 相同，仍为「常数阶」。</li>
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</ul>
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<p>“时间增长趋势”这个概念比较抽象，我们通过一个例子来加以理解。假设输入数据大小为 <span class="arithmatex">\(n\)</span> ，给定三个算法函数 <code>A</code> , <code>B</code> , <code>C</code> 。</p>
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<div class="tabbed-content">
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<div class="tabbed-block">
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@@ -3972,23 +3961,38 @@
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</code></pre></div>
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</div>
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<div class="tabbed-block">
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<div class="highlight"><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a>
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<div class="highlight"><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="c1">// 算法 A 时间复杂度：常数阶</span>
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<a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="k">fn</span> <span class="nf">algorithm_A</span><span class="p">(</span><span class="n">n</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w"> </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{}"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
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<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="p">}</span>
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<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="c1">// 算法 B 时间复杂度：线性阶</span>
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<a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="k">fn</span> <span class="nf">algorithm_B</span><span class="p">(</span><span class="n">n</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w"> </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{}"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
|
||||
<a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w"> </span><span class="p">}</span>
|
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<a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="p">}</span>
|
||||
<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="c1">// 算法 C 时间复杂度：常数阶</span>
|
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<a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="k">fn</span> <span class="nf">algorithm_C</span><span class="p">(</span><span class="n">n</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
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<a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="mi">1000000</span><span class="w"> </span><span class="p">{</span>
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<a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w"> </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{}"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
|
||||
<a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w"> </span><span class="p">}</span>
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<a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="p">}</span>
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</code></pre></div>
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<p>算法 <code>A</code> 只有 <span class="arithmatex">\(1\)</span> 个打印操作，算法运行时间不随着 <span class="arithmatex">\(n\)</span> 增大而增长。我们称此算法的时间复杂度为「常数阶」。</p>
|
||||
<p>算法 <code>B</code> 中的打印操作需要循环 <span class="arithmatex">\(n\)</span> 次，算法运行时间随着 <span class="arithmatex">\(n\)</span> 增大呈线性增长。此算法的时间复杂度被称为「线性阶」。</p>
|
||||
<p>算法 <code>C</code> 中的打印操作需要循环 <span class="arithmatex">\(1000000\)</span> 次，虽然运行时间很长，但它与输入数据大小 <span class="arithmatex">\(n\)</span> 无关。因此 <code>C</code> 的时间复杂度和 <code>A</code> 相同，仍为「常数阶」。</p>
|
||||
<p><img alt="算法 A, B, C 的时间增长趋势" src="../time_complexity.assets/time_complexity_simple_example.png" /></p>
|
||||
<p align="center"> Fig. 算法 A, B, C 的时间增长趋势 </p>
|
||||
|
||||
<p>相较于直接统计算法运行时间，时间复杂度分析有哪些优势和局限性呢？</p>
|
||||
<p><strong>时间复杂度能够有效评估算法效率</strong>。例如，算法 <code>B</code> 的运行时间呈线性增长，在 <span class="arithmatex">\(n > 1\)</span> 时比算法 <code>A</code> 慢，在 <span class="arithmatex">\(n > 1000000\)</span> 时比算法 <code>C</code> 慢。事实上，只要输入数据大小 <span class="arithmatex">\(n\)</span> 足够大，复杂度为「常数阶」的算法一定优于「线性阶」的算法，这正是时间增长趋势所表达的含义。</p>
|
||||
<p><strong>时间复杂度的推算方法更简便</strong>。显然，运行平台和计算操作类型都与算法运行时间的增长趋势无关。因此在时间复杂度分析中，我们可以简单地将所有计算操作的执行时间视为相同的“单位时间”，从而将“计算操作的运行时间的统计”简化为“计算操作的数量的统计”，这样的简化方法大大降低了估算难度。</p>
|
||||
<p>相较于直接统计算法运行时间，时间复杂度分析有哪些特点呢？</p>
|
||||
<p><strong>时间复杂度能够有效评估算法效率</strong>。例如，算法 <code>B</code> 的运行时间呈线性增长，在 <span class="arithmatex">\(n > 1\)</span> 时比算法 <code>A</code> 更慢，在 <span class="arithmatex">\(n > 1000000\)</span> 时比算法 <code>C</code> 更慢。事实上，只要输入数据大小 <span class="arithmatex">\(n\)</span> 足够大，复杂度为“常数阶”的算法一定优于“线性阶”的算法，这正是时间增长趋势所表达的含义。</p>
|
||||
<p><strong>时间复杂度的推算方法更简便</strong>。显然，运行平台和计算操作类型都与算法运行时间的增长趋势无关。因此在时间复杂度分析中，我们可以简单地将所有计算操作的执行时间视为相同的“单位时间”，从而将“计算操作的运行时间的统计”简化为“计算操作的数量的统计”，这样以来估算难度就大大降低了。</p>
|
||||
<p><strong>时间复杂度也存在一定的局限性</strong>。例如，尽管算法 <code>A</code> 和 <code>C</code> 的时间复杂度相同，但实际运行时间差别很大。同样，尽管算法 <code>B</code> 的时间复杂度比 <code>C</code> 高，但在输入数据大小 <span class="arithmatex">\(n\)</span> 较小时，算法 <code>B</code> 明显优于算法 <code>C</code> 。在这些情况下，我们很难仅凭时间复杂度判断算法效率高低。当然，尽管存在上述问题，复杂度分析仍然是评判算法效率最有效且常用的方法。</p>
|
||||
<h2 id="223">2.2.3. 函数渐近上界<a class="headerlink" href="#223" title="Permanent link">¶</a></h2>
|
||||
<p>设算法的计算操作数量是一个关于输入数据大小 <span class="arithmatex">\(n\)</span> 的函数，记为 <span class="arithmatex">\(T(n)\)</span> ，则以下算法的操作数量为</p>
|
||||
<div class="arithmatex">\[
|
||||
T(n) = 3 + 2n
|
||||
\]</div>
|
||||
<h2 id="222">2.2.2. 函数渐近上界<a class="headerlink" href="#222" title="Permanent link">¶</a></h2>
|
||||
<p>给定一个函数 <code>algorithm()</code> ：</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JS</label><label for="__tabbed_3_6">TS</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
@@ -4114,14 +4118,27 @@ T(n) = 3 + 2n
|
||||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a>
|
||||
<div class="highlight"><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="k">fn</span> <span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span>
|
||||
<a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span>
|
||||
<a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span>
|
||||
<a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a>
|
||||
<a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w"> </span><span class="c1">// 循环 n 次</span>
|
||||
<a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// +1（每轮都执行 i ++）</span>
|
||||
<a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w"> </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{}"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// +1</span>
|
||||
<a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p><span class="arithmatex">\(T(n)\)</span> 是一次函数，说明时间增长趋势是线性的，因此可以得出时间复杂度是线性阶。</p>
|
||||
<p>设算法的计算操作数量是一个关于输入数据大小 <span class="arithmatex">\(n\)</span> 的函数，记为 <span class="arithmatex">\(T(n)\)</span> ，则以上函数的的操作数量为：</p>
|
||||
<div class="arithmatex">\[
|
||||
T(n) = 3 + 2n
|
||||
\]</div>
|
||||
<p><span class="arithmatex">\(T(n)\)</span> 是一次函数，说明时间的增长趋势是线性的，因此其时间复杂度是线性阶。</p>
|
||||
<p>我们将线性阶的时间复杂度记为 <span class="arithmatex">\(O(n)\)</span> ，这个数学符号称为「大 <span class="arithmatex">\(O\)</span> 记号 Big-<span class="arithmatex">\(O\)</span> Notation」，表示函数 <span class="arithmatex">\(T(n)\)</span> 的「渐近上界 Asymptotic Upper Bound」。</p>
|
||||
<p>推算时间复杂度本质上是计算“操作数量函数 <span class="arithmatex">\(T(n)\)</span>”的渐近上界。接下来，我们来看函数渐近上界的数学定义。</p>
|
||||
<p>时间复杂度分析本质上是计算“操作数量函数 <span class="arithmatex">\(T(n)\)</span>”的渐近上界。接下来，我们来看函数渐近上界的数学定义。</p>
|
||||
<div class="admonition abstract">
|
||||
<p class="admonition-title">函数渐近上界</p>
|
||||
<p>若存在正实数 <span class="arithmatex">\(c\)</span> 和实数 <span class="arithmatex">\(n_0\)</span> ，使得对于所有的 <span class="arithmatex">\(n > n_0\)</span> ，均有
|
||||
@@ -4136,18 +4153,18 @@ $$</p>
|
||||
<p><img alt="函数的渐近上界" src="../time_complexity.assets/asymptotic_upper_bound.png" /></p>
|
||||
<p align="center"> Fig. 函数的渐近上界 </p>
|
||||
|
||||
<p>从本质上讲，计算渐近上界就是寻找一个函数 <span class="arithmatex">\(f(n)\)</span> ，使得当 <span class="arithmatex">\(n\)</span> 趋向于无穷大时，<span class="arithmatex">\(T(n)\)</span> 和 <span class="arithmatex">\(f(n)\)</span> 处于相同的增长级别，仅相差一个常数项 <span class="arithmatex">\(c\)</span> 的倍数。</p>
|
||||
<h2 id="224">2.2.4. 推算方法<a class="headerlink" href="#224" title="Permanent link">¶</a></h2>
|
||||
<p>也就是说，计算渐近上界就是寻找一个函数 <span class="arithmatex">\(f(n)\)</span> ，使得当 <span class="arithmatex">\(n\)</span> 趋向于无穷大时，<span class="arithmatex">\(T(n)\)</span> 和 <span class="arithmatex">\(f(n)\)</span> 处于相同的增长级别，仅相差一个常数项 <span class="arithmatex">\(c\)</span> 的倍数。</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>
|
||||
<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>忽略与 <span class="arithmatex">\(n\)</span> 无关的操作</strong>。因为它们都是 <span class="arithmatex">\(T(n)\)</span> 中的常数项，对时间复杂度不产生影响。</li>
|
||||
<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>
|
||||
<li><strong>省略所有系数</strong>。例如，循环 <span class="arithmatex">\(2n\)</span> 次、<span class="arithmatex">\(5n + 1\)</span> 次等，都可以简化记为 <span class="arithmatex">\(n\)</span> 次，因为 <span class="arithmatex">\(n\)</span> 前面的系数对时间复杂度没有影响。</li>
|
||||
<li><strong>循环嵌套时使用乘法</strong>。总操作数量等于外层循环和内层循环操作数量之积，每一层循环依然可以分别套用上述 <code>1.</code> 和 <code>2.</code> 技巧。</li>
|
||||
</ol>
|
||||
<p>以下示例展示了使用上述技巧前、后的统计结果。</p>
|
||||
<p>以下示例展示了使用上述技巧前、后的统计结果。两者推出的时间复杂度相同，即为 <span class="arithmatex">\(O(n^2)\)</span> 。</p>
|
||||
<div class="arithmatex">\[
|
||||
\begin{aligned}
|
||||
T(n) & = 2n(n + 1) + (5n + 1) + 2 & \text{完整统计 (-.-|||)} \newline
|
||||
@@ -4155,7 +4172,6 @@ T(n) & = 2n(n + 1) + (5n + 1) + 2 & \text{完整统计 (-.-|||)} \newlin
|
||||
T(n) & = n^2 + n & \text{偷懒统计 (o.O)}
|
||||
\end{aligned}
|
||||
\]</div>
|
||||
<p>最终，两者都能推出相同的时间复杂度结果，即 <span class="arithmatex">\(O(n^2)\)</span> 。</p>
|
||||
<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>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
@@ -4329,7 +4345,22 @@ T(n) & = n^2 + n & \text{偷懒统计 (o.O)}
|
||||
</code></pre></div>
|
||||
</div>
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a>
|
||||
<div class="highlight"><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="k">fn</span> <span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-47-2" name="__codelineno-47-2" href="#__codelineno-47-2"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// +0（技巧 1）</span>
|
||||
<a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="c1">// +0（技巧 1）</span>
|
||||
<a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a>
|
||||
<a id="__codelineno-47-5" name="__codelineno-47-5" href="#__codelineno-47-5"></a><span class="w"> </span><span class="c1">// +n（技巧 2）</span>
|
||||
<a id="__codelineno-47-6" name="__codelineno-47-6" href="#__codelineno-47-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="p">(</span><span class="mi">5</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-47-7" name="__codelineno-47-7" href="#__codelineno-47-7"></a><span class="w"> </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{}"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
|
||||
<a id="__codelineno-47-8" name="__codelineno-47-8" href="#__codelineno-47-8"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-47-9" name="__codelineno-47-9" href="#__codelineno-47-9"></a>
|
||||
<a id="__codelineno-47-10" name="__codelineno-47-10" href="#__codelineno-47-10"></a><span class="w"> </span><span class="c1">// +n*n（技巧 3）</span>
|
||||
<a id="__codelineno-47-11" name="__codelineno-47-11" href="#__codelineno-47-11"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-47-12" name="__codelineno-47-12" href="#__codelineno-47-12"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-47-13" name="__codelineno-47-13" href="#__codelineno-47-13"></a><span class="w"> </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{}"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
|
||||
<a id="__codelineno-47-14" name="__codelineno-47-14" href="#__codelineno-47-14"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-47-15" name="__codelineno-47-15" href="#__codelineno-47-15"></a><span class="w"> </span><span class="p">}</span>
|
||||
<a id="__codelineno-47-16" name="__codelineno-47-16" href="#__codelineno-47-16"></a><span class="p">}</span>
|
||||
</code></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -4369,7 +4400,7 @@ T(n) & = n^2 + n & \text{偷懒统计 (o.O)}
|
||||
</tbody>
|
||||
</table>
|
||||
</div>
|
||||
<h2 id="225">2.2.5. 常见类型<a class="headerlink" href="#225" title="Permanent link">¶</a></h2>
|
||||
<h2 id="224">2.2.4. 常见类型<a class="headerlink" href="#224" title="Permanent link">¶</a></h2>
|
||||
<p>设输入数据大小为 <span class="arithmatex">\(n\)</span> ，常见的时间复杂度类型包括（按照从低到高的顺序排列）：</p>
|
||||
<div class="arithmatex">\[
|
||||
\begin{aligned}
|
||||
@@ -4382,7 +4413,7 @@ O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!
|
||||
|
||||
<div class="admonition tip">
|
||||
<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、递归算法等。如果遇到不理解的部分，请不要担心，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>
|
||||
<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>
|
||||
<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>
|
||||
@@ -4661,10 +4692,6 @@ O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!
|
||||
</div>
|
||||
</div>
|
||||
<p>遍历数组和遍历链表等操作的时间复杂度均为 <span class="arithmatex">\(O(n)\)</span> ，其中 <span class="arithmatex">\(n\)</span> 为数组或链表的长度。</p>
|
||||
<div class="admonition question">
|
||||
<p class="admonition-title">如何确定输入数据大小 <span class="arithmatex">\(n\)</span> ？</p>
|
||||
<p><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>。例如，在上述示例中，我们直接将 <span class="arithmatex">\(n\)</span> 视为输入数据大小；在下面遍历数组的示例中，数据大小 <span class="arithmatex">\(n\)</span> 为数组的长度。</p>
|
||||
</div>
|
||||
<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>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
@@ -4811,6 +4838,7 @@ O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!
|
||||
</div>
|
||||
</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>
|
||||
<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>
|
||||
@@ -5244,18 +5272,15 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
</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>
|
||||
<div class="admonition note">
|
||||
<p class="admonition-title">Note</p>
|
||||
<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>
|
||||
</div>
|
||||
<p>指数阶增长非常迅速，在实际应用中通常是不可接受的。若一个问题使用「暴力枚举」求解的时间复杂度为 <span class="arithmatex">\(O(2^n)\)</span> ，那么通常需要使用「动态规划」或「贪心算法」等方法来解决。</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>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-108-1" name="__codelineno-108-1" href="#__codelineno-108-1"></a><span class="cm">/* 指数阶（循环实现） */</span>
|
||||
<a id="__codelineno-108-2" name="__codelineno-108-2" href="#__codelineno-108-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">exponential</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>
|
||||
<a id="__codelineno-108-3" name="__codelineno-108-3" href="#__codelineno-108-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</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">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
|
||||
<a id="__codelineno-108-4" name="__codelineno-108-4" href="#__codelineno-108-4"></a><span class="w"> </span><span class="c1">// cell 每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-108-4" name="__codelineno-108-4" href="#__codelineno-108-4"></a><span class="w"> </span><span class="c1">// 细胞每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-108-5" name="__codelineno-108-5" href="#__codelineno-108-5"></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"><</span><span class="w"> </span><span class="n">n</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-108-6" name="__codelineno-108-6" href="#__codelineno-108-6"></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">j</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">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">base</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-108-7" name="__codelineno-108-7" href="#__codelineno-108-7"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
|
||||
@@ -5271,7 +5296,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-109-1" name="__codelineno-109-1" href="#__codelineno-109-1"></a><span class="cm">/* 指数阶（循环实现） */</span>
|
||||
<a id="__codelineno-109-2" name="__codelineno-109-2" href="#__codelineno-109-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">exponential</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>
|
||||
<a id="__codelineno-109-3" name="__codelineno-109-3" href="#__codelineno-109-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</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">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
|
||||
<a id="__codelineno-109-4" name="__codelineno-109-4" href="#__codelineno-109-4"></a><span class="w"> </span><span class="c1">// cell 每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-109-4" name="__codelineno-109-4" href="#__codelineno-109-4"></a><span class="w"> </span><span class="c1">// 细胞每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-109-5" name="__codelineno-109-5" href="#__codelineno-109-5"></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"><</span><span class="w"> </span><span class="n">n</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-109-6" name="__codelineno-109-6" href="#__codelineno-109-6"></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">j</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">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">base</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-109-7" name="__codelineno-109-7" href="#__codelineno-109-7"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
|
||||
@@ -5288,7 +5313,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
<a id="__codelineno-110-2" name="__codelineno-110-2" href="#__codelineno-110-2"></a><span class="w"> </span><span class="sd">"""指数阶（循环实现）"""</span>
|
||||
<a id="__codelineno-110-3" name="__codelineno-110-3" href="#__codelineno-110-3"></a> <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
|
||||
<a id="__codelineno-110-4" name="__codelineno-110-4" href="#__codelineno-110-4"></a> <span class="n">base</span> <span class="o">=</span> <span class="mi">1</span>
|
||||
<a id="__codelineno-110-5" name="__codelineno-110-5" href="#__codelineno-110-5"></a> <span class="c1"># cell 每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-110-5" name="__codelineno-110-5" href="#__codelineno-110-5"></a> <span class="c1"># 细胞每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-110-6" name="__codelineno-110-6" href="#__codelineno-110-6"></a> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<a id="__codelineno-110-7" name="__codelineno-110-7" href="#__codelineno-110-7"></a> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">base</span><span class="p">):</span>
|
||||
<a id="__codelineno-110-8" name="__codelineno-110-8" href="#__codelineno-110-8"></a> <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
|
||||
@@ -5301,7 +5326,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-111-1" name="__codelineno-111-1" href="#__codelineno-111-1"></a><span class="cm">/* 指数阶（循环实现）*/</span>
|
||||
<a id="__codelineno-111-2" name="__codelineno-111-2" href="#__codelineno-111-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">exponential</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-111-3" name="__codelineno-111-3" href="#__codelineno-111-3"></a><span class="w"> </span><span class="nx">count</span><span class="p">,</span><span class="w"> </span><span class="nx">base</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="mi">1</span>
|
||||
<a id="__codelineno-111-4" name="__codelineno-111-4" href="#__codelineno-111-4"></a><span class="w"> </span><span class="c1">// cell 每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-111-4" name="__codelineno-111-4" href="#__codelineno-111-4"></a><span class="w"> </span><span class="c1">// 细胞每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-111-5" name="__codelineno-111-5" href="#__codelineno-111-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">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="nx">i</span><span class="w"> </span><span class="p"><</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-111-6" name="__codelineno-111-6" href="#__codelineno-111-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">j</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="nx">j</span><span class="w"> </span><span class="p"><</span><span class="w"> </span><span class="nx">base</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-111-7" name="__codelineno-111-7" href="#__codelineno-111-7"></a><span class="w"> </span><span class="nx">count</span><span class="o">++</span>
|
||||
@@ -5318,7 +5343,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
<a id="__codelineno-112-2" name="__codelineno-112-2" href="#__codelineno-112-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">exponential</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-112-3" name="__codelineno-112-3" href="#__codelineno-112-3"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
|
||||
<a id="__codelineno-112-4" name="__codelineno-112-4" href="#__codelineno-112-4"></a><span class="w"> </span><span class="nx">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
|
||||
<a id="__codelineno-112-5" name="__codelineno-112-5" href="#__codelineno-112-5"></a><span class="w"> </span><span class="c1">// cell 每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-112-5" name="__codelineno-112-5" href="#__codelineno-112-5"></a><span class="w"> </span><span class="c1">// 细胞每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-112-6" name="__codelineno-112-6" href="#__codelineno-112-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-112-7" name="__codelineno-112-7" href="#__codelineno-112-7"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="nx">base</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-112-8" name="__codelineno-112-8" href="#__codelineno-112-8"></a><span class="w"> </span><span class="nx">count</span><span class="o">++</span><span class="p">;</span>
|
||||
@@ -5335,7 +5360,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
<a id="__codelineno-113-2" name="__codelineno-113-2" href="#__codelineno-113-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">exponential</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-113-3" name="__codelineno-113-3" href="#__codelineno-113-3"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
|
||||
<a id="__codelineno-113-4" name="__codelineno-113-4" href="#__codelineno-113-4"></a><span class="w"> </span><span class="nx">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
|
||||
<a id="__codelineno-113-5" name="__codelineno-113-5" href="#__codelineno-113-5"></a><span class="w"> </span><span class="c1">// cell 每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-113-5" name="__codelineno-113-5" href="#__codelineno-113-5"></a><span class="w"> </span><span class="c1">// 细胞每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-113-6" name="__codelineno-113-6" href="#__codelineno-113-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-113-7" name="__codelineno-113-7" href="#__codelineno-113-7"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="nx">base</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-113-8" name="__codelineno-113-8" href="#__codelineno-113-8"></a><span class="w"> </span><span class="nx">count</span><span class="o">++</span><span class="p">;</span>
|
||||
@@ -5352,7 +5377,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
<a id="__codelineno-114-2" name="__codelineno-114-2" href="#__codelineno-114-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">exponential</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>
|
||||
<a id="__codelineno-114-3" name="__codelineno-114-3" href="#__codelineno-114-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
|
||||
<a id="__codelineno-114-4" name="__codelineno-114-4" href="#__codelineno-114-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">bas</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
|
||||
<a id="__codelineno-114-5" name="__codelineno-114-5" href="#__codelineno-114-5"></a><span class="w"> </span><span class="c1">// cell 每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-114-5" name="__codelineno-114-5" href="#__codelineno-114-5"></a><span class="w"> </span><span class="c1">// 细胞每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-114-6" name="__codelineno-114-6" href="#__codelineno-114-6"></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"><</span><span class="w"> </span><span class="n">n</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-114-7" name="__codelineno-114-7" href="#__codelineno-114-7"></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">j</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">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">bas</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-114-8" name="__codelineno-114-8" href="#__codelineno-114-8"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
|
||||
@@ -5368,7 +5393,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-115-1" name="__codelineno-115-1" href="#__codelineno-115-1"></a><span class="cm">/* 指数阶（循环实现） */</span>
|
||||
<a id="__codelineno-115-2" name="__codelineno-115-2" href="#__codelineno-115-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">exponential</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>
|
||||
<a id="__codelineno-115-3" name="__codelineno-115-3" href="#__codelineno-115-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">bas</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
|
||||
<a id="__codelineno-115-4" name="__codelineno-115-4" href="#__codelineno-115-4"></a><span class="w"> </span><span class="c1">// cell 每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-115-4" name="__codelineno-115-4" href="#__codelineno-115-4"></a><span class="w"> </span><span class="c1">// 细胞每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-115-5" name="__codelineno-115-5" href="#__codelineno-115-5"></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="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</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-115-6" name="__codelineno-115-6" href="#__codelineno-115-6"></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">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">bas</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-115-7" name="__codelineno-115-7" href="#__codelineno-115-7"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
|
||||
@@ -5385,7 +5410,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
<a id="__codelineno-116-2" name="__codelineno-116-2" href="#__codelineno-116-2"></a><span class="kd">func</span> <span class="nf">exponential</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">Int</span><span class="p">)</span> <span class="p">-></span> <span class="nb">Int</span> <span class="p">{</span>
|
||||
<a id="__codelineno-116-3" name="__codelineno-116-3" href="#__codelineno-116-3"></a> <span class="kd">var</span> <span class="nv">count</span> <span class="p">=</span> <span class="mi">0</span>
|
||||
<a id="__codelineno-116-4" name="__codelineno-116-4" href="#__codelineno-116-4"></a> <span class="kd">var</span> <span class="nv">base</span> <span class="p">=</span> <span class="mi">1</span>
|
||||
<a id="__codelineno-116-5" name="__codelineno-116-5" href="#__codelineno-116-5"></a> <span class="c1">// cell 每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-116-5" name="__codelineno-116-5" href="#__codelineno-116-5"></a> <span class="c1">// 细胞每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-116-6" name="__codelineno-116-6" href="#__codelineno-116-6"></a> <span class="k">for</span> <span class="kc">_</span> <span class="k">in</span> <span class="mi">0</span> <span class="p">..</span><span class="o"><</span> <span class="n">n</span> <span class="p">{</span>
|
||||
<a id="__codelineno-116-7" name="__codelineno-116-7" href="#__codelineno-116-7"></a> <span class="k">for</span> <span class="kc">_</span> <span class="k">in</span> <span class="mi">0</span> <span class="p">..</span><span class="o"><</span> <span class="n">base</span> <span class="p">{</span>
|
||||
<a id="__codelineno-116-8" name="__codelineno-116-8" href="#__codelineno-116-8"></a> <span class="bp">count</span> <span class="o">+=</span> <span class="mi">1</span>
|
||||
@@ -5403,7 +5428,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
<a id="__codelineno-117-3" name="__codelineno-117-3" href="#__codelineno-117-3"></a><span class="w"> </span><span class="kr">var</span><span class="w"> </span><span class="n">count</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
|
||||
<a id="__codelineno-117-4" name="__codelineno-117-4" href="#__codelineno-117-4"></a><span class="w"> </span><span class="kr">var</span><span class="w"> </span><span class="n">bas</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
|
||||
<a id="__codelineno-117-5" name="__codelineno-117-5" href="#__codelineno-117-5"></a><span class="w"> </span><span class="kr">var</span><span class="w"> </span><span class="n">i</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
|
||||
<a id="__codelineno-117-6" name="__codelineno-117-6" href="#__codelineno-117-6"></a><span class="w"> </span><span class="c1">// cell 每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-117-6" name="__codelineno-117-6" href="#__codelineno-117-6"></a><span class="w"> </span><span class="c1">// 细胞每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-117-7" name="__codelineno-117-7" href="#__codelineno-117-7"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-117-8" name="__codelineno-117-8" href="#__codelineno-117-8"></a><span class="w"> </span><span class="kr">var</span><span class="w"> </span><span class="n">j</span><span class="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
|
||||
<a id="__codelineno-117-9" name="__codelineno-117-9" href="#__codelineno-117-9"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">bas</span><span class="p">)</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
@@ -5420,7 +5445,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-118-1" name="__codelineno-118-1" href="#__codelineno-118-1"></a><span class="cm">/* 指数阶（循环实现） */</span>
|
||||
<a id="__codelineno-118-2" name="__codelineno-118-2" href="#__codelineno-118-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">exponential</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>
|
||||
<a id="__codelineno-118-3" name="__codelineno-118-3" href="#__codelineno-118-3"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
|
||||
<a id="__codelineno-118-4" name="__codelineno-118-4" href="#__codelineno-118-4"></a><span class="w"> </span><span class="c1">// cell 每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-118-4" name="__codelineno-118-4" href="#__codelineno-118-4"></a><span class="w"> </span><span class="c1">// 细胞每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-118-5" name="__codelineno-118-5" href="#__codelineno-118-5"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">n</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-118-6" name="__codelineno-118-6" href="#__codelineno-118-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">base</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-118-7" name="__codelineno-118-7" href="#__codelineno-118-7"></a><span class="w"> </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
|
||||
@@ -5437,7 +5462,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
<a id="__codelineno-119-2" name="__codelineno-119-2" href="#__codelineno-119-2"></a><span class="k">fn</span> <span class="nf">exponential</span><span class="p">(</span><span class="n">n</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span>-> <span class="kt">i32</span> <span class="p">{</span>
|
||||
<a id="__codelineno-119-3" name="__codelineno-119-3" href="#__codelineno-119-3"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
|
||||
<a id="__codelineno-119-4" name="__codelineno-119-4" href="#__codelineno-119-4"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
|
||||
<a id="__codelineno-119-5" name="__codelineno-119-5" href="#__codelineno-119-5"></a><span class="w"> </span><span class="c1">// cell 每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-119-5" name="__codelineno-119-5" href="#__codelineno-119-5"></a><span class="w"> </span><span class="c1">// 细胞每轮一分为二，形成数列 1, 2, 4, 8, ..., 2^(n-1)</span>
|
||||
<a id="__codelineno-119-6" name="__codelineno-119-6" href="#__codelineno-119-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-119-7" name="__codelineno-119-7" href="#__codelineno-119-7"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">base</span><span class="w"> </span><span class="p">{</span>
|
||||
<a id="__codelineno-119-8" name="__codelineno-119-8" href="#__codelineno-119-8"></a><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
|
||||
@@ -5454,7 +5479,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
<p><img alt="指数阶的时间复杂度" src="../time_complexity.assets/time_complexity_exponential.png" /></p>
|
||||
<p align="center"> Fig. 指数阶的时间复杂度 </p>
|
||||
|
||||
<p>在实际算法中，指数阶常出现于递归函数。例如以下代码，不断地一分为二，经过 <span class="arithmatex">\(n\)</span> 次分裂后停止。</p>
|
||||
<p>在实际算法中，指数阶常出现于递归函数。例如以下代码，其递归地一分为二，经过 <span class="arithmatex">\(n\)</span> 次分裂后停止。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="11:12"><input checked="checked" id="__tabbed_11_1" name="__tabbed_11" type="radio" /><input id="__tabbed_11_2" name="__tabbed_11" type="radio" /><input id="__tabbed_11_3" name="__tabbed_11" type="radio" /><input id="__tabbed_11_4" name="__tabbed_11" type="radio" /><input id="__tabbed_11_5" name="__tabbed_11" type="radio" /><input id="__tabbed_11_6" name="__tabbed_11" type="radio" /><input id="__tabbed_11_7" name="__tabbed_11" type="radio" /><input id="__tabbed_11_8" name="__tabbed_11" type="radio" /><input id="__tabbed_11_9" name="__tabbed_11" type="radio" /><input id="__tabbed_11_10" name="__tabbed_11" type="radio" /><input id="__tabbed_11_11" name="__tabbed_11" type="radio" /><input id="__tabbed_11_12" name="__tabbed_11" type="radio" /><div class="tabbed-labels"><label for="__tabbed_11_1">Java</label><label for="__tabbed_11_2">C++</label><label for="__tabbed_11_3">Python</label><label for="__tabbed_11_4">Go</label><label for="__tabbed_11_5">JS</label><label for="__tabbed_11_6">TS</label><label for="__tabbed_11_7">C</label><label for="__tabbed_11_8">C#</label><label for="__tabbed_11_9">Swift</label><label for="__tabbed_11_10">Zig</label><label for="__tabbed_11_11">Dart</label><label for="__tabbed_11_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
@@ -5564,10 +5589,9 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
</div>
|
||||
</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>
|
||||
<p>与指数阶相反，对数阶反映了“每轮缩减到一半的情况”。对数阶仅次于常数阶，时间增长缓慢，是理想的时间复杂度。</p>
|
||||
<p>对数阶常出现于「二分查找」和「分治算法」中，体现了“一分为多”和“化繁为简”的算法思想。</p>
|
||||
<p>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>
|
||||
<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">
|
||||
<div class="tabbed-block">
|
||||
@@ -5830,6 +5854,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
</div>
|
||||
</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>
|
||||
<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>
|
||||
@@ -5999,7 +6024,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
<p align="center"> Fig. 线性对数阶的时间复杂度 </p>
|
||||
|
||||
<h3 id="on_1">阶乘阶 <span class="arithmatex">\(O(n!)\)</span><a class="headerlink" href="#on_1" title="Permanent link">¶</a></h3>
|
||||
<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>
|
||||
<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">\[
|
||||
n! = n \times (n - 1) \times (n - 2) \times \cdots \times 2 \times 1
|
||||
\]</div>
|
||||
@@ -6175,13 +6200,14 @@ n! = n \times (n - 1) \times (n - 2) \times \cdots \times 2 \times 1
|
||||
<p><img alt="阶乘阶的时间复杂度" src="../time_complexity.assets/time_complexity_factorial.png" /></p>
|
||||
<p align="center"> Fig. 阶乘阶的时间复杂度 </p>
|
||||
|
||||
<h2 id="226">2.2.6. 最差、最佳、平均时间复杂度<a class="headerlink" href="#226" title="Permanent link">¶</a></h2>
|
||||
<p><strong>某些算法的时间复杂度不是固定的，而是与输入数据的分布有关</strong>。例如，假设输入一个长度为 <span class="arithmatex">\(n\)</span> 的数组 <code>nums</code> ，其中 <code>nums</code> 由从 <span class="arithmatex">\(1\)</span> 至 <span class="arithmatex">\(n\)</span> 的数字组成，但元素顺序是随机打乱的；算法的任务是返回元素 <span class="arithmatex">\(1\)</span> 的索引。我们可以得出以下结论：</p>
|
||||
<p>请注意，因为 <span class="arithmatex">\(n! > 2^n\)</span> ，所以阶乘阶比指数阶增长地更快，在 <span class="arithmatex">\(n\)</span> 较大时也是不可接受的。</p>
|
||||
<h2 id="225">2.2.5. 最差、最佳、平均时间复杂度<a class="headerlink" href="#225" title="Permanent link">¶</a></h2>
|
||||
<p><strong>算法的时间效率往往不是固定的，而是与输入数据的分布有关</strong>。假设输入一个长度为 <span class="arithmatex">\(n\)</span> 的数组 <code>nums</code> ，其中 <code>nums</code> 由从 <span class="arithmatex">\(1\)</span> 至 <span class="arithmatex">\(n\)</span> 的数字组成，但元素顺序是随机打乱的，任务目标是返回元素 <span class="arithmatex">\(1\)</span> 的索引。我们可以得出以下结论：</p>
|
||||
<ul>
|
||||
<li>当 <code>nums = [?, ?, ..., 1]</code> ，即当末尾元素是 <span class="arithmatex">\(1\)</span> 时，需要完整遍历数组，此时达到 <strong>最差时间复杂度 <span class="arithmatex">\(O(n)\)</span></strong> 。</li>
|
||||
<li>当 <code>nums = [1, ?, ?, ...]</code> ，即当首个数字为 <span class="arithmatex">\(1\)</span> 时，无论数组多长都不需要继续遍历，此时达到 <strong>最佳时间复杂度 <span class="arithmatex">\(\Omega(1)\)</span></strong> 。</li>
|
||||
<li>当 <code>nums = [?, ?, ..., 1]</code> ，即当末尾元素是 <span class="arithmatex">\(1\)</span> 时，需要完整遍历数组，<strong>达到最差时间复杂度 <span class="arithmatex">\(O(n)\)</span></strong> 。</li>
|
||||
<li>当 <code>nums = [1, ?, ?, ...]</code> ，即当首个数字为 <span class="arithmatex">\(1\)</span> 时，无论数组多长都不需要继续遍历，<strong>达到最佳时间复杂度 <span class="arithmatex">\(\Omega(1)\)</span></strong> 。</li>
|
||||
</ul>
|
||||
<p>“函数渐近上界”使用大 <span class="arithmatex">\(O\)</span> 记号表示，代表「最差时间复杂度」。相应地，“函数渐近下界”用 <span class="arithmatex">\(\Omega\)</span> 记号来表示，代表「最佳时间复杂度」。</p>
|
||||
<p>「最差时间复杂度」对应函数渐近上界，使用大 <span class="arithmatex">\(O\)</span> 记号表示。相应地，「最佳时间复杂度」对应函数渐近下界，用 <span class="arithmatex">\(\Omega\)</span> 记号表示。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="16:12"><input checked="checked" id="__tabbed_16_1" name="__tabbed_16" type="radio" /><input id="__tabbed_16_2" name="__tabbed_16" type="radio" /><input id="__tabbed_16_3" name="__tabbed_16" type="radio" /><input id="__tabbed_16_4" name="__tabbed_16" type="radio" /><input id="__tabbed_16_5" name="__tabbed_16" type="radio" /><input id="__tabbed_16_6" name="__tabbed_16" type="radio" /><input id="__tabbed_16_7" name="__tabbed_16" type="radio" /><input id="__tabbed_16_8" name="__tabbed_16" type="radio" /><input id="__tabbed_16_9" name="__tabbed_16" type="radio" /><input id="__tabbed_16_10" name="__tabbed_16" type="radio" /><input id="__tabbed_16_11" name="__tabbed_16" type="radio" /><input id="__tabbed_16_12" name="__tabbed_16" type="radio" /><div class="tabbed-labels"><label for="__tabbed_16_1">Java</label><label for="__tabbed_16_2">C++</label><label for="__tabbed_16_3">Python</label><label for="__tabbed_16_4">Go</label><label for="__tabbed_16_5">JS</label><label for="__tabbed_16_6">TS</label><label for="__tabbed_16_7">C</label><label for="__tabbed_16_8">C#</label><label for="__tabbed_16_9">Swift</label><label for="__tabbed_16_10">Zig</label><label for="__tabbed_16_11">Dart</label><label for="__tabbed_16_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
@@ -6512,13 +6538,10 @@ n! = n \times (n - 1) \times (n - 2) \times \cdots \times 2 \times 1
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="admonition tip">
|
||||
<p class="admonition-title">Tip</p>
|
||||
<p>实际应用中我们很少使用「最佳时间复杂度」，因为通常只有在很小概率下才能达到，可能会带来一定的误导性。相反，「最差时间复杂度」更为实用，因为它给出了一个“效率安全值”，让我们可以放心地使用算法。</p>
|
||||
</div>
|
||||
<p>从上述示例可以看出，最差或最佳时间复杂度只出现在“特殊分布的数据”中，这些情况的出现概率可能很小，因此并不能最真实地反映算法运行效率。相较之下，<strong>「平均时间复杂度」可以体现算法在随机输入数据下的运行效率</strong>，用 <span class="arithmatex">\(\Theta\)</span> 记号来表示。</p>
|
||||
<p>值得说明的是，我们在实际中很少使用「最佳时间复杂度」，因为通常只有在很小概率下才能达到，可能会带来一定的误导性。<strong>而「最差时间复杂度」更为实用，因为它给出了一个效率安全值</strong>，让我们可以放心地使用算法。</p>
|
||||
<p>从上述示例可以看出，最差或最佳时间复杂度只出现于“特殊的数据分布”，这些情况的出现概率可能很小，并不能真实地反映算法运行效率。相比之下，<strong>「平均时间复杂度」可以体现算法在随机输入数据下的运行效率</strong>，用 <span class="arithmatex">\(\Theta\)</span> 记号来表示。</p>
|
||||
<p>对于部分算法，我们可以简单地推算出随机数据分布下的平均情况。比如上述示例，由于输入数组是被打乱的，因此元素 <span class="arithmatex">\(1\)</span> 出现在任意索引的概率都是相等的，那么算法的平均循环次数则是数组长度的一半 <span class="arithmatex">\(\frac{n}{2}\)</span> ，平均时间复杂度为 <span class="arithmatex">\(\Theta(\frac{n}{2}) = \Theta(n)\)</span> 。</p>
|
||||
<p>但在实际应用中，尤其是较为复杂的算法，计算平均时间复杂度比较困难，因为很难简便地分析出在数据分布下的整体数学期望。在这种情况下，我们通常使用最差时间复杂度作为算法效率的评判标准。</p>
|
||||
<p>但对于较为复杂的算法，计算平均时间复杂度往往是比较困难的，因为很难分析出在数据分布下的整体数学期望。在这种情况下，我们通常使用最差时间复杂度作为算法效率的评判标准。</p>
|
||||
<div class="admonition question">
|
||||
<p class="admonition-title">为什么很少看到 <span class="arithmatex">\(\Theta\)</span> 符号？</p>
|
||||
<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>
|
||||
|
||||
Reference in New Issue
Block a user