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<h1 id="33">3.3 &nbsp; 数字编码 *<a class="headerlink" href="#33" title="Permanent link">&para;</a></h1>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>在本书中，标题带有的 * 符号的是选读章节。如果你时间有限或感到理解困难，可以先跳过，等学完必读章节后再单独攻克。</p>
</div>
<h2 id="331">3.3.1 &nbsp; 原码、反码和补码<a class="headerlink" href="#331" title="Permanent link">&para;</a></h2>
<p>在上一节的表格中我们发现，所有整数类型能够表示的负数都比正数多一个，例如 <code>byte</code> 的取值范围是 <span class="arithmatex">\([-128, 127]\)</span> 。这个现象比较反直觉，它的内在原因涉及到原码、反码、补码的相关知识。</p>
<p>首先需要指出，<strong>数字是以“补码”的形式存储在计算机中的</strong>。在分析这样做的原因之前，我们首先给出三者的定义。</p>
<ul>
<li><strong>原码</strong>：我们将数字的二进制表示的最高位视为符号位，其中 <span class="arithmatex">\(0\)</span> 表示正数，<span class="arithmatex">\(1\)</span> 表示负数，其余位表示数字的值。</li>
<li><strong>反码</strong>：正数的反码与其原码相同，负数的反码是对其原码除符号位外的所有位取反。</li>
<li><strong>补码</strong>：正数的补码与其原码相同，负数的补码是在其反码的基础上加 <span class="arithmatex">\(1\)</span> 。</li>
</ul>
<p>图 3-4 展示了原码、反码和补码之间的转换方法。</p>
<p><a class="glightbox" href="../number_encoding.assets/1s_2s_complement.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="原码、反码与补码之间的相互转换" src="../number_encoding.assets/1s_2s_complement.png" /></a></p>
<p align="center"> 图 3-4 &nbsp; 原码、反码与补码之间的相互转换 </p>

<p>「原码 true form」虽然最直观，但存在一些局限性。一方面，<strong>负数的原码不能直接用于运算</strong>。例如在原码下计算 <span class="arithmatex">\(1 + (-2)\)</span> ，得到的结果是 <span class="arithmatex">\(-3\)</span> ，这显然是不对的。</p>
<div class="arithmatex">\[
\begin{aligned}
&amp; 1 + (-2) \newline
&amp; \rightarrow 0000 \; 0001 + 1000 \; 0010 \newline
&amp; = 1000 \; 0011 \newline
&amp; \rightarrow -3
\end{aligned}
\]</div>
<p>为了解决此问题，计算机引入了「反码 1's complement code」。如果我们先将原码转换为反码，并在反码下计算 <span class="arithmatex">\(1 + (-2)\)</span> ，最后将结果从反码转化回原码，则可得到正确结果 <span class="arithmatex">\(-1\)</span> 。</p>
<div class="arithmatex">\[
\begin{aligned}
&amp; 1 + (-2) \newline
&amp; \rightarrow 0000 \; 0001 \; \text{(原码)} + 1000 \; 0010 \; \text{(原码)} \newline
&amp; = 0000 \; 0001 \; \text{(反码)} + 1111  \; 1101 \; \text{(反码)} \newline
&amp; = 1111 \; 1110 \; \text{(反码)} \newline
&amp; = 1000 \; 0001 \; \text{(原码)} \newline
&amp; \rightarrow -1
\end{aligned}
\]</div>
<p>另一方面，<strong>数字零的原码有 <span class="arithmatex">\(+0\)</span> 和 <span class="arithmatex">\(-0\)</span> 两种表示方式</strong>。这意味着数字零对应着两个不同的二进制编码，其可能会带来歧义。比如在条件判断中，如果没有区分正零和负零，则可能会导致判断结果出错。而如果我们想要处理正零和负零歧义，则需要引入额外的判断操作，其可能会降低计算机的运算效率。</p>
<div class="arithmatex">\[
\begin{aligned}
+0 &amp; \rightarrow 0000 \; 0000 \newline
-0 &amp; \rightarrow 1000 \; 0000
\end{aligned}
\]</div>
<p>与原码一样，反码也存在正负零歧义问题，因此计算机进一步引入了「补码 2's complement code」。我们先来观察一下负零的原码、反码、补码的转换过程：</p>
<div class="arithmatex">\[
\begin{aligned}
-0 \rightarrow \; &amp; 1000 \; 0000 \; \text{(原码)} \newline
= \; &amp; 1111 \; 1111 \; \text{(反码)} \newline
= 1 \; &amp; 0000 \; 0000 \; \text{(补码)} \newline
\end{aligned}
\]</div>
<p>在负零的反码基础上加 <span class="arithmatex">\(1\)</span> 会产生进位，但 <code>byte</code> 类型的长度只有 8 位，因此溢出到第 9 位的 <span class="arithmatex">\(1\)</span> 会被舍弃。也就是说，<strong>负零的补码为 <span class="arithmatex">\(0000 \; 0000\)</span> ，与正零的补码相同</strong>。这意味着在补码表示中只存在一个零，正负零歧义从而得到解决。</p>
<p>还剩余最后一个疑惑：<code>byte</code> 类型的取值范围是 <span class="arithmatex">\([-128, 127]\)</span> ，多出来的一个负数 <span class="arithmatex">\(-128\)</span> 是如何得到的呢？我们注意到，区间 <span class="arithmatex">\([-127, +127]\)</span> 内的所有整数都有对应的原码、反码和补码，并且原码和补码之间是可以互相转换的。</p>
<p>然而，<strong>补码 <span class="arithmatex">\(1000 \; 0000\)</span> 是一个例外，它并没有对应的原码</strong>。根据转换方法，我们得到该补码的原码为 <span class="arithmatex">\(0000 \; 0000\)</span> 。这显然是矛盾的，因为该原码表示数字 <span class="arithmatex">\(0\)</span> ，它的补码应该是自身。计算机规定这个特殊的补码 <span class="arithmatex">\(1000 \; 0000\)</span> 代表 <span class="arithmatex">\(-128\)</span> 。实际上，<span class="arithmatex">\((-1) + (-127)\)</span> 在补码下的计算结果就是 <span class="arithmatex">\(-128\)</span> 。</p>
<div class="arithmatex">\[
\begin{aligned}
&amp; (-127) + (-1) \newline
&amp; \rightarrow 1111 \; 1111 \; \text{(原码)} + 1000 \; 0001 \; \text{(原码)} \newline
&amp; = 1000 \; 0000 \; \text{(反码)} + 1111  \; 1110 \; \text{(反码)} \newline
&amp; = 1000 \; 0001 \; \text{(补码)} + 1111  \; 1111 \; \text{(补码)} \newline
&amp; = 1000 \; 0000 \; \text{(补码)} \newline
&amp; \rightarrow -128
\end{aligned}
\]</div>
<p>你可能已经发现，上述的所有计算都是加法运算。这暗示着一个重要事实：<strong>计算机内部的硬件电路主要是基于加法运算设计的</strong>。这是因为加法运算相对于其他运算（比如乘法、除法和减法）来说，硬件实现起来更简单，更容易进行并行化处理，运算速度更快。</p>
<p>请注意，这并不意味着计算机只能做加法。<strong>通过将加法与一些基本逻辑运算结合，计算机能够实现各种其他的数学运算</strong>。例如，计算减法 <span class="arithmatex">\(a - b\)</span> 可以转换为计算加法 <span class="arithmatex">\(a + (-b)\)</span> ；计算乘法和除法可以转换为计算多次加法或减法。</p>
<p>现在我们可以总结出计算机使用补码的原因：基于补码表示，计算机可以用同样的电路和操作来处理正数和负数的加法，不需要设计特殊的硬件电路来处理减法，并且无须特别处理正负零的歧义问题。这大大简化了硬件设计，提高了运算效率。</p>
<p>补码的设计非常精妙，因篇幅关系我们就先介绍到这里，建议有兴趣的读者进一步深度了解。</p>
<h2 id="332">3.3.2 &nbsp; 浮点数编码<a class="headerlink" href="#332" title="Permanent link">&para;</a></h2>
<p>细心的你可能会发现：<code>int</code> 和 <code>float</code> 长度相同，都是 4 bytes ，但为什么 <code>float</code> 的取值范围远大于 <code>int</code> ？这非常反直觉，因为按理说 <code>float</code> 需要表示小数，取值范围应该变小才对。</p>
<p>实际上，<strong>这是因为浮点数 <code>float</code> 采用了不同的表示方式</strong>。记一个 32-bit 长度的二进制数为：</p>
<div class="arithmatex">\[
b_{31} b_{30} b_{29} \ldots b_2 b_1 b_0
\]</div>
<p>根据 IEEE 754 标准，32-bit 长度的 <code>float</code> 由以下三个部分构成。</p>
<ul>
<li>符号位 <span class="arithmatex">\(\mathrm{S}\)</span> ：占 1 bit ，对应 <span class="arithmatex">\(b_{31}\)</span> 。</li>
<li>指数位 <span class="arithmatex">\(\mathrm{E}\)</span> ：占 8 bits ，对应 <span class="arithmatex">\(b_{30} b_{29} \ldots b_{23}\)</span> 。</li>
<li>分数位 <span class="arithmatex">\(\mathrm{N}\)</span> ：占 23 bits ，对应 <span class="arithmatex">\(b_{22} b_{21} \ldots b_0\)</span> 。</li>
</ul>
<p>二进制数 <code>float</code> 对应的值的计算方法：</p>
<div class="arithmatex">\[
\text {val} = (-1)^{b_{31}} \times 2^{\left(b_{30} b_{29} \ldots b_{23}\right)_2-127} \times\left(1 . b_{22} b_{21} \ldots b_0\right)_2
\]</div>
<p>转化到十进制下的计算公式：</p>
<div class="arithmatex">\[
\text {val}=(-1)^{\mathrm{S}} \times 2^{\mathrm{E} -127} \times (1 + \mathrm{N})
\]</div>
<p>其中各项的取值范围：</p>
<div class="arithmatex">\[
\begin{aligned}
\mathrm{S} \in &amp; \{ 0, 1\}, \quad \mathrm{E} \in \{ 1, 2, \dots, 254 \} \newline
(1 + \mathrm{N}) = &amp; (1 + \sum_{i=1}^{23} b_{23-i} 2^{-i}) \subset [1, 2 - 2^{-23}]
\end{aligned}
\]</div>
<p><a class="glightbox" href="../number_encoding.assets/ieee_754_float.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="IEEE 754 标准下的 float 的计算示例" src="../number_encoding.assets/ieee_754_float.png" /></a></p>
<p align="center"> 图 3-5 &nbsp; IEEE 754 标准下的 float 的计算示例 </p>

<p>观察图 3-5 ，给定一个示例数据 <span class="arithmatex">\(\mathrm{S} = 0\)</span> ， <span class="arithmatex">\(\mathrm{E} = 124\)</span> ，<span class="arithmatex">\(\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\)</span> ，则有：</p>
<div class="arithmatex">\[
\text { val } = (-1)^0 \times 2^{124 - 127} \times (1 + 0.375) = 0.171875
\]</div>
<p>现在我们可以回答最初的问题：<strong><code>float</code> 的表示方式包含指数位，导致其取值范围远大于 <code>int</code></strong> 。根据以上计算，<code>float</code> 可表示的最大正数为 <span class="arithmatex">\(2^{254 - 127} \times (2 - 2^{-23}) \approx 3.4 \times 10^{38}\)</span> ，切换符号位便可得到最小负数。</p>
<p><strong>尽管浮点数 <code>float</code> 扩展了取值范围，但其副作用是牺牲了精度</strong>。整数类型 <code>int</code> 将全部 32 位用于表示数字，数字是均匀分布的；而由于指数位的存在，浮点数 <code>float</code> 的数值越大，相邻两个数字之间的差值就会趋向越大。</p>
<p>如表 3-2 所示，指数位 <span class="arithmatex">\(E = 0\)</span> 和 <span class="arithmatex">\(E = 255\)</span> 具有特殊含义，<strong>用于表示零、无穷大、<span class="arithmatex">\(\mathrm{NaN}\)</span> 等</strong>。</p>
<p align="center"> 表 3-2 &nbsp; 指数位含义 </p>

<div class="center-table">
<table>
<thead>
<tr>
<th>指数位 E</th>
<th>分数位 <span class="arithmatex">\(\mathrm{N} = 0\)</span></th>
<th>分数位 <span class="arithmatex">\(\mathrm{N} \ne 0\)</span></th>
<th>计算公式</th>
</tr>
</thead>
<tbody>
<tr>
<td><span class="arithmatex">\(0\)</span></td>
<td><span class="arithmatex">\(\pm 0\)</span></td>
<td>次正规数</td>
<td><span class="arithmatex">\((-1)^{\mathrm{S}} \times 2^{-126} \times (0.\mathrm{N})\)</span></td>
</tr>
<tr>
<td><span class="arithmatex">\(1, 2, \dots, 254\)</span></td>
<td>正规数</td>
<td>正规数</td>
<td><span class="arithmatex">\((-1)^{\mathrm{S}} \times 2^{(\mathrm{E} -127)} \times (1.\mathrm{N})\)</span></td>
</tr>
<tr>
<td><span class="arithmatex">\(255\)</span></td>
<td><span class="arithmatex">\(\pm \infty\)</span></td>
<td><span class="arithmatex">\(\mathrm{NaN}\)</span></td>
<td></td>
</tr>
</tbody>
</table>
</div>
<p>值得说明的是，次正规数显著提升了浮点数的精度。最小正正规数为 <span class="arithmatex">\(2^{-126}\)</span> ，最小正次正规数为 <span class="arithmatex">\(2^{-126} \times 2^{-23}\)</span> 。</p>
<p>双精度 <code>double</code> 也采用类似 <code>float</code> 的表示方法，在此不做赘述。</p>

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 								            <input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_1">
 								              <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
-												deploy

											
										
										
											2023-07-19 01:44:47 +08:00
+								                <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07 19.07 3.5m-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13.51 1.67m-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87.4 1.31M19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35-2.53-2.2m4.33-2.7 1.15-2.77 2.2 2.54-3.35.23m1.15-4.96-1.14-2.78 3.34.24-2.2 2.54M9.63 18.93l2.77 1.15-2.53 2.19-.24-3.34Z"/></svg>
-												deploy

											
										
										
											2023-05-31 01:03:26 +08:00
+								              </label>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-05-31 01:03:26 +08:00
-												deploy

											
										
										
											2023-06-18 03:47:44 +08:00
+								            <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_2">
-												deploy

											
										
										
											2023-05-31 01:03:26 +08:00
 								              <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
-												deploy

											
										
										
											2023-07-19 01:44:47 +08:00
+								                <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07 19.07 3.5m-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13.51 1.67m-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87.4 1.31M19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35-2.53-2.2m4.33-2.7 1.15-2.77 2.2 2.54-3.35.23m1.15-4.96-1.14-2.78 3.34.24-2.2 2.54M9.63 18.93l2.77 1.15-2.53 2.19-.24-3.34Z"/></svg>
-												deploy

											
										
										
											2023-05-31 01:03:26 +08:00
+								              </label>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-05-31 01:03:26 +08:00
+								        </form>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								      <label class="md-header__button md-icon" for="__search">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.516 6.516 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5Z"/></svg>
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 								      <div class="md-search" data-md-component="search" role="dialog">
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 								    <form class="md-search__form" name="search">
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 								      <label class="md-search__icon md-icon" for="__search">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.516 6.516 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5Z"/></svg>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11h12Z"/></svg>
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-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								            <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M18 16.08c-.76 0-1.44.3-1.96.77L8.91 12.7c.05-.23.09-.46.09-.7 0-.24-.04-.47-.09-.7l7.05-4.11c.54.5 1.25.81 2.04.81a3 3 0 0 0 3-3 3 3 0 0 0-3-3 3 3 0 0 0-3 3c0 .24.04.47.09.7L8.04 9.81C7.5 9.31 6.79 9 6 9a3 3 0 0 0-3 3 3 3 0 0 0 3 3c.79 0 1.5-.31 2.04-.81l7.12 4.15c-.05.21-.08.43-.08.66 0 1.61 1.31 2.91 2.92 2.91 1.61 0 2.92-1.3 2.92-2.91A2.92 2.92 0 0 0 18 16.08Z"/></svg>
 								          </a>
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-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
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+								          <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 6.41 17.59 5 12 10.59 6.41 5 5 6.41 10.59 12 5 17.59 6.41 19 12 13.41 17.59 19 19 17.59 13.41 12 19 6.41Z"/></svg>
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 								            正在初始化搜索引擎
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 								          <ol class="md-search-result__list" role="presentation"></ol>
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 								    </div>
 								  </div>
 								</div>
 								      <div class="md-header__source">
 								        <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
 								  <div class="md-source__icon md-icon">
-												deploy

											
										
										
											2023-04-16 16:40:06 +08:00
+								    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512"><!--! Font Awesome Free 6.4.0 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"/></svg>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								  </div>
 								  <div class="md-source__repository">
 								    krahets/hello-algo
 								  </div>
 								</a>
 								      </div>
 								  </nav>
 								</header>
 								    <div class="md-container" data-md-component="container">
 								      <main class="md-main" data-md-component="main">
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 								              <div class="md-sidebar md-sidebar--primary" data-md-component="sidebar" data-md-type="navigation" >
 								                <div class="md-sidebar__scrollwrap">
 								                  <div class="md-sidebar__inner">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								<nav class="md-nav md-nav--primary" aria-label="导航栏" data-md-level="0">
 								  <label class="md-nav__title" for="__drawer">
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 								      <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
 								  <div class="md-source__icon md-icon">
-												deploy

											
										
										
											2023-04-16 16:40:06 +08:00
+								    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512"><!--! Font Awesome Free 6.4.0 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"/></svg>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								  </div>
 								  <div class="md-source__repository">
 								    krahets/hello-algo
 								  </div>
 								</a>
 								    </div>
 								  <ul class="md-nav__list" data-md-scrollfix>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-03-03 20:34:57 +08:00
-												First commit

											
										
										
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-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item md-nav__item--nested">
-												First commit

											
										
										
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-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
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-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          <div class="md-nav__link md-nav__container">
 								            <a href="../../chapter_preface/" class="md-nav__link ">
 								    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M21 4H3a2 2 0 0 0-2 2v13a2 2 0 0 0 2 2h18a2 2 0 0 0 2-2V6a2 2 0 0 0-2-2M3 19V6h8v13H3m18 0h-8V6h8v13m-7-9.5h6V11h-6V9.5m0 2.5h6v1.5h-6V12m0 2.5h6V16h-6v-1.5Z"/></svg>
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								    第 0 章 &nbsp; 前言
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
 								            </a>
-												First commit

											
										
										
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-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								              <label class="md-nav__link " for="__nav_1">
 								                <span class="md-nav__icon md-icon"></span>
 								              </label>
 								          </div>
 								        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
 								          <label class="md-nav__title" for="__nav_1">
 								            <span class="md-nav__icon md-icon"></span>
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								            第 0 章 &nbsp; 前言
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </label>
 								          <ul class="md-nav__list" data-md-scrollfix>
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
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-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
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 								    <li class="md-nav__item">
 								      <a href="../../chapter_preface/about_the_book/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.1 &nbsp; 关于本书
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
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+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
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-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
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 								    <li class="md-nav__item">
 								      <a href="../../chapter_preface/suggestions/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.2 &nbsp; 如何使用本书
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-03-03 20:34:57 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-03-03 20:34:57 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_preface/summary/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.3 &nbsp; 小结
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-03-03 20:34:57 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-03-03 20:34:57 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </ul>
 								        </nav>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-02-26 22:45:57 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item md-nav__item--nested">
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          <div class="md-nav__link md-nav__container">
 								            <a href="../../chapter_introduction/" class="md-nav__link ">
-												deploy

											
										
										
											2023-07-19 16:10:17 +08:00
+								    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								    第 1 章 &nbsp; 初识算法
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
 								            </a>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								              <label class="md-nav__link " for="__nav_2">
 								                <span class="md-nav__icon md-icon"></span>
 								              </label>
 								          </div>
 								        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
 								          <label class="md-nav__title" for="__nav_2">
 								            <span class="md-nav__icon md-icon"></span>
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								            第 1 章 &nbsp; 初识算法
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </label>
 								          <ul class="md-nav__list" data-md-scrollfix>
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_introduction/algorithms_are_everywhere/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.1 &nbsp; 算法无处不在
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_introduction/what_is_dsa/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.2 &nbsp; 算法是什么
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-02-26 22:45:57 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-02-26 22:45:57 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_introduction/summary/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.3 &nbsp; 小结
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-02-26 22:45:57 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-02-26 22:45:57 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </ul>
 								        </nav>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-08-24 17:47:31 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item md-nav__item--nested">
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          <div class="md-nav__link md-nav__container">
 								            <a href="../../chapter_computational_complexity/" class="md-nav__link ">
 								    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-28 04:36:05 +08:00
+								    第 2 章 &nbsp; 复杂度分析
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
 								            </a>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								              <label class="md-nav__link " for="__nav_3">
 								                <span class="md-nav__icon md-icon"></span>
 								              </label>
 								          </div>
 								        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
 								          <label class="md-nav__title" for="__nav_3">
 								            <span class="md-nav__icon md-icon"></span>
-												deploy

											
										
										
											2023-08-28 04:36:05 +08:00
+								            第 2 章 &nbsp; 复杂度分析
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </label>
 								          <ul class="md-nav__list" data-md-scrollfix>
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_computational_complexity/performance_evaluation/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.1 &nbsp; 算法效率评估
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-08-24 17:47:31 +08:00
+								    <li class="md-nav__item">
 								      <a href="../../chapter_computational_complexity/iteration_and_recursion/" class="md-nav__link">
 								  <span class="md-ellipsis">
 .2 &nbsp; 迭代与递归
 								  </span>
-												deploy

											
										
										
											2023-08-25 02:22:36 +08:00
 								    <span class="md-status md-status--new" title="最近添加">
 								    </span>
-												deploy

											
										
										
											2023-08-24 17:47:31 +08:00
 								      </a>
 								    </li>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    <li class="md-nav__item">
 								      <a href="../../chapter_computational_complexity/time_complexity/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-24 17:47:31 +08:00
+.3 &nbsp; 时间复杂度
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_computational_complexity/space_complexity/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-24 17:47:31 +08:00
+.4 &nbsp; 空间复杂度
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-24 17:47:31 +08:00
+.5 &nbsp; 小结
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </ul>
 								        </nav>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item md-nav__item--active md-nav__item--nested">
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" checked>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          <div class="md-nav__link md-nav__container">
 								            <a href="../" class="md-nav__link ">
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-19 16:10:17 +08:00
+								    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z"/></svg>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								    第 3 章 &nbsp; 数据结构
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								            </a>
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								              <label class="md-nav__link " for="__nav_4">
 								                <span class="md-nav__icon md-icon"></span>
 								              </label>
 								          </div>
 								        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="true">
 								          <label class="md-nav__title" for="__nav_4">
 								            <span class="md-nav__icon md-icon"></span>
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								            第 3 章 &nbsp; 数据结构
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </label>
 								          <ul class="md-nav__list" data-md-scrollfix>
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								      <a href="../classification_of_data_structure/" class="md-nav__link">
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.1 &nbsp; 数据结构分类
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item">
 								      <a href="../basic_data_types/" class="md-nav__link">
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.2 &nbsp; 基本数据类型
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
 								      </a>
 								    </li>
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item md-nav__item--active">
 								      <input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
-												deploy

											
										
										
											2023-03-07 23:59:27 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								        <label class="md-nav__link md-nav__link--active" for="__toc">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.3 &nbsp; 数字编码 *
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								          <span class="md-nav__icon md-icon"></span>
 								        </label>
 								      <a href="./" class="md-nav__link md-nav__link--active">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.3 &nbsp; 数字编码 *
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								<nav class="md-nav md-nav--secondary" aria-label="目录">
-												deploy

											
										
										
											2023-03-07 23:59:27 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <label class="md-nav__title" for="__toc">
 								      <span class="md-nav__icon md-icon"></span>
 								      目录
 								    </label>
 								    <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 								        <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
+								  <a href="#331" class="md-nav__link">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.3.1 &nbsp; 原码、反码和补码
-												deploy

											
										
										
											2023-02-22 19:01:30 +08:00
+								  </a>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								</li>
-												deploy

											
										
										
											2023-03-07 23:59:27 +08:00
 								        <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
+								  <a href="#332" class="md-nav__link">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.3.2 &nbsp; 浮点数编码
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								  </a>
 								</li>
 								    </ul>
 								</nav>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
+								      <a href="../character_encoding/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.4 &nbsp; 字符编码 *
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
 								      <a href="../summary/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.5 &nbsp; 小结
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </ul>
 								        </nav>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item md-nav__item--nested">
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          <div class="md-nav__link md-nav__container">
 								            <a href="../../chapter_array_and_linkedlist/" class="md-nav__link ">
-												deploy

											
										
										
											2023-07-19 16:10:17 +08:00
+								    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								    第 4 章 &nbsp; 数组与链表
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
 								            </a>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								              <label class="md-nav__link " for="__nav_5">
 								                <span class="md-nav__icon md-icon"></span>
 								              </label>
 								          </div>
 								        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
 								          <label class="md-nav__title" for="__nav_5">
 								            <span class="md-nav__icon md-icon"></span>
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								            第 4 章 &nbsp; 数组与链表
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </label>
 								          <ul class="md-nav__list" data-md-scrollfix>
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_array_and_linkedlist/array/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.1 &nbsp; 数组
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.2 &nbsp; 链表
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.3 &nbsp; 列表
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.4 &nbsp; 小结
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </ul>
 								        </nav>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item md-nav__item--nested">
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          <div class="md-nav__link md-nav__container">
 								            <a href="../../chapter_stack_and_queue/" class="md-nav__link ">
 								    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								    第 5 章 &nbsp; 栈与队列
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
 								            </a>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								              <label class="md-nav__link " for="__nav_6">
 								                <span class="md-nav__icon md-icon"></span>
 								              </label>
 								          </div>
 								        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
 								          <label class="md-nav__title" for="__nav_6">
 								            <span class="md-nav__icon md-icon"></span>
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								            第 5 章 &nbsp; 栈与队列
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </label>
 								          <ul class="md-nav__list" data-md-scrollfix>
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.1 &nbsp; 栈
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.2 &nbsp; 队列
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.3 &nbsp; 双向队列
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.4 &nbsp; 小结
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </ul>
 								        </nav>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:07:42 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-06-16 21:22:53 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item md-nav__item--nested">
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          <div class="md-nav__link md-nav__container">
 								            <a href="../../chapter_hashing/" class="md-nav__link ">
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z"/></svg>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-21 03:56:52 +08:00
+								    第 6 章 &nbsp; 哈希表
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								            </a>
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								              <label class="md-nav__link " for="__nav_7">
 								                <span class="md-nav__icon md-icon"></span>
 								              </label>
 								          </div>
 								        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
 								          <label class="md-nav__title" for="__nav_7">
 								            <span class="md-nav__icon md-icon"></span>
-												deploy

											
										
										
											2023-08-21 03:56:52 +08:00
+								            第 6 章 &nbsp; 哈希表
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </label>
 								          <ul class="md-nav__list" data-md-scrollfix>
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								      <a href="../../chapter_hashing/hash_map/" class="md-nav__link">
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.1 &nbsp; 哈希表
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-05-21 19:07:42 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:07:42 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.2 &nbsp; 哈希冲突
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
 								      </a>
 								    </li>
-												deploy

											
										
										
											2023-05-21 19:07:42 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:07:42 +08:00
-												deploy

											
										
										
											2023-06-16 21:22:53 +08:00
+								    <li class="md-nav__item">
 								      <a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.3 &nbsp; 哈希算法
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-06-16 21:22:53 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-06-16 21:22:53 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-06-16 21:22:53 +08:00
-												deploy

											
										
										
											2023-05-21 19:07:42 +08:00
+								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      <a href="../../chapter_hashing/summary/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.4 &nbsp; 小结
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-05-21 19:07:42 +08:00
+								      </a>
 								    </li>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </ul>
 								        </nav>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item md-nav__item--nested">
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          <div class="md-nav__link md-nav__container">
 								            <a href="../../chapter_tree/" class="md-nav__link ">
 								    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.5 17c-.14 0-.26 0-.39.04L17.5 13.8c.45-.45.75-1.09.75-1.8a2.5 2.5 0 0 0-2.5-2.5c-.14 0-.25 0-.4.04L13.74 6.3c.47-.46.76-1.09.76-1.8a2.5 2.5 0 0 0-5 0c0 .7.29 1.34.76 1.79L8.65 9.54c-.15-.04-.26-.04-.4-.04a2.5 2.5 0 0 0-2.5 2.5c0 .71.29 1.34.75 1.79l-1.61 3.25C4.76 17 4.64 17 4.5 17a2.5 2.5 0 0 0 0 5A2.5 2.5 0 0 0 7 19.5c0-.7-.29-1.34-.76-1.79l1.62-3.25c.14.04.26.04.39.04s.25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0A2.5 2.5 0 0 0 12 17c-.13 0-.26 0-.39.04L10 13.8c.45-.45.75-1.09.75-1.8 0-.7-.29-1.33-.75-1.79l1.61-3.25c.13.04.26.04.39.04s.26 0 .39-.04L14 10.21a2.5 2.5 0 0 0 1.75 4.29c.13 0 .25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0 2.5 2.5 0 0 0-2.5-2.5m-15 3.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1m8.5-1c0 .55-.45 1-1 1s-1-.45-1-1 .45-1 1-1 1 .45 1 1M7.25 12c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1M11 4.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m3.75 7.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m4.75 8.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1Z"/></svg>
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								    第 7 章 &nbsp; 树
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
 								            </a>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								              <label class="md-nav__link " for="__nav_8">
 								                <span class="md-nav__icon md-icon"></span>
 								              </label>
 								          </div>
 								        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
 								          <label class="md-nav__title" for="__nav_8">
 								            <span class="md-nav__icon md-icon"></span>
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								            第 7 章 &nbsp; 树
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </label>
 								          <ul class="md-nav__list" data-md-scrollfix>
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      <a href="../../chapter_tree/binary_tree/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.1 &nbsp; 二叉树
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      <a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.2 &nbsp; 二叉树遍历
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      <a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.3 &nbsp; 二叉树数组表示
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.4 &nbsp; 二叉搜索树
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_tree/avl_tree/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.5 &nbsp; AVL 树 *
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_tree/summary/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.6 &nbsp; 小结
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </ul>
 								        </nav>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-02-26 20:11:54 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-02-26 22:45:57 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-06-12 23:05:10 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item md-nav__item--nested">
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          <div class="md-nav__link md-nav__container">
 								            <a href="../../chapter_heap/" class="md-nav__link ">
 								    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 1a2.5 2.5 0 0 0-2.5 2.5A2.5 2.5 0 0 0 11 5.79V7H7a2 2 0 0 0-2 2v.71A2.5 2.5 0 0 0 3.5 12 2.5 2.5 0 0 0 5 14.29V15H4a2 2 0 0 0-2 2v1.21A2.5 2.5 0 0 0 .5 20.5 2.5 2.5 0 0 0 3 23a2.5 2.5 0 0 0 2.5-2.5A2.5 2.5 0 0 0 4 18.21V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 9 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2H7v-.71A2.5 2.5 0 0 0 8.5 12 2.5 2.5 0 0 0 7 9.71V9h10v.71A2.5 2.5 0 0 0 15.5 12a2.5 2.5 0 0 0 1.5 2.29V15h-1a2 2 0 0 0-2 2v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 15 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 21 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2h-1v-.71A2.5 2.5 0 0 0 20.5 12 2.5 2.5 0 0 0 19 9.71V9a2 2 0 0 0-2-2h-4V5.79a2.5 2.5 0 0 0 1.5-2.29A2.5 2.5 0 0 0 12 1m0 1.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M6 11a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m12 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M3 19.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1Z"/></svg>
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								    第 8 章 &nbsp; 堆
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
 								            </a>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								              <label class="md-nav__link " for="__nav_9">
 								                <span class="md-nav__icon md-icon"></span>
 								              </label>
 								          </div>
 								        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
 								          <label class="md-nav__title" for="__nav_9">
 								            <span class="md-nav__icon md-icon"></span>
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								            第 8 章 &nbsp; 堆
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </label>
 								          <ul class="md-nav__list" data-md-scrollfix>
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      <a href="../../chapter_heap/heap/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.1 &nbsp; 堆
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-04-17 18:24:27 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-04-17 18:24:27 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-04-17 18:24:27 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      <a href="../../chapter_heap/build_heap/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.2 &nbsp; 建堆操作
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-06-12 23:05:10 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-06-12 23:05:10 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-06-12 23:05:10 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_heap/top_k/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.3 &nbsp; Top-K 问题
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-04-17 18:24:27 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-04-17 18:24:27 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-04-17 18:24:27 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      <a href="../../chapter_heap/summary/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.4 &nbsp; 小结
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-02-26 22:45:57 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-02-26 22:45:57 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </ul>
 								        </nav>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-02-15 03:42:47 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item md-nav__item--nested">
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          <div class="md-nav__link md-nav__container">
 								            <a href="../../chapter_graph/" class="md-nav__link ">
 								    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12 5.37-.44-.06L6 14.9c.24.21.4.48.47.78h11.06c.07-.3.23-.57.47-.78l-5.56-9.59-.44.06M6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43.43 0 .83.16 1.12.43l4.28-2.53H6.6M12 22a1.68 1.68 0 0 1-1.68-1.68l.09-.56-4.3-2.55c-.31.36-.76.58-1.27.58a1.68 1.68 0 0 1-1.68-1.68c0-.79.53-1.45 1.26-1.64V9.36c-.83-.11-1.47-.82-1.47-1.68A1.68 1.68 0 0 1 4.63 6c.55 0 1.03.26 1.34.66l4.41-2.53-.06-.45c0-.93.75-1.68 1.68-1.68.93 0 1.68.75 1.68 1.68l-.06.45 4.41 2.53c.31-.4.79-.66 1.34-.66a1.68 1.68 0 0 1 1.68 1.68c0 .86-.64 1.57-1.47 1.68v5.11c.73.19 1.26.85 1.26 1.64a1.68 1.68 0 0 1-1.68 1.68c-.51 0-.96-.22-1.27-.58l-4.3 2.55.09.56A1.68 1.68 0 0 1 12 22M10.8 4.86 6.3 7.44l.02.24c0 .71-.44 1.32-1.06 1.57l.03 5.25 5.51-9.64m2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24-4.5-2.58Z"/></svg>
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								    第 9 章 &nbsp; 图
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
 								            </a>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								              <label class="md-nav__link " for="__nav_10">
 								                <span class="md-nav__icon md-icon"></span>
 								              </label>
 								          </div>
 								        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
 								          <label class="md-nav__title" for="__nav_10">
 								            <span class="md-nav__icon md-icon"></span>
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								            第 9 章 &nbsp; 图
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </label>
 								          <ul class="md-nav__list" data-md-scrollfix>
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      <a href="../../chapter_graph/graph/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.1 &nbsp; 图
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-02-15 03:42:47 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-02-15 03:42:47 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      <a href="../../chapter_graph/graph_operations/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.2 &nbsp; 图基础操作
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-02-15 03:42:47 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-02-26 22:45:57 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-02-26 22:45:57 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      <a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.3 &nbsp; 图的遍历
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_graph/summary/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.4 &nbsp; 小结
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-02-26 22:45:57 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-02-26 22:45:57 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </ul>
 								        </nav>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-08-04 05:25:00 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item md-nav__item--nested">
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          <div class="md-nav__link md-nav__container">
 								            <a href="../../chapter_searching/" class="md-nav__link ">
 								    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4h18M3 16v-2h6v2H3m0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1H3Z"/></svg>
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								    第 10 章 &nbsp; 搜索
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
 								            </a>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								              <label class="md-nav__link " for="__nav_11">
 								                <span class="md-nav__icon md-icon"></span>
 								              </label>
 								          </div>
 								        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
 								          <label class="md-nav__title" for="__nav_11">
 								            <span class="md-nav__icon md-icon"></span>
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								            第 10 章 &nbsp; 搜索
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </label>
 								          <ul class="md-nav__list" data-md-scrollfix>
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      <a href="../../chapter_searching/binary_search/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.1 &nbsp; 二分查找
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-08-04 05:25:00 +08:00
+								    <li class="md-nav__item">
 								      <a href="../../chapter_searching/binary_search_insertion/" class="md-nav__link">
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.2 &nbsp; 二分查找插入点
-												deploy

											
										
										
											2023-08-04 05:25:00 +08:00
+								  </span>
 								      </a>
 								    </li>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      <a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.3 &nbsp; 二分查找边界
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      <a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.4 &nbsp; 哈希优化策略
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      <a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.5 &nbsp; 重识搜索算法
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_searching/summary/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.6 &nbsp; 小结
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </ul>
 								        </nav>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-03-19 23:38:24 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-24 00:32:47 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-26 04:48:01 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-03-26 22:05:43 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-03-26 04:45:30 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item md-nav__item--nested">
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          <div class="md-nav__link md-nav__container">
 								            <a href="../../chapter_sorting/" class="md-nav__link ">
 								    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 17h3l-4 4-4-4h3V3h2M2 17h10v2H2M6 5v2H2V5m0 6h7v2H2v-2Z"/></svg>
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								    第 11 章 &nbsp; 排序
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
 								            </a>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								              <label class="md-nav__link " for="__nav_12">
 								                <span class="md-nav__icon md-icon"></span>
 								              </label>
 								          </div>
 								        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
 								          <label class="md-nav__title" for="__nav_12">
 								            <span class="md-nav__icon md-icon"></span>
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								            第 11 章 &nbsp; 排序
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </label>
 								          <ul class="md-nav__list" data-md-scrollfix>
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-04-16 04:55:58 +08:00
+								      <a href="../../chapter_sorting/sorting_algorithm/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.1 &nbsp; 排序算法
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-05-24 00:32:47 +08:00
+								    <li class="md-nav__item">
 								      <a href="../../chapter_sorting/selection_sort/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.2 &nbsp; 选择排序
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-05-24 00:32:47 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-24 00:32:47 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-24 00:32:47 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    <li class="md-nav__item">
 								      <a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.3 &nbsp; 冒泡排序
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.4 &nbsp; 插入排序
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.5 &nbsp; 快速排序
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.6 &nbsp; 归并排序
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-05-26 04:48:01 +08:00
+								    <li class="md-nav__item">
 								      <a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.7 &nbsp; 堆排序
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-05-26 04:48:01 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-26 04:48:01 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-05-26 04:48:01 +08:00
-												deploy

											
										
										
											2023-03-26 22:05:43 +08:00
+								    <li class="md-nav__item">
 								      <a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
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											2023-08-19 22:07:40 +08:00
+.8 &nbsp; 桶排序
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
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-												deploy

											
										
										
											2023-03-26 22:05:43 +08:00
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-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-03-26 22:05:43 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-03-26 22:05:43 +08:00
-												deploy

											
										
										
											2023-03-19 23:38:24 +08:00
+								    <li class="md-nav__item">
 								      <a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.9 &nbsp; 计数排序
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-03-19 23:38:24 +08:00
+								      </a>
 								    </li>
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											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-03-19 23:38:24 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-03-19 23:38:24 +08:00
-												deploy

											
										
										
											2023-03-26 04:45:30 +08:00
+								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-03-26 22:05:43 +08:00
+								      <a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
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											2023-08-19 22:07:40 +08:00
+.10 &nbsp; 基数排序
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
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-												deploy

											
										
										
											2023-03-26 04:45:30 +08:00
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											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-03-26 04:45:30 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-03-26 04:45:30 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    <li class="md-nav__item">
 								      <a href="../../chapter_sorting/summary/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.11 &nbsp; 小结
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
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 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </ul>
 								        </nav>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-02-27 03:45:08 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-04-17 18:24:27 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-17 02:17:51 +08:00
-												deploy

											
										
										
											2023-07-17 20:19:46 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-17 04:21:02 +08:00
-												deploy

											
										
										
											2023-07-16 04:19:01 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item md-nav__item--nested">
-												deploy

											
										
										
											2023-07-16 04:19:01 +08:00
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											2023-07-17 17:51:12 +08:00
 								        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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											2023-07-16 04:19:01 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          <div class="md-nav__link md-nav__container">
 								            <a href="../../chapter_divide_and_conquer/" class="md-nav__link ">
-												deploy

											
										
										
											2023-07-19 16:10:17 +08:00
+								    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17 7v2h5V7h-5M2 9v6h5V9H2m10 0v2H9v2h3v2l3-3-3-3m5 2v2h5v-2h-5m0 4v2h5v-2h-5Z"/></svg>
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											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								    第 12 章 &nbsp; 分治
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
 								            </a>
-												deploy

											
										
										
											2023-07-16 04:19:01 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								              <label class="md-nav__link " for="__nav_13">
 								                <span class="md-nav__icon md-icon"></span>
 								              </label>
 								          </div>
 								        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
 								          <label class="md-nav__title" for="__nav_13">
 								            <span class="md-nav__icon md-icon"></span>
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								            第 12 章 &nbsp; 分治
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </label>
 								          <ul class="md-nav__list" data-md-scrollfix>
-												deploy

											
										
										
											2023-07-16 04:19:01 +08:00
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											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-16 04:19:01 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_divide_and_conquer/divide_and_conquer/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.1 &nbsp; 分治算法
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-07-16 04:19:01 +08:00
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-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
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											2023-07-16 04:19:01 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-16 04:19:01 +08:00
-												deploy

											
										
										
											2023-07-17 04:21:02 +08:00
+								    <li class="md-nav__item">
 								      <a href="../../chapter_divide_and_conquer/binary_search_recur/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
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											2023-08-19 22:07:40 +08:00
+.2 &nbsp; 分治搜索策略
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-07-17 04:21:02 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-17 04:21:02 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-17 04:21:02 +08:00
-												deploy

											
										
										
											2023-07-16 04:19:01 +08:00
+								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-07-17 02:17:51 +08:00
+								      <a href="../../chapter_divide_and_conquer/build_binary_tree_problem/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
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											2023-08-19 22:07:40 +08:00
+.3 &nbsp; 构建树问题
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-07-16 04:19:01 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-17 02:17:51 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-17 02:17:51 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_divide_and_conquer/hanota_problem/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.4 &nbsp; 汉诺塔问题
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-07-17 02:17:51 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-17 02:17:51 +08:00
-												deploy

											
										
										
											2023-07-17 20:19:46 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_divide_and_conquer/summary/" class="md-nav__link">
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.5 &nbsp; 小结
-												deploy

											
										
										
											2023-07-17 20:19:46 +08:00
+								  </span>
 								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </ul>
 								        </nav>
-												deploy

											
										
										
											2023-07-16 04:19:01 +08:00
+								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-16 04:19:01 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-16 04:19:01 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-16 04:19:01 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-04-17 18:24:27 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-06-21 02:55:52 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-06-22 00:23:01 +08:00
-												deploy

											
										
										
											2023-02-27 03:45:08 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item md-nav__item--nested">
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											2023-02-27 03:45:08 +08:00
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-04-16 04:55:58 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          <div class="md-nav__link md-nav__container">
 								            <a href="../../chapter_backtracking/" class="md-nav__link ">
 								    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M18 15a3 3 0 0 1 3 3 3 3 0 0 1-3 3 2.99 2.99 0 0 1-2.83-2H14v-2h1.17c.41-1.17 1.52-2 2.83-2m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m0-9a1.43 1.43 0 0 0 1.43-1.43 1.43 1.43 0 1 0-2.86 0A1.43 1.43 0 0 0 18 8m0-5.43a4 4 0 0 1 4 4C22 9.56 18 14 18 14s-4-4.44-4-7.43a4 4 0 0 1 4-4M8.83 17H10v2H8.83A2.99 2.99 0 0 1 6 21a3 3 0 0 1-3-3c0-1.31.83-2.42 2-2.83V14h2v1.17c.85.3 1.53.98 1.83 1.83M6 17a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1M6 3a3 3 0 0 1 3 3c0 1.31-.83 2.42-2 2.83V10H5V8.83A2.99 2.99 0 0 1 3 6a3 3 0 0 1 3-3m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m5 14v-2h2v2h-2m-4-6H5v-2h2v2Z"/></svg>
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								    第 13 章 &nbsp; 回溯
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
 								            </a>
-												deploy

											
										
										
											2023-04-16 04:55:58 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								              <label class="md-nav__link " for="__nav_14">
 								                <span class="md-nav__icon md-icon"></span>
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 								        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
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 								            <span class="md-nav__icon md-icon"></span>
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								            第 13 章 &nbsp; 回溯
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </label>
 								          <ul class="md-nav__list" data-md-scrollfix>
-												deploy

											
										
										
											2023-06-02 01:35:09 +08:00
-												deploy

											
										
										
											2023-04-16 04:55:58 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-04-16 04:55:58 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      <a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.1 &nbsp; 回溯算法
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-04-17 18:24:27 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-04-17 18:24:27 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-04-17 18:24:27 +08:00
 								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      <a href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.2 &nbsp; 全排列问题
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-04-17 18:24:27 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-04-17 18:24:27 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-04-17 18:24:27 +08:00
-												deploy

											
										
										
											2023-06-21 02:55:52 +08:00
+								    <li class="md-nav__item">
 								      <a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.3 &nbsp; 子集和问题
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-06-21 02:55:52 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-06-21 02:55:52 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-06-21 02:55:52 +08:00
-												deploy

											
										
										
											2023-04-17 18:24:27 +08:00
+								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-05-21 19:29:51 +08:00
+								      <a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.4 &nbsp; N 皇后问题
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-04-16 04:55:58 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-06-22 00:23:01 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-06-22 00:23:01 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_backtracking/summary/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
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											2023-08-19 22:07:40 +08:00
+.5 &nbsp; 小结
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-06-22 00:23:01 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-06-22 00:23:01 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </ul>
 								        </nav>
-												deploy

											
										
										
											2023-04-16 04:55:58 +08:00
+								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-04-16 04:55:58 +08:00
-												deploy

											
										
										
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-												deploy

											
										
										
											2023-04-24 03:30:35 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-01 22:39:28 +08:00
-												deploy

											
										
										
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-												deploy

											
										
										
											2023-07-06 00:08:29 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-07 21:39:04 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-11 19:21:38 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-14 02:56:32 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-13 05:26:03 +08:00
-												deploy

											
										
										
											2023-04-16 04:55:58 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item md-nav__item--nested">
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											2023-04-16 04:55:58 +08:00
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											2023-06-30 04:33:52 +08:00
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											2023-07-17 17:51:12 +08:00
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											2023-06-30 04:33:52 +08:00
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											2023-07-17 17:51:12 +08:00
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 								            <a href="../../chapter_dynamic_programming/" class="md-nav__link ">
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											2023-08-19 22:07:40 +08:00
+								    第 14 章 &nbsp; 动态规划
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
 								            </a>
-												deploy

											
										
										
											2023-06-30 04:33:52 +08:00
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											2023-07-17 17:51:12 +08:00
+								              <label class="md-nav__link " for="__nav_15">
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 								            <span class="md-nav__icon md-icon"></span>
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								            第 14 章 &nbsp; 动态规划
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </label>
 								          <ul class="md-nav__list" data-md-scrollfix>
-												deploy

											
										
										
											2023-06-30 04:33:52 +08:00
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											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-06-30 04:33:52 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_dynamic_programming/intro_to_dynamic_programming/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
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											2023-08-19 22:07:40 +08:00
+.1 &nbsp; 初探动态规划
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-07-01 22:39:28 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
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											2023-07-01 22:39:28 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-01 22:39:28 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_dynamic_programming/dp_problem_features/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
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											2023-08-19 22:07:40 +08:00
+.2 &nbsp; DP 问题特性
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-06-30 04:33:52 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-06 00:08:29 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-06 00:08:29 +08:00
-												deploy

											
										
										
											2023-07-07 21:39:04 +08:00
+								    <li class="md-nav__item">
-												deploy

											
										
										
											2023-07-09 03:02:15 +08:00
+								      <a href="../../chapter_dynamic_programming/dp_solution_pipeline/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.3 &nbsp; DP 解题思路
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-07-07 21:39:04 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-07 21:39:04 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-07 21:39:04 +08:00
-												deploy

											
										
										
											2023-07-06 00:08:29 +08:00
+								    <li class="md-nav__item">
 								      <a href="../../chapter_dynamic_programming/knapsack_problem/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
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											2023-08-19 22:07:40 +08:00
+.4 &nbsp; 0-1 背包问题
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-07-06 00:08:29 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-11 19:21:38 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-11 19:21:38 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_dynamic_programming/unbounded_knapsack_problem/" class="md-nav__link">
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											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
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											2023-08-19 22:07:40 +08:00
+.5 &nbsp; 完全背包问题
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-07-11 19:21:38 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-13 05:26:03 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-13 05:26:03 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_dynamic_programming/edit_distance_problem/" class="md-nav__link">
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
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											2023-08-19 22:07:40 +08:00
+.6 &nbsp; 编辑距离问题
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-07-13 05:26:03 +08:00
+								      </a>
 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-14 02:56:32 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-14 02:56:32 +08:00
 								    <li class="md-nav__item">
 								      <a href="../../chapter_dynamic_programming/summary/" class="md-nav__link">
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											2023-07-17 17:51:12 +08:00
 								  <span class="md-ellipsis">
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											2023-08-19 22:07:40 +08:00
+.7 &nbsp; 小结
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
-												deploy

											
										
										
											2023-07-14 02:56:32 +08:00
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 								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-14 02:56:32 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </ul>
 								        </nav>
-												deploy

											
										
										
											2023-06-30 04:33:52 +08:00
+								    </li>
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-06-30 04:33:52 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-06-30 04:33:52 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
-												deploy

											
										
										
											2023-07-21 21:53:15 +08:00
-												deploy

											
										
										
											2023-07-21 22:50:40 +08:00
-												deploy

											
										
										
											2023-07-21 15:14:51 +08:00
-												deploy

											
										
										
											2023-06-30 04:33:52 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								    <li class="md-nav__item md-nav__item--nested">
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											2023-06-30 04:33:52 +08:00
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											2023-07-17 17:51:12 +08:00
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											2023-02-27 03:45:08 +08:00
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											2023-07-17 17:51:12 +08:00
 								          <div class="md-nav__link md-nav__container">
-												deploy

											
										
										
											2023-07-20 18:24:35 +08:00
+								            <a href="../../chapter_greedy/" class="md-nav__link ">
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											2023-07-17 17:51:12 +08:00
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											2023-07-20 18:24:35 +08:00
+								    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 3c3.88 0 7 3.14 7 7 0 2.8-1.63 5.19-4 6.31V21H9v-3H8c-1.11 0-2-.89-2-2v-3H4.5c-.42 0-.66-.5-.42-.81L6 9.66A7.003 7.003 0 0 1 13 3m0-2C8.41 1 4.61 4.42 4.06 8.9L2.5 11h-.03l-.02.03c-.55.76-.62 1.76-.19 2.59.36.69 1 1.17 1.74 1.32V16c0 1.85 1.28 3.42 3 3.87V23h11v-5.5c2.5-1.67 4-4.44 4-7.5 0-4.97-4.04-9-9-9m4 7.83c0 1.54-1.36 2.77-3.42 4.64L13 14l-.58-.53C10.36 11.6 9 10.37 9 8.83c0-1.2.96-2.19 2.16-2.2h.04c.69 0 1.35.31 1.8.83.45-.52 1.11-.83 1.8-.83 1.2-.01 2.2.96 2.2 2.16v.04Z"/></svg>
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											2023-07-17 17:51:12 +08:00
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											2023-08-19 22:07:40 +08:00
+								    第 15 章 &nbsp; 贪心
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								  </span>
 								            </a>
 								              <label class="md-nav__link " for="__nav_16">
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-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								            第 15 章 &nbsp; 贪心
-												deploy

											
										
										
											2023-07-20 18:24:35 +08:00
+								          </label>
 								          <ul class="md-nav__list" data-md-scrollfix>
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 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.1 &nbsp; 贪心算法
-												deploy

											
										
										
											2023-07-20 18:24:35 +08:00
+								  </span>
 								      </a>
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 								    <li class="md-nav__item">
 								      <a href="../../chapter_greedy/fractional_knapsack_problem/" class="md-nav__link">
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-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.2 &nbsp; 分数背包问题
-												deploy

											
										
										
											2023-07-20 18:24:35 +08:00
+								  </span>
-												deploy

											
										
										
											2023-07-21 15:14:51 +08:00
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 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.3 &nbsp; 最大容量问题
-												deploy

											
										
										
											2023-07-21 15:14:51 +08:00
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-												deploy

											
										
										
											2023-07-21 21:53:15 +08:00
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 								      <a href="../../chapter_greedy/max_product_cutting_problem/" class="md-nav__link">
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-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.4 &nbsp; 最大切分乘积问题
-												deploy

											
										
										
											2023-07-21 21:53:15 +08:00
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-												deploy

											
										
										
											2023-07-21 22:50:40 +08:00
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 								    <li class="md-nav__item">
 								      <a href="../../chapter_greedy/summary/" class="md-nav__link">
 								  <span class="md-ellipsis">
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+.5 &nbsp; 小结
-												deploy

											
										
										
											2023-07-21 22:50:40 +08:00
+								  </span>
-												deploy

											
										
										
											2023-07-20 18:24:35 +08:00
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											2023-08-19 22:07:40 +08:00
+								    第 16 章 &nbsp; 附录
-												deploy

											
										
										
											2023-07-20 18:24:35 +08:00
+								  </span>
 								            </a>
 								              <label class="md-nav__link " for="__nav_17">
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-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								            第 16 章 &nbsp; 附录
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								          </label>
 								          <ul class="md-nav__list" data-md-scrollfix>
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-												deploy

											
										
										
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 								  <span class="md-ellipsis">
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+.1 &nbsp; 编程环境安装
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+								  </span>
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+								      </a>
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+.2 &nbsp; 一起参与创作
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+								  </span>
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+								      </a>
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+								          </ul>
 								        </nav>
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+								    </li>
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-												deploy

											
										
										
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-												First commit

											
										
										
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+								        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
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-												First commit

											
										
										
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-												deploy

											
										
										
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+								          <div class="md-nav__link md-nav__container">
 								            <a href="../../chapter_reference/" class="md-nav__link ">
 								  <span class="md-ellipsis">
 								    参考文献
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-												First commit

											
										
										
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+								          </div>
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+								        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
 								          <label class="md-nav__title" for="__nav_18">
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+								            <span class="md-nav__icon md-icon"></span>
 								            参考文献
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+								    </li>
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 								                <div class="md-sidebar__scrollwrap">
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 								<nav class="md-nav md-nav--secondary" aria-label="目录">
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-												First commit

											
										
										
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 								    <label class="md-nav__title" for="__toc">
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 								      目录
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-												deploy

											
										
										
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+								  <a href="#331" class="md-nav__link">
-												deploy

											
										
										
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+.3.1 &nbsp; 原码、反码和补码
-												deploy

											
										
										
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+								  </a>
-												First commit

											
										
										
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+								</li>
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 								        <li class="md-nav__item">
-												deploy

											
										
										
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+								  <a href="#332" class="md-nav__link">
-												deploy

											
										
										
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+.3.2 &nbsp; 浮点数编码
-												First commit

											
										
										
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+								  </a>
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 								              <article class="md-content__inner md-typeset">
-												deploy

											
										
										
											2023-09-21 20:44:52 +08:00
+								                  <!--
 								  Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>
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-												deploy

											
										
										
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+								    <a href="https://github.com/krahets/hello-algo/tree/main/docs/chapter_data_structure/number_encoding.md" title="编辑此页" class="md-content__button md-icon">
-												First commit

											
										
										
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 								      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M10 20H6V4h7v5h5v3.1l2-2V8l-6-6H6c-1.1 0-2 .9-2 2v16c0 1.1.9 2 2 2h4v-2m10.2-7c.1 0 .3.1.4.2l1.3 1.3c.2.2.2.6 0 .8l-1 1-2.1-2.1 1-1c.1-.1.2-.2.4-.2m0 3.9L14.1 23H12v-2.1l6.1-6.1 2.1 2.1Z"/></svg>
 								    </a>
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+								<!--
 								  Hack: check whether the content contains a h1 headline. If it doesn't, the
 								  page title (or respectively site name) is used as the main headline.
 								-->
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+								<h1 id="33">3.3 &nbsp; 数字编码 *<a class="headerlink" href="#33" title="Permanent link">&para;</a></h1>
-												deploy

											
										
										
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+								<div class="admonition note">
 								<p class="admonition-title">Note</p>
-												deploy

											
										
										
											2023-08-13 19:36:16 +08:00
+								<p>在本书中，标题带有的 * 符号的是选读章节。如果你时间有限或感到理解困难，可以先跳过，等学完必读章节后再单独攻克。</p>
-												deploy

											
										
										
											2023-02-24 11:43:41 +08:00
+								</div>
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								<h2 id="331">3.3.1 &nbsp; 原码、反码和补码<a class="headerlink" href="#331" title="Permanent link">&para;</a></h2>
-												deploy

											
										
										
											2023-08-20 23:28:04 +08:00
+								<p>在上一节的表格中我们发现，所有整数类型能够表示的负数都比正数多一个，例如 <code>byte</code> 的取值范围是 <span class="arithmatex">\([-128, 127]\)</span> 。这个现象比较反直觉，它的内在原因涉及到原码、反码、补码的相关知识。</p>
-												deploy

											
										
										
											2023-08-27 23:41:10 +08:00
+								<p>首先需要指出，<strong>数字是以“补码”的形式存储在计算机中的</strong>。在分析这样做的原因之前，我们首先给出三者的定义。</p>
-												deploy

											
										
										
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+								<ul>
 								<li><strong>原码</strong>：我们将数字的二进制表示的最高位视为符号位，其中 <span class="arithmatex">\(0\)</span> 表示正数，<span class="arithmatex">\(1\)</span> 表示负数，其余位表示数字的值。</li>
 								<li><strong>反码</strong>：正数的反码与其原码相同，负数的反码是对其原码除符号位外的所有位取反。</li>
 								<li><strong>补码</strong>：正数的补码与其原码相同，负数的补码是在其反码的基础上加 <span class="arithmatex">\(1\)</span> 。</li>
 								</ul>
-												deploy

											
										
										
											2023-08-28 23:52:27 +08:00
+								<p>图 3-4 展示了原码、反码和补码之间的转换方法。</p>
-												deploy

											
										
										
											2023-09-22 13:08:10 +08:00
+								<p><a class="glightbox" href="../number_encoding.assets/1s_2s_complement.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="原码、反码与补码之间的相互转换" src="../number_encoding.assets/1s_2s_complement.png" /></a></p>
-												deploy

											
										
										
											2023-08-22 13:50:24 +08:00
+								<p align="center"> 图 3-4 &nbsp; 原码、反码与补码之间的相互转换 </p>
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
-												deploy

											
										
										
											2023-08-20 23:28:04 +08:00
+								<p>「原码 true form」虽然最直观，但存在一些局限性。一方面，<strong>负数的原码不能直接用于运算</strong>。例如在原码下计算 <span class="arithmatex">\(1 + (-2)\)</span> ，得到的结果是 <span class="arithmatex">\(-3\)</span> ，这显然是不对的。</p>
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
+								<div class="arithmatex">\[
 								\begin{aligned}
 								&amp; 1 + (-2) \newline
-												deploy

											
										
										
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+								&amp; \rightarrow 0000 \; 0001 + 1000 \; 0010 \newline
 								&amp; = 1000 \; 0011 \newline
-												deploy

											
										
										
											2023-08-21 19:32:49 +08:00
+								&amp; \rightarrow -3
-												deploy

											
										
										
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+								\end{aligned}
 								\]</div>
-												deploy

											
										
										
											2023-08-20 23:28:04 +08:00
+								<p>为了解决此问题，计算机引入了「反码 1's complement code」。如果我们先将原码转换为反码，并在反码下计算 <span class="arithmatex">\(1 + (-2)\)</span> ，最后将结果从反码转化回原码，则可得到正确结果 <span class="arithmatex">\(-1\)</span> 。</p>
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
+								<div class="arithmatex">\[
 								\begin{aligned}
 								&amp; 1 + (-2) \newline
-												deploy

											
										
										
											2023-08-30 15:28:58 +08:00
+								&amp; \rightarrow 0000 \; 0001 \; \text{(原码)} + 1000 \; 0010 \; \text{(原码)} \newline
 								&amp; = 0000 \; 0001 \; \text{(反码)} + 1111  \; 1101 \; \text{(反码)} \newline
 								&amp; = 1111 \; 1110 \; \text{(反码)} \newline
 								&amp; = 1000 \; 0001 \; \text{(原码)} \newline
-												deploy

											
										
										
											2023-08-13 19:36:16 +08:00
+								&amp; \rightarrow -1
-												deploy

											
										
										
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+								\end{aligned}
 								\]</div>
-												deploy

											
										
										
											2023-08-13 19:36:16 +08:00
+								<p>另一方面，<strong>数字零的原码有 <span class="arithmatex">\(+0\)</span> 和 <span class="arithmatex">\(-0\)</span> 两种表示方式</strong>。这意味着数字零对应着两个不同的二进制编码，其可能会带来歧义。比如在条件判断中，如果没有区分正零和负零，则可能会导致判断结果出错。而如果我们想要处理正零和负零歧义，则需要引入额外的判断操作，其可能会降低计算机的运算效率。</p>
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
+								<div class="arithmatex">\[
-												deploy

											
										
										
											2023-05-26 20:37:56 +08:00
+								\begin{aligned}
-												deploy

											
										
										
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+								+0 &amp; \rightarrow 0000 \; 0000 \newline
 								-0 &amp; \rightarrow 1000 \; 0000
-												deploy

											
										
										
											2023-05-26 20:37:56 +08:00
+								\end{aligned}
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
+								\]</div>
-												deploy

											
										
										
											2023-08-20 23:28:04 +08:00
+								<p>与原码一样，反码也存在正负零歧义问题，因此计算机进一步引入了「补码 2's complement code」。我们先来观察一下负零的原码、反码、补码的转换过程：</p>
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
+								<div class="arithmatex">\[
 								\begin{aligned}
-												deploy

											
										
										
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+								-0 \rightarrow \; &amp; 1000 \; 0000 \; \text{(原码)} \newline
 								= \; &amp; 1111 \; 1111 \; \text{(反码)} \newline
 								= 1 \; &amp; 0000 \; 0000 \; \text{(补码)} \newline
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
+								\end{aligned}
 								\]</div>
-												deploy

											
										
										
											2023-08-30 15:28:58 +08:00
+								<p>在负零的反码基础上加 <span class="arithmatex">\(1\)</span> 会产生进位，但 <code>byte</code> 类型的长度只有 8 位，因此溢出到第 9 位的 <span class="arithmatex">\(1\)</span> 会被舍弃。也就是说，<strong>负零的补码为 <span class="arithmatex">\(0000 \; 0000\)</span> ，与正零的补码相同</strong>。这意味着在补码表示中只存在一个零，正负零歧义从而得到解决。</p>
-												deploy

											
										
										
											2023-08-13 19:36:16 +08:00
+								<p>还剩余最后一个疑惑：<code>byte</code> 类型的取值范围是 <span class="arithmatex">\([-128, 127]\)</span> ，多出来的一个负数 <span class="arithmatex">\(-128\)</span> 是如何得到的呢？我们注意到，区间 <span class="arithmatex">\([-127, +127]\)</span> 内的所有整数都有对应的原码、反码和补码，并且原码和补码之间是可以互相转换的。</p>
-												deploy

											
										
										
											2023-08-30 15:28:58 +08:00
+								<p>然而，<strong>补码 <span class="arithmatex">\(1000 \; 0000\)</span> 是一个例外，它并没有对应的原码</strong>。根据转换方法，我们得到该补码的原码为 <span class="arithmatex">\(0000 \; 0000\)</span> 。这显然是矛盾的，因为该原码表示数字 <span class="arithmatex">\(0\)</span> ，它的补码应该是自身。计算机规定这个特殊的补码 <span class="arithmatex">\(1000 \; 0000\)</span> 代表 <span class="arithmatex">\(-128\)</span> 。实际上，<span class="arithmatex">\((-1) + (-127)\)</span> 在补码下的计算结果就是 <span class="arithmatex">\(-128\)</span> 。</p>
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
+								<div class="arithmatex">\[
 								\begin{aligned}
 								&amp; (-127) + (-1) \newline
-												deploy

											
										
										
											2023-08-30 15:28:58 +08:00
+								&amp; \rightarrow 1111 \; 1111 \; \text{(原码)} + 1000 \; 0001 \; \text{(原码)} \newline
 								&amp; = 1000 \; 0000 \; \text{(反码)} + 1111  \; 1110 \; \text{(反码)} \newline
 								&amp; = 1000 \; 0001 \; \text{(补码)} + 1111  \; 1111 \; \text{(补码)} \newline
 								&amp; = 1000 \; 0000 \; \text{(补码)} \newline
-												deploy

											
										
										
											2023-08-13 19:36:16 +08:00
+								&amp; \rightarrow -128
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
+								\end{aligned}
 								\]</div>
-												deploy

											
										
										
											2023-08-13 19:36:16 +08:00
+								<p>你可能已经发现，上述的所有计算都是加法运算。这暗示着一个重要事实：<strong>计算机内部的硬件电路主要是基于加法运算设计的</strong>。这是因为加法运算相对于其他运算（比如乘法、除法和减法）来说，硬件实现起来更简单，更容易进行并行化处理，运算速度更快。</p>
 								<p>请注意，这并不意味着计算机只能做加法。<strong>通过将加法与一些基本逻辑运算结合，计算机能够实现各种其他的数学运算</strong>。例如，计算减法 <span class="arithmatex">\(a - b\)</span> 可以转换为计算加法 <span class="arithmatex">\(a + (-b)\)</span> ；计算乘法和除法可以转换为计算多次加法或减法。</p>
-												deploy

											
										
										
											2023-08-20 14:52:42 +08:00
+								<p>现在我们可以总结出计算机使用补码的原因：基于补码表示，计算机可以用同样的电路和操作来处理正数和负数的加法，不需要设计特殊的硬件电路来处理减法，并且无须特别处理正负零的歧义问题。这大大简化了硬件设计，提高了运算效率。</p>
-												deploy

											
										
										
											2023-08-13 19:36:16 +08:00
+								<p>补码的设计非常精妙，因篇幅关系我们就先介绍到这里，建议有兴趣的读者进一步深度了解。</p>
-												deploy

											
										
										
											2023-08-19 22:07:40 +08:00
+								<h2 id="332">3.3.2 &nbsp; 浮点数编码<a class="headerlink" href="#332" title="Permanent link">&para;</a></h2>
-												deploy

											
										
										
											2023-09-17 01:11:45 +08:00
+								<p>细心的你可能会发现：<code>int</code> 和 <code>float</code> 长度相同，都是 4 bytes ，但为什么 <code>float</code> 的取值范围远大于 <code>int</code> ？这非常反直觉，因为按理说 <code>float</code> 需要表示小数，取值范围应该变小才对。</p>
-												deploy

											
										
										
											2023-07-29 04:48:39 +08:00
+								<p>实际上，<strong>这是因为浮点数 <code>float</code> 采用了不同的表示方式</strong>。记一个 32-bit 长度的二进制数为：</p>
 								<div class="arithmatex">\[
 								b_{31} b_{30} b_{29} \ldots b_2 b_1 b_0
 								\]</div>
-												deploy

											
										
										
											2023-08-27 23:41:10 +08:00
+								<p>根据 IEEE 754 标准，32-bit 长度的 <code>float</code> 由以下三个部分构成。</p>
-												deploy

											
										
										
											2023-02-22 19:01:30 +08:00
+								<ul>
-												deploy

											
										
										
											2023-07-29 04:48:39 +08:00
+								<li>符号位 <span class="arithmatex">\(\mathrm{S}\)</span> ：占 1 bit ，对应 <span class="arithmatex">\(b_{31}\)</span> 。</li>
 								<li>指数位 <span class="arithmatex">\(\mathrm{E}\)</span> ：占 8 bits ，对应 <span class="arithmatex">\(b_{30} b_{29} \ldots b_{23}\)</span> 。</li>
 								<li>分数位 <span class="arithmatex">\(\mathrm{N}\)</span> ：占 23 bits ，对应 <span class="arithmatex">\(b_{22} b_{21} \ldots b_0\)</span> 。</li>
-												deploy

											
										
										
											2023-02-22 19:01:30 +08:00
+								</ul>
-												deploy

											
										
										
											2023-07-29 04:48:39 +08:00
+								<p>二进制数 <code>float</code> 对应的值的计算方法：</p>
-												deploy

											
										
										
											2023-02-22 19:01:30 +08:00
+								<div class="arithmatex">\[
-												deploy

											
										
										
											2023-07-29 04:48:39 +08:00
+								\text {val} = (-1)^{b_{31}} \times 2^{\left(b_{30} b_{29} \ldots b_{23}\right)_2-127} \times\left(1 . b_{22} b_{21} \ldots b_0\right)_2
-												deploy

											
										
										
											2023-02-22 19:01:30 +08:00
+								\]</div>
-												deploy

											
										
										
											2023-07-29 04:48:39 +08:00
+								<p>转化到十进制下的计算公式：</p>
-												deploy

											
										
										
											2023-02-22 19:01:30 +08:00
+								<div class="arithmatex">\[
-												deploy

											
										
										
											2023-07-29 04:48:39 +08:00
+								\text {val}=(-1)^{\mathrm{S}} \times 2^{\mathrm{E} -127} \times (1 + \mathrm{N})
-												deploy

											
										
										
											2023-02-22 19:01:30 +08:00
+								\]</div>
-												deploy

											
										
										
											2023-07-29 04:48:39 +08:00
+								<p>其中各项的取值范围：</p>
-												deploy

											
										
										
											2023-02-22 19:28:44 +08:00
+								<div class="arithmatex">\[
 								\begin{aligned}
-												deploy

											
										
										
											2023-08-27 23:41:10 +08:00
+								\mathrm{S} \in &amp; \{ 0, 1\}, \quad \mathrm{E} \in \{ 1, 2, \dots, 254 \} \newline
-												deploy

											
										
										
											2023-02-22 19:28:44 +08:00
+								(1 + \mathrm{N}) = &amp; (1 + \sum_{i=1}^{23} b_{23-i} 2^{-i}) \subset [1, 2 - 2^{-23}]
 								\end{aligned}
 								\]</div>
-												deploy

											
										
										
											2023-09-22 13:08:10 +08:00
+								<p><a class="glightbox" href="../number_encoding.assets/ieee_754_float.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="IEEE 754 标准下的 float 的计算示例" src="../number_encoding.assets/ieee_754_float.png" /></a></p>
-												deploy

											
										
										
											2023-08-22 13:50:24 +08:00
+								<p align="center"> 图 3-5 &nbsp; IEEE 754 标准下的 float 的计算示例 </p>
-												deploy

											
										
										
											2023-02-26 19:53:32 +08:00
-												deploy

											
										
										
											2023-08-22 13:50:24 +08:00
+								<p>观察图 3-5 ，给定一个示例数据 <span class="arithmatex">\(\mathrm{S} = 0\)</span> ， <span class="arithmatex">\(\mathrm{E} = 124\)</span> ，<span class="arithmatex">\(\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\)</span> ，则有：</p>
-												deploy

											
										
										
											2023-02-22 19:01:30 +08:00
+								<div class="arithmatex">\[
 								\text { val } = (-1)^0 \times 2^{124 - 127} \times (1 + 0.375) = 0.171875
 								\]</div>
-												deploy

											
										
										
											2023-06-12 23:05:10 +08:00
+								<p>现在我们可以回答最初的问题：<strong><code>float</code> 的表示方式包含指数位，导致其取值范围远大于 <code>int</code></strong> 。根据以上计算，<code>float</code> 可表示的最大正数为 <span class="arithmatex">\(2^{254 - 127} \times (2 - 2^{-23}) \approx 3.4 \times 10^{38}\)</span> ，切换符号位便可得到最小负数。</p>
-												deploy

											
										
										
											2023-05-17 04:37:57 +08:00
+								<p><strong>尽管浮点数 <code>float</code> 扩展了取值范围，但其副作用是牺牲了精度</strong>。整数类型 <code>int</code> 将全部 32 位用于表示数字，数字是均匀分布的；而由于指数位的存在，浮点数 <code>float</code> 的数值越大，相邻两个数字之间的差值就会趋向越大。</p>
-												deploy

											
										
										
											2023-08-22 13:50:24 +08:00
+								<p>如表 3-2 所示，指数位 <span class="arithmatex">\(E = 0\)</span> 和 <span class="arithmatex">\(E = 255\)</span> 具有特殊含义，<strong>用于表示零、无穷大、<span class="arithmatex">\(\mathrm{NaN}\)</span> 等</strong>。</p>
 								<p align="center"> 表 3-2 &nbsp; 指数位含义 </p>
-												deploy

											
										
										
											2023-08-19 19:21:30 +08:00
-												deploy

											
										
										
											2023-03-07 01:16:02 +08:00
+								<div class="center-table">
-												deploy

											
										
										
											2023-02-22 19:01:30 +08:00
+								<table>
 								<thead>
 								<tr>
 								<th>指数位 E</th>
 								<th>分数位 <span class="arithmatex">\(\mathrm{N} = 0\)</span></th>
 								<th>分数位 <span class="arithmatex">\(\mathrm{N} \ne 0\)</span></th>
 								<th>计算公式</th>
 								</tr>
 								</thead>
 								<tbody>
 								<tr>
 								<td><span class="arithmatex">\(0\)</span></td>
 								<td><span class="arithmatex">\(\pm 0\)</span></td>
-												deploy

											
										
										
											2023-03-01 22:22:09 +08:00
+								<td>次正规数</td>
-												deploy

											
										
										
											2023-02-22 19:01:30 +08:00
+								<td><span class="arithmatex">\((-1)^{\mathrm{S}} \times 2^{-126} \times (0.\mathrm{N})\)</span></td>
 								</tr>
 								<tr>
 								<td><span class="arithmatex">\(1, 2, \dots, 254\)</span></td>
 								<td>正规数</td>
 								<td>正规数</td>
 								<td><span class="arithmatex">\((-1)^{\mathrm{S}} \times 2^{(\mathrm{E} -127)} \times (1.\mathrm{N})\)</span></td>
 								</tr>
 								<tr>
 								<td><span class="arithmatex">\(255\)</span></td>
 								<td><span class="arithmatex">\(\pm \infty\)</span></td>
 								<td><span class="arithmatex">\(\mathrm{NaN}\)</span></td>
 								<td></td>
 								</tr>
 								</tbody>
 								</table>
-												deploy

											
										
										
											2023-03-07 01:16:02 +08:00
+								</div>
-												deploy

											
										
										
											2023-08-27 23:41:10 +08:00
+								<p>值得说明的是，次正规数显著提升了浮点数的精度。最小正正规数为 <span class="arithmatex">\(2^{-126}\)</span> ，最小正次正规数为 <span class="arithmatex">\(2^{-126} \times 2^{-23}\)</span> 。</p>
 								<p>双精度 <code>double</code> 也采用类似 <code>float</code> 的表示方法，在此不做赘述。</p>
-												First commit

											
										
										
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+								      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><!--! Font Awesome Free 6.4.0 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M459.37 151.716c.325 4.548.325 9.097.325 13.645 0 138.72-105.583 298.558-298.558 298.558-59.452 0-114.68-17.219-161.137-47.106 8.447.974 16.568 1.299 25.34 1.299 49.055 0 94.213-16.568 130.274-44.832-46.132-.975-84.792-31.188-98.112-72.772 6.498.974 12.995 1.624 19.818 1.624 9.421 0 18.843-1.3 27.614-3.573-48.081-9.747-84.143-51.98-84.143-102.985v-1.299c13.969 7.797 30.214 12.67 47.431 13.319-28.264-18.843-46.781-51.005-46.781-87.391 0-19.492 5.197-37.36 14.294-52.954 51.655 63.675 129.3 105.258 216.365 109.807-1.624-7.797-2.599-15.918-2.599-24.04 0-57.828 46.782-104.934 104.934-104.934 30.213 0 57.502 12.67 76.67 33.137 23.715-4.548 46.456-13.32 66.599-25.34-7.798 24.366-24.366 44.833-46.132 57.827 21.117-2.273 41.584-8.122 60.426-16.243-14.292 20.791-32.161 39.308-52.628 54.253z"/></svg>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    </a>
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-												deploy

											
										
										
											2023-04-16 16:40:06 +08:00
+								      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 640 512"><!--! Font Awesome Free 6.4.0 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3l89.3 89.4-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z"/></svg>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
+								    </a>
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-												deploy

											
										
										
											2023-07-17 18:38:57 +08:00
+								    <script id="__config" type="application/json">{"base": "../..", "features": ["content.action.edit", "content.code.annotate", "content.code.copy", "content.tabs.link", "content.tooltips", "navigation.indexes", "navigation.instant", "navigation.top", "navigation.footer", "navigation.tracking", "search.highlight", "search.share", "search.suggest", "toc.follow"], "search": "../../assets/javascripts/workers/search.780af0f4.min.js", "translations": {"clipboard.copied": "\u5df2\u590d\u5236", "clipboard.copy": "\u590d\u5236", "search.result.more.one": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.more.other": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 # \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.none": "\u6ca1\u6709\u627e\u5230\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.one": "\u627e\u5230 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.other": "# \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.placeholder": "\u952e\u5165\u4ee5\u5f00\u59cb\u641c\u7d22", "search.result.term.missing": "\u7f3a\u5c11", "select.version": "\u9009\u62e9\u5f53\u524d\u7248\u672c"}}</script>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
+								      <script src="../../assets/javascripts/bundle.f11ae8b1.min.js"></script>
-												First commit

											
										
										
											2023-02-08 14:54:41 +08:00
-												deploy

											
										
										
											2023-07-17 17:51:12 +08:00
 								          <script src="../../javascripts/mathjax.js"></script>
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 								          <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
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											2023-07-17 17:51:12 +08:00
 								          <script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
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-												deploy

											
										
										
											2023-09-22 13:08:10 +08:00
+								  <script>document$.subscribe(() => {const lightbox = GLightbox({"touchNavigation": true, "loop": false, "zoomable": true, "draggable": false, "openEffect": "zoom", "closeEffect": "zoom", "slideEffect": "none"});})</script></body>
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