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<h1 id="113">11.3 &nbsp; 冒泡排序<a class="headerlink" href="#113" title="Permanent link">&para;</a></h1>
<p>「冒泡排序 bubble sort」通过连续地比较与交换相邻元素实现排序。这个过程就像气泡从底部升到顶部一样，因此得名冒泡排序。</p>
<p>如图 11-4 所示，冒泡过程可以利用元素交换操作来模拟：从数组最左端开始向右遍历，依次比较相邻元素大小，如果“左元素 &gt; 右元素”就交换它俩。遍历完成后，最大的元素会被移动到数组的最右端。</p>
@@ -3975,11 +4007,73 @@
<li><strong>稳定排序</strong>：由于在“冒泡”中遇到相等元素不交换。</li>
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<h1 id="118">11.8 &nbsp; 桶排序<a class="headerlink" href="#118" title="Permanent link">&para;</a></h1>
<p>前述的几种排序算法都属于“基于比较的排序算法”，它们通过比较元素间的大小来实现排序。此类排序算法的时间复杂度无法超越 <span class="arithmatex">\(O(n \log n)\)</span> 。接下来，我们将探讨几种“非比较排序算法”，它们的时间复杂度可以达到线性阶。</p>
<p>「桶排序 bucket sort」是分治策略的一个典型应用。它通过设置一些具有大小顺序的桶，每个桶对应一个数据范围，将数据平均分配到各个桶中；然后，在每个桶内部分别执行排序；最终按照桶的顺序将所有数据合并。</p>
@@ -3840,11 +3872,73 @@
<p><img alt="根据概率分布划分桶" src="../bucket_sort.assets/scatter_in_buckets_distribution.png" /></p>
<p align="center"> 图 11-15 &nbsp; 根据概率分布划分桶 </p>
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<h1 id="119">11.9 &nbsp; 计数排序<a class="headerlink" href="#119" title="Permanent link">&para;</a></h1>
<p>「计数排序 counting sort」通过统计元素数量来实现排序，通常应用于整数数组。</p>
<h2 id="1191">11.9.1 &nbsp; 简单实现<a class="headerlink" href="#1191" title="Permanent link">&para;</a></h2>
@@ -4204,11 +4236,73 @@
<p><strong>计数排序只适用于非负整数</strong>。若想要将其用于其他类型的数据，需要确保这些数据可以被转换为非负整数，并且在转换过程中不能改变各个元素之间的相对大小关系。例如，对于包含负数的整数数组，可以先给所有数字加上一个常数，将全部数字转化为正数，排序完成后再转换回去即可。</p>
<p><strong>计数排序适用于数据量大但数据范围较小的情况</strong>。比如，在上述示例中 <span class="arithmatex">\(m\)</span> 不能太大，否则会占用过多空间。而当 <span class="arithmatex">\(n \ll m\)</span> 时，计数排序使用 <span class="arithmatex">\(O(m)\)</span> 时间，可能比 <span class="arithmatex">\(O(n \log n)\)</span> 的排序算法还要慢。</p>
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@@ -3968,11 +4000,73 @@
<li><strong>非稳定排序</strong>：在交换堆顶元素和堆底元素时，相等元素的相对位置可能发生变化。</li>
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<h1 id="114">11.4 &nbsp; 插入排序<a class="headerlink" href="#114" title="Permanent link">&para;</a></h1>
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@@ -3691,11 +3723,73 @@
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@@ -4072,11 +4104,73 @@
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<h1 id="115">11.5 &nbsp; 快速排序<a class="headerlink" href="#115" title="Permanent link">&para;</a></h1>
<p>「快速排序 quick sort」是一种基于分治策略的排序算法，运行高效，应用广泛。</p>
<p>快速排序的核心操作是“哨兵划分”，其目标是：选择数组中的某个元素作为“基准数”，将所有小于基准数的元素移到其左侧，而大于基准数的元素移到其右侧。具体来说，哨兵划分的流程如图 11-8 所示。</p>
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<h1 id="1110">11.10 &nbsp; 基数排序<a class="headerlink" href="#1110" title="Permanent link">&para;</a></h1>
<p>上一节我们介绍了计数排序，它适用于数据量 <span class="arithmatex">\(n\)</span> 较大但数据范围 <span class="arithmatex">\(m\)</span> 较小的情况。假设我们需要对 <span class="arithmatex">\(n = 10^6\)</span> 个学号进行排序，而学号是一个 <span class="arithmatex">\(8\)</span> 位数字，这意味着数据范围 <span class="arithmatex">\(m = 10^8\)</span> 非常大，使用计数排序需要分配大量内存空间，而基数排序可以避免这种情况。</p>
<p>「基数排序 radix sort」的核心思想与计数排序一致，也通过统计个数来实现排序。在此基础上，基数排序利用数字各位之间的递进关系，依次对每一位进行排序，从而得到最终的排序结果。</p>
@@ -4106,11 +4138,73 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
<li><strong>稳定排序</strong>：与计数排序相同。</li>
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<h1 id="112">11.2 &nbsp; 选择排序<a class="headerlink" href="#112" title="Permanent link">&para;</a></h1>
<p>「选择排序 selection sort」的工作原理非常直接：开启一个循环，每轮从未排序区间选择最小的元素，将其放到已排序区间的末尾。</p>
<p>设数组的长度为 <span class="arithmatex">\(n\)</span> ，选择排序的算法流程如图 11-2 所示。</p>
@@ -3699,11 +3731,73 @@
<p><img alt="选择排序非稳定示例" src="../selection_sort.assets/selection_sort_instability.png" /></p>
<p align="center"> 图 11-3 &nbsp; 选择排序非稳定示例 </p>
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<h1 id="111">11.1 &nbsp; 排序算法<a class="headerlink" href="#111" title="Permanent link">&para;</a></h1>
<p>「排序算法 sorting algorithm」用于对一组数据按照特定顺序进行排列。排序算法有着广泛的应用，因为有序数据通常能够被更有效地查找、分析和处理。</p>
<p>如图 11-1 所示，排序算法中的数据类型可以是整数、浮点数、字符或字符串等。排序的判断规则可根据需求设定，如数字大小、字符 ASCII 码顺序或自定义规则。</p>
@@ -3476,11 +3508,73 @@
<p><strong>运行快、原地、稳定、正向自适应、通用性好</strong>。显然，迄今为止尚未发现兼具以上所有特性的排序算法。因此，在选择排序算法时，需要根据具体的数据特点和问题需求来决定。</p>
<p>接下来，我们将共同学习各种排序算法，并基于上述评价维度对各个排序算法的优缺点进行分析。</p>
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