Chemmy's Blog

C#排序算法的比较 - r163 - 博客园

Excerpt

首先通过图表比较不同排序算法的时间复杂度和稳定性。排序方法平均时间最坏情况最好情况辅助空间稳定性直接插入排序O(n2)O(n2)O(n)O(1)是冒泡排序O(n2)O(n2)O(n)O(1)是简单选择排序O(n2)O(n2)O(n2)O(1)是希尔排序-O(nlog2n)~O(n2)O(nlog2n

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首先通过图表比较不同排序算法的时间复杂度和稳定性。

排序方法	平均时间	最坏情况	最好情况	辅助空间	稳定性
直接插入排序	O(n2)	O(n2)	O(n)	O(1)	是
冒泡排序	O(n2)	O(n2)	O(n)	O(1)	是
简单选择排序	O(n2)	O(n2)	O(n2)	O(1)	是
希尔排序	-	O(nlog2n)~O(n2)	O(nlog2n)~O(n2)	O(1)	否
快速排序	O(nlog2n)	O(n2)	O(nlog2n)	O(log2n)	否
堆排序	O(nlog2n)	O(nlog2n)	O(nlog2n)	O(1)	否
2-路归并排序	O(nlog2n)	O(nlog2n)	O(nlog2n)	O(n)	是
基数排序	O(d(n + rd))	O(d(n + rd))	O(d(n + rd))	O(rd)	是

注：1. 算法的时间复杂度一般情况下指最坏情况下的渐近时间复杂度。

        2. 排序算法的稳定性会对多关键字排序产生影响。

下面通过C#代码说明不同的排序算法

插入排序

时间复杂度：平均情况—O(n²) 最坏情况—O(n²) 辅助空间：O(1) 稳定性：稳定

插入排序是在一个已经有序的小序列的基础上，一次插入一个元素。当然，刚开始这个有序的小序列只有1个元素，就是第一个元素。比较是从有序序列的末尾开始，也就是想要插入的元素和已经有序的最大者开始比起，如果比它大则直接插入在其后面，否则一直往前找直到找到它该插入的位置。如果碰见一个和插入元素相等的，那么插入元素把想插入的元素放在相等元素的后面。所以，相等元素的前后顺序没有改变，从原无序序列出去的顺序就是排好序后的顺序，所以插入排序是稳定的。

希尔排序(shell)

时间复杂度：理想情况—O(nlog₂n) 最坏情况—O(n²) 稳定性：不稳定

希尔排序是按照不同步长对元素进行插入排序，当刚开始元素很无序的时候，步长最大，所以插入排序的元素个数很少，速度很快；当元素基本有序了，步长很小，插入排序对于有序的序列效率很高。所以，希尔排序的时间复杂度会比o(n^2)好一些。由于多次插入排序，我们知道一次插入排序是稳定的，不会改变相同元素的相对顺序，但在不同的插入排序过程中，相同的元素可能在各自的插入排序中移动，最后其稳定性就会被打乱，所以shell排序是不稳定的。

冒泡排序

时间复杂度：平均情况—O(n²) 最坏情况—O(n²) 辅助空间：O(1) 稳定性：稳定

冒泡排序就是把小的元素往前调或者把大的元素往后调。比较是相邻的两个元素比较，交换也发生在这两个元素之间。所以，如果两个元素相等，我想你是不会再无聊地把他们俩交换一下的；如果两个相等的元素没有相邻，那么即使通过前面的两两交换把两个相邻起来，这时候也不会交换，所以相同元素的前后顺序并没有改变，所以冒泡排序是一种稳定排序算法。

快速排序

时间复杂度：平均情况—O(nlog₂n) 最坏情况—O(n²) 辅助空间：O(log₂n) 稳定性：不稳定

快速排序有两个方向，左边的i下标一直往右走，当a[i] <= a[center_index]，其中center_index是中枢元素的数组下标，一般取为数组第0个元素。而右边的j下标一直往左走，当a[j] > a[center_index]。如果i和j都走不动了，i <= j, 交换a[i]和a[j],重复上面的过程，直到i>j。 交换a[j]和a[center_index]，完成一趟快速排序。在中枢元素和a[j]交换的时候，很有可能把前面的元素的稳定性打乱，比如序列为 5 3 3 4 3 8 9 10 11， 现在中枢元素5和3(第5个元素，下标从1开始计)交换就会把元素3的稳定性打乱，所以快速排序是一个不稳定的排序算法，不稳定发生在中枢元素和a[j]交换的时刻。

C# 代码   复制

<span><img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/None.gif" alt=""><span>
<img id="Highlighter_45_648_Open_Image" src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedBlockStart.gif" alt=""><span>int<span> Partition(SqList <span>&amp;<span>L, <span>int<span> low, <span>int<span> high) <span id="Highlighter_45_648_Open_Text"><span>{
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""> <span>//<span> 交换顺序表L中子序列L.r[low..high]的记录，使枢轴记录到位，
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">   <span>//<span> 并返回其所在位置，此时，在它之前（后）的记录均不大（小）于它<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">   KeyType pivotkey;
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">   RedType temp;
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">   pivotkey <span>=<span> L.r[low].key;     <span>//<span> 用子表的第一个记录作枢轴记录<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img id="Highlighter_241_600_Open_Image" src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedSubBlockStart.gif" alt="">   <span>while<span> (low <span>&lt;<span> high) <span id="Highlighter_241_600_Open_Text"><span>{           <span>//<span> 从表的两端交替地向中间扫描<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">      <span>while<span> (low <span>&lt;<span> high <span>&amp;&amp;<span> L.r[high].key<span>&gt;=<span>pivotkey) <span>--<span>high;
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">      temp<span>=<span>L.r[low];
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">      L.r[low]<span>=<span>L.r[high];
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">      L.r[high]<span>=<span>temp;           <span>//<span> 将比枢轴记录小的记录交换到低端<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">      <span>while<span> (low  <span>&lt;<span> high <span>&amp;&amp;<span> L.r[low].key <span>&lt;<span> <span>=<span>pivotkey) <span>++<span>low;
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">      temp<span>=<span>L.r[low];
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">      L.r[low]<span>=<span>L.r[high];
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">      L.r[high]<span>=<span>temp;           <span>//<span> 将比枢轴记录大的记录交换到高端<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedSubBlockEnd.gif" alt="">   }<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">   <span>return<span> low;                  <span>//<span> 返回枢轴所在位置<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedBlockEnd.gif" alt="">}<span> <span>//<span> Partition        <span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/None.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/None.gif" alt="">
<img id="Highlighter_713_1006_Open_Image" src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedBlockStart.gif" alt=""><span>void<span> QSort(SqList <span>&amp;<span>L, <span>int<span> low, <span>int<span> high) <span id="Highlighter_713_1006_Open_Text"><span>{
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">  <span>//<span> 对顺序表L中的子序列L.r[low..high]进行快速排序<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">  <span>int<span> pivotloc;
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img id="Highlighter_790_1003_Open_Image" src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedSubBlockStart.gif" alt="">  <span>if<span> (low  <span>&lt;<span>  high) <span id="Highlighter_790_1003_Open_Text"><span>{                      <span>//<span> 长度大于1<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">    pivotloc <span>=<span> Partition(L, low, high);  <span>//<span> 将L.r[low..high]一分为二<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">    QSort(L, low, pivotloc<span>-<span>1<span>); <span>//<span> 对低子表递归排序，pivotloc是枢轴位置<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">    QSort(L, pivotloc<span>+<span>1<span>, high);          <span>//<span> 对高子表递归排序<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedSubBlockEnd.gif" alt="">  }<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedBlockEnd.gif" alt="">}<span> <span>//<span> QSort     <span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/None.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/None.gif" alt="">  
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/None.gif" alt="">
<img id="Highlighter_1052_1102_Open_Image" src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedBlockStart.gif" alt=""><span>void<span> QuickSort(SqList <span>&amp;<span>L) <span id="Highlighter_1052_1102_Open_Text"><span>{
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">   <span>//<span> 对顺序表L进行快速排序<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">   QSort(L, <span>1<span>, L.length);
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedBlockEnd.gif" alt="">}<span> <span>//<span> QuickSort </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>

选择排序

时间复杂度：平均情况—O(n²) 最坏情况—O(n²) 辅助空间：O(1) 稳定性：不稳定

选择排序是给每个位置选择当前元素最小的，比如给第一个位置选择最小的，在剩余元素里面给第二个元素选择第二小的，依次类推，直到第n-1个元素，第n个元素不用选择了，因为只剩下它一个最大的元素了。那么，在一趟选择，如果当前元素比一个元素小，而该小的元素又出现在一个和当前元素相等的元素后面，那么交换后稳定性就被破坏了。比较拗口，举个例子，序列5 8 5 2 9， 我们知道第一遍选择第1个元素5会和2交换，那么原序列中2个5的相对前后顺序就被破坏了，所以选择排序不是一个稳定的排序算法。

堆排序

时间复杂度：平均情况—O(nlog₂n) 最坏情况—O(nlog₂n) 辅助空间：O(1) 稳定性：不稳定

我们知道堆的结构是节点i的孩子为2*i和2*i+1节点，大顶堆要求父节点大于等于其2个子节点，小顶堆要求父节点小于等于其2个子节点。在一个长为n的序列，堆排序的过程是从第n/2开始和其子节点共3个值选择最大(大顶堆)或者最小(小顶堆),这3个元素之间的选择当然不会破坏稳定性。但当为n/2-1, n/2-2, …1这些个父节点选择元素时，就会破坏稳定性。有可能第n/2个父节点交换把后面一个元素交换过去了，而第n/2-1个父节点把后面一个相同的元素没有交换，那么这2个相同的元素之间的稳定性就被破坏了。所以，堆排序不是稳定的排序算法

归并排序

时间复杂度：平均情况—O(nlog₂n) 最坏情况—O(nlog₂n) 辅助空间：O(n) 稳定性：稳定

归并排序是把序列递归地分成短序列，递归出口是短序列只有1个元素(认为直接有序)或者2个序列(1次比较和交换),然后把各个有序的段序列合并成一个有序的长序列，不断合并直到原序列全部排好序。可以发现，在1个或2个元素时，1个元素不会交换，2个元素如果大小相等也没有人故意交换，这不会破坏稳定性。那么，在短的有序序列合并的过程中，稳定是是否受到破坏？没有，合并过程中我们可以保证如果两个当前元素相等时，我们把处在前面的序列的元素保存在结果序列的前面，这样就保证了稳定性。所以，归并排序也是稳定的排序算法。

C# 代码   复制

<span>
<img id="Highlighter_61_479_Open_Image" src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedBlockStart.gif" alt=""><span>void<span> Merge (RedType SR[], RedType TR[], <span>int<span> i, <span>int<span> m, <span>int<span> n) <span id="Highlighter_61_479_Open_Text"><span>{
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">   <span>//<span> 将有序的SR[i..m]和SR[m+1..n]归并为有序的TR[i..n]<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">   <span>int<span> j,k;
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img id="Highlighter_167_280_Open_Image" src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedSubBlockStart.gif" alt="">   <span>for<span> (j<span>=<span>m<span>+<span>1<span>, k<span>=<span>i;  i <span>&lt;<span> <span>=<span>m <span>&amp;&amp;<span> j <span>&lt;<span> <span>=<span>n;  <span>++<span>k) <span id="Highlighter_167_280_Open_Text"><span>{
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">      <span>//<span> 将SR中记录由小到大地并入TR<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">      <span>if<span> LQ(SR[i].key,SR[j].key) TR[k] <span>=<span> SR[i<span>++<span>];
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">      <span>else<span> TR[k] <span>=<span> SR[j<span>++<span>];
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedSubBlockEnd.gif" alt="">   }<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">   <span>if<span> (i <span>&lt;<span> <span>=<span>m)  <span>//<span> TR[k..n] = SR[i..m];  将剩余的SR[i..m]复制到TR<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">      <span>while<span> (k <span>&lt;<span> <span>=<span>n <span>&amp;&amp;<span> i <span>&lt;<span> <span>=<span>m) TR[k<span>++<span>]<span>=<span>SR[i<span>++<span>];
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">   <span>if<span> (j <span>&lt;<span> <span>=<span>n)  <span>//<span> 将剩余的SR[j..n]复制到TR<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">      <span>while<span> (k <span>&lt;<span> <span>=<span>n <span>&amp;&amp;<span>j  <span>&lt;<span> <span>=<span>n) TR[k<span>++<span>]<span>=<span>SR[j<span>++<span>];
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedBlockEnd.gif" alt="">}<span> <span>//<span> Merge    <span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/None.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/None.gif" alt="">  
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/None.gif" alt="">
<img id="Highlighter_552_919_Open_Image" src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedBlockStart.gif" alt=""><span>void<span> MSort(RedType SR[], RedType TR1[], <span>int<span> s, <span>int<span> t) <span id="Highlighter_552_919_Open_Text"><span>{
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">   <span>//<span> 将SR[s..t]归并排序为TR1[s..t]。<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">   <span>int<span> m;
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">   RedType TR2[<span>20<span>];
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">   <span>if<span> (s<span>==<span>t) TR1[t] <span>=<span> SR[s];
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img id="Highlighter_657_916_Open_Image" src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedSubBlockStart.gif" alt="">   <span>else<span> <span id="Highlighter_657_916_Open_Text"><span>{
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">      m<span>=<span>(s<span>+<span>t)<span>/<span>2<span>;            <span>//<span> 将SR[s..t]平分为SR[s..m]和SR[m+1..t]<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">      MSort(SR,TR2,s,m);    <span>//<span> 递归地将SR[s..m]归并为有序的TR2[s..m]<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">      MSort(SR,TR2,m<span>+<span>1<span>,t);  <span>//<span> 将SR[m+1..t]归并为有序的TR2[m+1..t]<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">      Merge(TR2,TR1,s,m,t); <span>//<span> 将TR2[s..m]和TR2[m+1..t]归并到TR1[s..t]<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedSubBlockEnd.gif" alt="">   }<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedBlockEnd.gif" alt="">}<span> <span>//<span> MSort    <span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/None.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/None.gif" alt="">  
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/None.gif" alt="">
<img id="Highlighter_964_1019_Open_Image" src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedBlockStart.gif" alt=""><span>void<span> MergeSort(SqList <span>&amp;<span>L) <span id="Highlighter_964_1019_Open_Text"><span>{
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">  <span>//<span> 对顺序表L作归并排序。<span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt=""><span>
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">  MSort(L.r, L.r, <span>1<span>, L.length);
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/InBlock.gif" alt="">
<img src="C%E6%8E%92%E5%BA%8F%E7%AE%97%E6%B3%95%E7%9A%84%E6%AF%94%E8%BE%83%20-%20r163%20-%20%E5%8D%9A%E5%AE%A2%E5%9B%AD/ExpandedBlockEnd.gif" alt="">}<span> <span>//<span> MergeSort  </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>