java实现几种常见的排序算法
2016-05-09 21:17
876 查看
几种常见排序算法
几种常见排序算法写在前面
基础介绍
初级排序算法
selection sort选择排序
insertion sort插入排序
ShellSort希尔排序
shuffing不是排序算法
merge sort归并排序
Abstract in-place merge原地归并的抽象方法
Top-down mergesort自顶向下的归并排序
Bottom-up mergesort自底向上的归并排序
quicksort
三向切分的快速排序
Heapsort堆排序
总结和比较
命题
本文介绍几种常见排序算法(选择排序,插入排序,希尔排序,归并排序,快速排序,堆排序),对算法的思路、性质、特点、具体步骤、java实现以及trace图解进行了全面的说明。最后对几种排序算法进行了比较和总结。
写在前面
本文所有图片均截图自coursera上由普林斯顿的课程《Algorithms, Part I》中的Slides相关命题的证明可参考《算法(第4版)》
源码可在官网下载,也可以在我的github仓库 algorithms-learning下载,已经使用maven构建
仓库下载:
git clone git@github.com:brianway/algorithms-learning.git
基础介绍
java: Interface Comparable<T>
Java中很多类已经实现了
Comparable<T>接口,用户也可自定义类型实现该接口
total order:
Antisymmetry(反对称性): if v ≤ w and w ≤ v, then v = w.
Transitivity(传递性): if v ≤ w and w ≤ x, then v ≤ x.
Totality: either v ≤ w or w ≤ v or both.
注意: The
<=operator for double is not a total order ,violates totality: (Double.NaN
<=Double.NaN)
is false
通用代码:
<code class="language-java hljs has-numbering" style="font-family: 'Source Code Pro', monospace; display: block; padding: 0px; color: inherit; box-sizing: border-box;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// Less. Is item v less than w ?</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">private</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">static</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">boolean</span> <span class="hljs-title" style="font-family: 宋体 !important; box-sizing: border-box;">less</span>(Comparable v, Comparable w) { <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">return</span> v.compareTo(w) < <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">0</span>; } <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">//Exchange. Swap item in array a[] at index i with the one at index j</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">private</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">static</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">void</span> <span class="hljs-title" style="font-family: 宋体 !important; box-sizing: border-box;">exch</span>(Comparable[] a,, <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i, <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> j) { Comparable swap = a[i]; a[i] = a[j]; a[j] = swap; } </code><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li></ul><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li></ul>
初级排序算法
selection sort(选择排序)
思路:在第i次迭代中,在剩下的(即未排序的)元素中找到最小的元素
将第i个元素与最小的元素交换位置
现象:
设已排序的和未排序的交界处为 ↑,则每次循环, ↑ 从左往右移动一个位置
↑ 左边的元素(包括↑)固定了,且升序
↑ 右边的任一元素全部比左边的所有元素都大
步骤:
move the pointer to the right
indentify index of minimun entry on right
exchange into positon
java实现:
<code class="language-java hljs has-numbering" style="font-family: 'Source Code Pro', monospace; display: block; padding: 0px; color: inherit; box-sizing: border-box;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">public</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">static</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">void</span> <span class="hljs-title" style="font-family: 宋体 !important; box-sizing: border-box;">sort</span>(Comparable[] a) { <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> N = a.length; <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">for</span> (<span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i = <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">0</span>; i < N; i++) { <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> min = i; <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">for</span> (<span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> j = i+<span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>; j < N; j++) { <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">if</span> (less(a[j], a[min])) min = j; } exch(a, i, min); } }</code><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li></ul><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li></ul>
特点:
运行时间和输入无关,无论输入是已排序,时间复杂度都是O(n^2)
数据移动最少,交换的次数和数组大小是线性关系
insertion sort(插入排序)
思路:在第i次迭代中,将第i个元素与每一个它左边且比它大的的元素交换位置
现象:
设已排序的和未排序的交界处为 ↑,则每次循环, ↑ 从左往右移动一个位置
↑ 左边的元素(包括↑)且升序,但位置不固定(因为后续可能会因插入而移动)
↑ 右边的元素还不可见
步骤:
Move the pointer to the right.
Moving from right to left, exchange
a[i]with each larger entry to its left.
java实现:
<code class="language-java hljs has-numbering" style="font-family: 'Source Code Pro', monospace; display: block; padding: 0px; color: inherit; box-sizing: border-box;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">public</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">static</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">void</span> <span class="hljs-title" style="font-family: 宋体 !important; box-sizing: border-box;">sort</span>(Comparable[] a) { <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> N = a.length; <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">for</span> (<span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i = <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">0</span>; i < N; i++) { <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">for</span> (<span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> j = i; j > <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">0</span> && less(a[j], a[j-<span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>]); j--) { exch(a, j, j-<span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>); } } }</code><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li></ul><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li></ul>
inversion(倒置):An inversion is a pair of keys that are out of order
部分有序:An array is partially sorted if the number of inversions is ≤ c N.
特点:
运行时间和输入有关,当输入已排序时,时间复杂度是O(n);
For partially-sorted arrays, insertion sort runs in linear time.(交换的次数等于输入中倒置(inversion)的个数)
插入排序适合部分有序数组,也适合小规模数组
ShellSort(希尔排序)
希尔排序是基于插入排序的。思路:
Move entries more than one position at a time by h-sorting the array
按照h的步长进行插入排序
现象:
数组中任意间隔为h的元素都是有序的
A g-sorted array remains g-sorted after h-sorting it.
性质:
递增数列一般采用3x+1:1,4,13,40,121,364…..,使用这种递增数列的希尔排序所需的比较次数不会超过N的若干倍乘以递增数列的长度。
最坏情况下,使用3x+1递增数列的希尔排序的比较次数是O(N^(3/2))
java实现:
<code class="language-java hljs has-numbering" style="font-family: 'Source Code Pro', monospace; display: block; padding: 0px; color: inherit; box-sizing: border-box;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">public</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">static</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">void</span> <span class="hljs-title" style="font-family: 宋体 !important; box-sizing: border-box;">sort</span>(Comparable[] a) { <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> N = a.length; <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// 3x+1 increment sequence: 1, 4, 13, 40, 121, 364, 1093, ... </span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> h = <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>; <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">while</span> (h < N/<span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">3</span>) h = <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">3</span>*h + <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>; <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">while</span> (h >= <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>) { <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// h-sort the array</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">for</span> (<span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i = h; i < N; i++) { <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">for</span> (<span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> j = i; j >= h && less(a[j], a[j-h]); j -= h) { exch(a, j, j-h); } } h /= <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">3</span>; } }</code><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li></ul><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li></ul>
shuffing(不是排序算法)
目标:Rearrange array so that result is a uniformly random permutationshuffle sort思路
为数组的每一个位置生成一个随机实数
排序这个生成的数组
Knuth shuffle demo
In iteration i, pick integer r between 0 and i uniformly at random.
Swap
a[i]and
a[r].
correct variant: between i and N – 1
Mergesort–Java sort for objects.
Quicksort–Java sort for primitive types.
下面看看这两种排序算法
merge sort(归并排序)
思路:Divide array into two halves.
Recursively sort each half.
Merge two halves.
Abstract in-place merge(原地归并的抽象方法)
Given two sorted subarrays a[lo] to a[mid] and a[mid+1] to a[hi],replace with sorted subarray a[lo] to a[hi]步骤:
先将所有元素复制到
aux[]中,再归并回
a[]中。
归并时的四个判断:
左半边用尽(取右半边元素)
右半边用尽(取左半边元素)
右半边的当前元素小于左半边的当前元素(取右半边的元素)
右半边的当前元素大于/等于左半边的当前元素(取左半边的元素)
merging java实现:
<code class="language-java hljs has-numbering" style="font-family: 'Source Code Pro', monospace; display: block; padding: 0px; color: inherit; box-sizing: border-box;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background: transparent;"> <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// stably merge a[lo .. mid] with a[mid+1 ..hi] using aux[lo .. hi]</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">private</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">static</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">void</span> <span class="hljs-title" style="font-family: 宋体 !important; box-sizing: border-box;">merge</span>(Comparable[] a, Comparable[] aux, <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> lo, <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> mid, <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> hi) { <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// precondition: a[lo .. mid] and a[mid+1 .. hi] are sorted subarrays</span> <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// copy to aux[]</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">for</span> (<span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> k = lo; k <= hi; k++) { aux[k] = a[k]; } <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// merge back to a[]</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i = lo, j = mid+<span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>; <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">for</span> (<span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> k = lo; k <= hi; k++) { <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">if</span> (i > mid) a[k] = aux[j++]; <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">else</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">if</span> (j > hi) a[k] = aux[i++]; <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">else</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">if</span> (less(aux[j], aux[i])) a[k] = aux[j++]; <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">else</span> a[k] = aux[i++]; } }</code><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li><li style="box-sizing: border-box; padding: 0px 5px;">18</li></ul><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li><li style="box-sizing: border-box; padding: 0px 5px;">18</li></ul>
Top-down mergesort(自顶向下的归并排序)
mergesort java实现:<code class="language-java hljs has-numbering" style="font-family: 'Source Code Pro', monospace; display: block; padding: 0px; color: inherit; box-sizing: border-box;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// mergesort a[lo..hi] using auxiliary array aux[lo..hi]</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">private</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">static</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">void</span> <span class="hljs-title" style="font-family: 宋体 !important; box-sizing: border-box;">sort</span>(Comparable[] a, Comparable[] aux, <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> lo, <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> hi) { <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">if</span> (hi <= lo) <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">return</span>; <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> mid = lo + (hi - lo) / <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">2</span>; sort(a, aux, lo, mid); <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">//将左边排序</span> sort(a, aux, mid + <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>, hi); <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">//将右边排序</span> merge(a, aux, lo, mid, hi); <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">//归并结果</span> }</code><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li></ul><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li></ul>
自顶向下的归并排序的轨迹图
由图可知,原地归并排序的大致趋势是,先局部排序,再扩大规模;先左边排序,再右边排序;每次都是左边一半局部排完且merge了,右边一半才开始从最局部的地方开始排序。
改进
对小规模子数组使用插入排序
测试数组是否已经有序(左边最大<右边最小时,直接返回)
不将元素复制到辅助数组(节省时间而非空间)
Bottom-up mergesort(自底向上的归并排序)
思路:先归并微型数组,从两两归并开始(每个元素理解为大小为1的数组)
重复上述步骤,逐步扩大归并的规模,2,4,8…..
java实现:
<code class="language-java hljs has-numbering" style="font-family: 'Source Code Pro', monospace; display: block; padding: 0px; color: inherit; box-sizing: border-box;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">public</span> <span class="hljs-class" style="font-family: 宋体 !important; box-sizing: border-box;"><span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">class</span> <span class="hljs-title" style="box-sizing: border-box; color: rgb(102, 0, 102);">MergeBU</span>{</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">private</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">static</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">void</span> <span class="hljs-title" style="font-family: 宋体 !important; box-sizing: border-box;">merge</span>(...){ <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">/* as before */</span> } <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">public</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">static</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">void</span> <span class="hljs-title" style="font-family: 宋体 !important; box-sizing: border-box;">sort</span>(Comparable[] a){ <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> N = a.length; Comparable[] aux = <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">new</span> Comparable ; <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">for</span> (<span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> sz = <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>; sz < N; sz = sz+sz) <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">for</span> (<span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> lo = <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">0</span>; lo < N-sz; lo += sz+sz) merge(a, aux, lo, lo+sz-<span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>, Math.min(lo+sz+sz-<span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>, N-<span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>)); } }</code><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li></ul><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li></ul>
自底向上的归并排序的轨迹图
由图可知,自底向上归并排序的大致趋势是,先局部排序,逐步扩大到全局排序;步调均匀,稳步扩大
quicksort
思路:Shuffle the array.
Partition(切分) so that, for some j
entry a[j] is in place
no larger entry to the left of j
no smaller entry to the right of j
Sort each piece recursively.
其中很重要的一步就是Partition(切分),这个过程使得满足以下三个条件:
对于某个j,a[j]已经排定;
a[lo]到a[j-1]中的所有元素都不大于a[j];
a[j+1]到a[hi]中的所有元素都不小于a[j];
partition java实现
<code class="language-java hljs has-numbering" style="font-family: 'Source Code Pro', monospace; display: block; padding: 0px; color: inherit; box-sizing: border-box;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// partition the subarray a[lo..hi] so that a[lo..j-1] <= a[j] <= a[j+1..hi]</span> <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// and return the index j.</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">private</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">static</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> <span class="hljs-title" style="font-family: 宋体 !important; box-sizing: border-box;">partition</span>(Comparable[] a, <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> lo, <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> hi) { <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i = lo; <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> j = hi + <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>; Comparable v = a[lo]; <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">while</span> (<span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">true</span>) { <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// find item on lo to swap</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">while</span> (less(a[++i], v)) <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">if</span> (i == hi) <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">break</span>; <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// find item on hi to swap</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">while</span> (less(v, a[--j])) <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">if</span> (j == lo) <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">break</span>; <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// redundant since a[lo] acts as sentinel</span> <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// check if pointers cross</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">if</span> (i >= j) <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">break</span>; exch(a, i, j); } <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// put partitioning item v at a[j]</span> exch(a, lo, j); <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// now, a[lo .. j-1] <= a[j] <= a[j+1 .. hi]</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">return</span> j; }</code><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li><li style="box-sizing: border-box; padding: 0px 5px;">18</li><li style="box-sizing: border-box; padding: 0px 5px;">19</li><li style="box-sizing: border-box; padding: 0px 5px;">20</li><li style="box-sizing: border-box; padding: 0px 5px;">21</li><li style="box-sizing: border-box; padding: 0px 5px;">22</li><li style="box-sizing: border-box; padding: 0px 5px;">23</li><li style="box-sizing: border-box; padding: 0px 5px;">24</li><li style="box-sizing: border-box; padding: 0px 5px;">25</li><li style="box-sizing: border-box; padding: 0px 5px;">26</li><li style="box-sizing: border-box; padding: 0px 5px;">27</li><li style="box-sizing: border-box; padding: 0px 5px;">28</li></ul><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li><li style="box-sizing: border-box; padding: 0px 5px;">18</li><li style="box-sizing: border-box; padding: 0px 5px;">19</li><li style="box-sizing: border-box; padding: 0px 5px;">20</li><li style="box-sizing: border-box; padding: 0px 5px;">21</li><li style="box-sizing: border-box; padding: 0px 5px;">22</li><li style="box-sizing: border-box; padding: 0px 5px;">23</li><li style="box-sizing: border-box; padding: 0px 5px;">24</li><li style="box-sizing: border-box; padding: 0px 5px;">25</li><li style="box-sizing: border-box; padding: 0px 5px;">26</li><li style="box-sizing: border-box; padding: 0px 5px;">27</li><li style="box-sizing: border-box; padding: 0px 5px;">28</li></ul>
快排java实现:
<code class="language-java hljs has-numbering" style="font-family: 'Source Code Pro', monospace; display: block; padding: 0px; color: inherit; box-sizing: border-box;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">public</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">static</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">void</span> <span class="hljs-title" style="font-family: 宋体 !important; box-sizing: border-box;">sort</span>(Comparable[] a) { StdRandom.shuffle(a); sort(a, <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">0</span>, a.length - <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>); } <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// quicksort the subarray from a[lo] to a[hi]</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">private</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">static</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">void</span> <span class="hljs-title" style="font-family: 宋体 !important; box-sizing: border-box;">sort</span>(Comparable[] a, <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> lo, <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> hi) { <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">if</span> (hi <= lo) <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">return</span>; <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> j = partition(a, lo, hi); sort(a, lo, j-<span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>); sort(a, j+<span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>, hi); <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">assert</span> isSorted(a, lo, hi); }</code><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li></ul><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li></ul>
快排的轨迹图
由图可知,和归并排序不同,快排的大致趋势是,先全局大体有个走势——左边比右边小,逐步细化到局部;也是先左后右;局部完成时全部排序也就完成了。
一些实现的细节:
原地切分:不使用辅助数组
别越界:测试条件(j == lo)是冗余的(a[lo]不可能比自己小);
保持随机性:初始时的随机打乱跟重要
终止循环
处理切分元素值有重复的情况:这里可能出问题
性质:
快排是in-place的
快排不稳定
改进
对小规模子数组使用插入排序
三取样切分
三向切分的快速排序
思路:Let v be partitioning item a[lo].
Scan i from left to right.
(a[i] < v): exchange a[lt] with a[i]; increment both lt and i
(a[i] > v): exchange a[gt] with a[i]; decrement gt
(a[i] == v): increment i
主要是通过增加一个指针来实现的。普通的快拍只有lo和high两个指针,故只能记录
大于(high右边)和
小于(lo左边)两个区间,
等于只能并入其中一个;这里增加了使用了lt,i,gt三个指针,从而达到记录
大于(gt右边)、
小于(lt左边)和
等于(lt和i之间)三个区间。
三切分的示意图
三向切分的java实现:
<code class="language-java hljs has-numbering" style="font-family: 'Source Code Pro', monospace; display: block; padding: 0px; color: inherit; box-sizing: border-box;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// quicksort the subarray a[lo .. hi] using 3-way partitioning</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">private</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">static</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">void</span> <span class="hljs-title" style="font-family: 宋体 !important; box-sizing: border-box;">sort</span>(Comparable[] a, <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> lo, <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> hi) { <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">if</span> (hi <= lo) <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">return</span>; <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> lt = lo, gt = hi; Comparable v = a[lo]; <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i = lo; <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">while</span> (i <= gt) { <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> cmp = a[i].compareTo(v); <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">if</span> (cmp < <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">0</span>) exch(a, lt++, i++); <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">else</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">if</span> (cmp > <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">0</span>) exch(a, i, gt--); <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">else</span> i++; } <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">// a[lo..lt-1] < v = a[lt..gt] < a[gt+1..hi]. </span> sort(a, lo, lt-<span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>); sort(a, gt+<span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>, hi); }</code><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li></ul><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li></ul>
Heapsort(堆排序)
思路:Create max-heap with all N keys.
Repeatedly remove the maximum key.
swim:由下至上的堆有序化
sink:由上至下的对有序化
堆排序主要分为两个阶段:
堆的构造
下沉排序
java实现如下:
<code class="language-java hljs has-numbering" style="font-family: 'Source Code Pro', monospace; display: block; padding: 0px; color: inherit; box-sizing: border-box;font-size:undefined; white-space: pre; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">public</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">static</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">void</span> <span class="hljs-title" style="font-family: 宋体 !important; box-sizing: border-box;">sort</span>(Comparable[] pq) { <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> N = pq.length; <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">//堆的构造</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">for</span> (<span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">int</span> k = N/<span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">2</span>; k >= <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>; k--) sink(pq, k, N); <span class="hljs-comment" style="font-family: 宋体 !important; color: rgb(136, 0, 0); box-sizing: border-box;">//下沉排序</span> <span class="hljs-keyword" style="font-family: 宋体 !important; color: rgb(0, 0, 136); box-sizing: border-box;">while</span> (N > <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>) { exch(pq, <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>, N--); sink(pq, <span class="hljs-number" style="font-family: 宋体 !important; color: rgb(0, 102, 102); box-sizing: border-box;">1</span>, N); } }</code><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li></ul><ul class="pre-numbering" style="font-family: 宋体 !important; box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li></ul>
堆排序的轨迹图
由图看出,堆排序的趋势是,堆构造阶段,大致是降序的走势,到了下沉阶段,从右到左(或者说从后往前)逐步有序
Significance: In-place sorting algorithm with N log N worst-case.
Mergesort: no, linear extra space.
Quicksort: no, quadratic time in worst case
缺点
Inner loop longer than quicksort’s.
Makes poor use of cache memory.
Not stable(不稳定)
总结和比较
排序算法总结表最好情况和最坏情况:参见上面的表格
关于稳定性:
稳定性,插入排序,归并排序
不稳定:选择排序,快排,希尔排序,堆排序
原因: Long-distance exchange
关于额外空间:除了归并排序需要线性的额外空间,其他都是in-place的
命题
对于长度为N的数组,选择排序需要N^2/2次比较和N次交换(pf见P156)对于随机排列的长度为N的且主键不重复的数组(pf见P157)
平均情况下插入排序需要~N^2/4次比较和~N^2/4次交换
最坏情况下需要~N^2/2次比较和~N^2/2次交换,
最好情况下需要N-1次比较和0次交换。
Mergesort uses at most N lg N compares and 6 N lg N array accesses to sort any array of size N. (pf见P173)
Mergesort uses extra space proportional to N.(The array
aux[]needs to be of size N for the last merge.)
Any compare-based sorting algorithm must use at least lg ( N ! ) ~ N lg N compares in the worst-case.(pf见P177)
长度为N的无重复数组排序,快速排序平均需要~2N ln N 次比较(以及1/6即1/3 N ln N的交换)
最多需要约N^2/2次比较
最少需要~N lg N 次比较
用下沉操作由N个元素构造堆只需少于2N次比较以及少于N次交换(pf见P206)
将N个元素排序,堆排序只需少于(2NlgN+2N)次比较以及一半次数的交换(pf见P208)
相关文章推荐
- spring的annotation-driven
- 将eclipse项目导入Android studio
- Java JDBC基本操作(增,删,改,查)总结
- spring4 4000 +springmvc4+hibernate4集成框架流程:
- JAVA事务处理 之 JDBC事务
- JAVA之路002-运算符
- Maven学习5之eclipse下 maven package运行和scope理解
- 重新学javaweb---多文件上传及文件名/位置 重编
- 第一章 JAVA入门(导出参数之DumpArgs类)
- Java核心技术(五) —— 泛型程序设计(2)
- 第一章 JAVA入门(Main函数简介续)
- JAVA框架——Spring Mvc与struts2区别
- java selector
- javaweb分布式事务
- VLC for android Windows平台上基于 Eclipse的开发
- Java基础-第4天
- Spring 拦截器的使用
- SpringIOC容器-对象依赖
- eclipse中怎么生成API帮助文档
- struts中文转换