上一篇提到了计数排序，它在输入序列元素的取值范围较小时，表现不俗。但是，现实生活中不总是满足这个条件，比如最大整形数据可以达到231-1，这样就存在2个问题：

1）因为m的值很大，不再满足m=O(n)，计数排序的时间复杂也就不再是线性的；

2）当m很大时，为计数数组申请的内存空间会很大；

为解决这两个问题，本篇讨论基数排序（Radix sort），基数排列的思想是：

1）将先按照某个基数将输入序列的每个元素划分成若干部分，每个部分对排序结果的影响是有优先级的；

2）先按低优先级排序，再按高优先级排序，依次递推。这里要注意，每个部分进行排序时，必须选用稳定排序算法，例如基数排序。

3）最后的次序就是高优先级高的在前，高优先级相同的，低优先级高的在前。

 （一）算法实现

 1     @Override
 2     protected void sort(int[] toSort) {
 3         // number to sort, n integers
 4         int n = toSort.length;
 5         // b bits each integer
 6         int b = Integer.SIZE;
 7         /*
 8          * Split each integer into b/r digits, and each r bits long. So average
 9          * running time is O(b/r(2^r+n)). It is proved that running time is
10          * close to least time while choosing r to lgn.
11          */
12         int r = (int) Math.ceil(Math.log(n) / Math.log(2));
13         // considering the space cost, the maximum of r is 16.
14         r = Math.min(r, 16);
15 
16         int upperLimit = 1 << r;
17         int loopCount = b / r;
18         int j = 0;
19         int[] resultArray = new int[toSort.length];
20         int[] countingArray = new int[upperLimit];
21         while (j < loopCount) {
22             int rightShift = j * r;
23             radixSort(toSort, upperLimit, rightShift, resultArray,
24                     countingArray);
25             Arrays.fill(countingArray, 0);
26             j++;
27         }
28         int mod = b % r;
29         if (mod != 0) {
30             upperLimit = 1 << mod;
31             int rightShift = r * loopCount;
32             countingArray = new int[upperLimit];
33             radixSort(toSort, upperLimit, rightShift, resultArray,
34                     countingArray);
35         }
36     }
37 
38     private void radixSort(int[] toSort, int upperLimit, int rightShift,
39             int[] resultArray, int[] countingArray) {
40         int allOnes = upperLimit - 1;
41         for (int i = 0; i < toSort.length; i++) {
42             int radix = (toSort[i] >> rightShift) & allOnes;
43             countingArray[radix]++;
44         }
45         for (int i = 1; i < countingArray.length; i++) {
46             countingArray[i] += countingArray[i - 1];
47         }
48 
49         for (int i = toSort.length - 1; i >= 0; i--) {
50             int radix = (toSort[i] >> rightShift) & allOnes;
51             resultArray[countingArray[radix] - 1] = toSort[i];
52             countingArray[radix]--;
53         }
54         System.arraycopy(resultArray, 0, toSort, 0, resultArray.length);
55     }
radixSort

1）算法属于分配排序

2）平均时间复杂度是O(b/r(2r+n)), b-每个元素的bit数，r-每个元素划分成b/r个数字，每个数字r个bit。当r=log2n时，复杂度是O(2bn/log2n)，也就是说，当b=O（log2n）时，时间复杂度是O(n).

3) 空间复杂度是O(2r+n)

4)算法属于稳定排序

（二）算法仿真

下面对随机化快速排序和基数排序，针对不同输入整数序列长度，仿真结果如下，从结果看，当输入序列长度越大，基数排序性能越优越。

**************************************************
Number to Sort is:2500
Array to sort is:{1642670374,460719485,1773719101,2140462092,1260791250,199719453,1290828881,1946941575,2032337910,643536338...}
Cost time of 【RadixSort】 is(milliseconds):48
Sort result of 【RadixSort】:{217942,491656,1389218,2642908,3608001,3976751,4905471,5094692,6340348,7693772...}
Cost time of 【RandomizedQuickSort】 is(milliseconds):1
Sort result of 【RandomizedQuickSort】:{217942,491656,1389218,2642908,3608001,3976751,4905471,5094692,6340348,7693772...}
**************************************************
Number to Sort is:25000
Array to sort is:{987947608,1181521142,1240568028,373349221,289183678,2051121943,1257313984,745646081,1414556623,1859315040...}
Cost time of 【RadixSort】 is(milliseconds):1
Sort result of 【RadixSort】:{47434,109303,240122,255093,448360,526046,526445,628228,837987,966240...}
Cost time of 【RandomizedQuickSort】 is(milliseconds):2
Sort result of 【RandomizedQuickSort】:{47434,109303,240122,255093,448360,526046,526445,628228,837987,966240...}
**************************************************
Number to Sort is:250000
Array to sort is:{1106960922,1965236858,1114033657,1196235697,2083563075,1994568819,1185250879,670222217,1386040268,1316674615...}
Cost time of 【RadixSort】 is(milliseconds):7
Sort result of 【RadixSort】:{466,884,8722,35382,37181,44708,53396,55770,67518,74898...}
Cost time of 【RandomizedQuickSort】 is(milliseconds):27
Sort result of 【RandomizedQuickSort】:{466,884,8722,35382,37181,44708,53396,55770,67518,74898...}
**************************************************
Number to Sort is:2500000
Array to sort is:{1903738012,485657780,1747057138,2082998554,1658643001,91586227,2127717572,557705232,533021562,1322007386...}
Cost time of 【RadixSort】 is(milliseconds):81
Sort result of 【RadixSort】:{369,392,1316,1378,2301,3819,4013,4459,5922,6423...}
Cost time of 【RandomizedQuickSort】 is(milliseconds):340
Sort result of 【RandomizedQuickSort】:{369,392,1316,1378,2301,3819,4013,4459,5922,6423...}
**************************************************
Number to Sort is:25000000
Array to sort is:{2145921976,298753549,11187940,410746614,503122524,1951513957,1760836125,2141838979,1702951573,1402856280...}
Cost time of 【RadixSort】 is(milliseconds):1,022
Sort result of 【RadixSort】:{130,145,406,601,683,688,736,865,869,954...}
Cost time of 【RandomizedQuickSort】 is(milliseconds):3,667
Sort result of 【RandomizedQuickSort】:{130,145,406,601,683,688,736,865,869,954...}

 相关源码：

 1 package com.cnblogs.riyueshiwang.sort;
 2 
 3 import java.util.Arrays;
 4 
 5 public class RadixSort extends abstractSort {
 6 
 7     @Override
 8     protected void sort(int[] toSort) {
 9         // number to sort, n integers
10         int n = toSort.length;
11         // b bits each integer
12         int b = Integer.SIZE;
13         /*
14          * Split each integer into b/r digits, and each r bits long. So average
15          * running time is O(b/r(2^r+n)). It is proved that running time is
16          * close to least time while choosing r to lgn.
17          */
18         int r = (int) Math.ceil(Math.log(n) / Math.log(2));
19         // considering the space cost, the maximum of r is 16.
20         r = Math.min(r, 16);
21 
22         int upperLimit = 1 << r;
23         int loopCount = b / r;
24         int j = 0;
25         int[] resultArray = new int[toSort.length];
26         int[] countingArray = new int[upperLimit];
27         while (j < loopCount) {
28             int rightShift = j * r;
29             radixSort(toSort, upperLimit, rightShift, resultArray,
30                     countingArray);
31             Arrays.fill(countingArray, 0);
32             j++;
33         }
34         int mod = b % r;
35         if (mod != 0) {
36             upperLimit = 1 << mod;
37             int rightShift = r * loopCount;
38             countingArray = new int[upperLimit];
39             radixSort(toSort, upperLimit, rightShift, resultArray,
40                     countingArray);
41         }
42     }
43 
44     private void radixSort(int[] toSort, int upperLimit, int rightShift,
45             int[] resultArray, int[] countingArray) {
46         int allOnes = upperLimit - 1;
47         for (int i = 0; i < toSort.length; i++) {
48             int radix = (toSort[i] >> rightShift) & allOnes;
49             countingArray[radix]++;
50         }
51         for (int i = 1; i < countingArray.length; i++) {
52             countingArray[i] += countingArray[i - 1];
53         }
54 
55         for (int i = toSort.length - 1; i >= 0; i--) {
56             int radix = (toSort[i] >> rightShift) & allOnes;
57             resultArray[countingArray[radix] - 1] = toSort[i];
58             countingArray[radix]--;
59         }
60         System.arraycopy(resultArray, 0, toSort, 0, resultArray.length);
61     }
62 
63     public static void main(String[] args) {
64         for (int j = 0, n = 2500; j < 5; j++, n = n * 10) {
65             System.out
66                     .println("**************************************************");
67             System.out.println("Number to Sort is:" + n);
68             int upperLimit = Integer.MAX_VALUE;
69             int[] array = CommonUtils.getRandomIntArray(n, upperLimit);
70             System.out.print("Array to sort is:");
71             CommonUtils.printIntArray(array);
72 
73             int[] array1 = Arrays.copyOf(array, n);
74             new RadixSort().sortAndprint(array1);
75 
76             int[] array2 = Arrays.copyOf(array, n);
77             new RandomizedQuickSort().sortAndprint(array2);
78         }
79     }
80 }
RadixSort.java