排序算法七：基数排序（Radix sort）

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

     @Override
     protected void sort(int[] toSort) {
         // number to sort, n integers
         int n = toSort.length;
         // b bits each integer
         int b = Integer.SIZE;
         /*
          * Split each integer into b/r digits, and each r bits long. So average
          * running time is O(b/r(2^r+n)). It is proved that running time is
          * close to least time while choosing r to lgn.
          */
         int r = (int) Math.ceil(Math.log(n) / Math.log(2));
         // considering the space cost, the maximum of r is 16.
         r = Math.min(r, 16);

         int upperLimit = 1 << r;
         int loopCount = b / r;
         int j = 0;
         int[] resultArray = new int[toSort.length];
         int[] countingArray = new int[upperLimit];
         while (j < loopCount) {
             int rightShift = j * r;
             radixSort(toSort, upperLimit, rightShift, resultArray,
                     countingArray);
             Arrays.fill(countingArray, 0);
             j++;
         }
         int mod = b % r;
         if (mod != 0) {
             upperLimit = 1 << mod;
             int rightShift = r * loopCount;
             countingArray = new int[upperLimit];
             radixSort(toSort, upperLimit, rightShift, resultArray,
                     countingArray);
         }
     }

     private void radixSort(int[] toSort, int upperLimit, int rightShift,
             int[] resultArray, int[] countingArray) {
         int allOnes = upperLimit - 1;
         for (int i = 0; i < toSort.length; i++) {
             int radix = (toSort[i] >> rightShift) & allOnes;
             countingArray[radix]++;
         }
         for (int i = 1; i < countingArray.length; i++) {
             countingArray[i] += countingArray[i - 1];
         }

         for (int i = toSort.length - 1; i >= 0; i--) {
             int radix = (toSort[i] >> rightShift) & allOnes;
             resultArray[countingArray[radix] - 1] = toSort[i];
             countingArray[radix]--;
         }
         System.arraycopy(resultArray, 0, toSort, 0, resultArray.length);
     }

radixSort

1）算法属于分配排序

2）平均时间复杂度是O(b/r(2^r+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(2^r+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：

 package com.cnblogs.riyueshiwang.sort;

 import java.util.Arrays;

 public class RadixSort extends abstractSort {

     @Override
     protected void sort(int[] toSort) {
         // number to sort, n integers
         int n = toSort.length;
         // b bits each integer
         int b = Integer.SIZE;
         /*
          * Split each integer into b/r digits, and each r bits long. So average
          * running time is O(b/r(2^r+n)). It is proved that running time is
          * close to least time while choosing r to lgn.
          */
         int r = (int) Math.ceil(Math.log(n) / Math.log(2));
         // considering the space cost, the maximum of r is 16.
         r = Math.min(r, 16);

         int upperLimit = 1 << r;
         int loopCount = b / r;
         int j = 0;
         int[] resultArray = new int[toSort.length];
         int[] countingArray = new int[upperLimit];
         while (j < loopCount) {
             int rightShift = j * r;
             radixSort(toSort, upperLimit, rightShift, resultArray,
                     countingArray);
             Arrays.fill(countingArray, 0);
             j++;
         }
         int mod = b % r;
         if (mod != 0) {
             upperLimit = 1 << mod;
             int rightShift = r * loopCount;
             countingArray = new int[upperLimit];
             radixSort(toSort, upperLimit, rightShift, resultArray,
                     countingArray);
         }
     }

     private void radixSort(int[] toSort, int upperLimit, int rightShift,
             int[] resultArray, int[] countingArray) {
         int allOnes = upperLimit - 1;
         for (int i = 0; i < toSort.length; i++) {
             int radix = (toSort[i] >> rightShift) & allOnes;
             countingArray[radix]++;
         }
         for (int i = 1; i < countingArray.length; i++) {
             countingArray[i] += countingArray[i - 1];
         }

         for (int i = toSort.length - 1; i >= 0; i--) {
             int radix = (toSort[i] >> rightShift) & allOnes;
             resultArray[countingArray[radix] - 1] = toSort[i];
             countingArray[radix]--;
         }
         System.arraycopy(resultArray, 0, toSort, 0, resultArray.length);
     }

     public static void main(String[] args) {
         for (int j = 0, n = 2500; j < 5; j++, n = n * 10) {
             System.out
                     .println("**************************************************");
             System.out.println("Number to Sort is:" + n);
             int upperLimit = Integer.MAX_VALUE;
             int[] array = CommonUtils.getRandomIntArray(n, upperLimit);
             System.out.print("Array to sort is:");
             CommonUtils.printIntArray(array);

             int[] array1 = Arrays.copyOf(array, n);
             new RadixSort().sortAndprint(array1);

             int[] array2 = Arrays.copyOf(array, n);
             new RandomizedQuickSort().sortAndprint(array2);
         }
     }
 }

RadixSort.java