常见排序算法导读(10)[基数排序]

与前面介绍的7种排序算法不同,基数排序(Radix Sort)是基于多关键字排序的一种排序算法。也就是说,前面介绍的7种排序算法是建立在对单关键字进行比较的基础之上,而基数排序则是采用”分配”与”收集”的办法,将单关键字按照一定策略拆分成多个关键字,进而对多个关键字进行排序,最终完成对单关键字的排序。

基数排序的典型例子当然就是扑克牌排序啦,几乎所有的数据结构教科书都会讲到,原因是形象易懂。每张扑克牌都有两个关键字:花色和面值。假定有序关系为:

  • 花色: 黑桃 < 红桃 < 梅花 < 方块
  • 面值: 2 < 3 < 4 < 5 < 6 < 7 < 8 < 9 < 10 < J < Q < K < A

如果把所有扑克牌排成下列次序:

  • 黑桃2, …, 黑桃A, 红桃2, …, 红桃A, 梅花2, …, 梅花A, 方块2, …, 方块A。

那么这就是多关键字排序。排序后形成的有序序列叫做词典有序序列。 对于扑克牌排序,有两种方法。可以先按花色排序,之后再按面值排序;也可以先按面值排序,再按花色排序。

再举个对数字进行基数排序的例子,

《常见排序算法导读(10)[基数排序]》

利用visualgo.net创建一个动画演示(01)如下:

《常见排序算法导读(10)[基数排序]》

注意: 后面给出的第一个基数排序C代码实现(radixsort.c)并未完全基于上面的动画演示,而是基于David Galles在RadixSort上给出的动画演示(个人觉得David给出的组织bin[]的方法更酷一些,但是可读性稍差一些)。 补充说明中给出的代码实现radixsort2.c则是完全基于上面的动画演示。

基数排序的基本思想

设待排记录中的多个关键字为K1, K2, …, Kr。K1第1关键字,Kr为第r关键字。由记录R1, R2, …, Rn组成的表关于关键字K1, K2, …, Kr有序,当且仅当对每一对记录Ri < Rj 都有(K1i, K2i, …, Kri) <= (K1j, K2j, …, Krj)。 排序方法如下:

  1. 首先按照关键字K1对R1, R2, …, Rn进行排序,得到新的序列LIST1;
  2. 按照关键字K2对LIST1进行排序,得到新的序列LIST2; 如此循环往复直到完成对关键字Kr的排序为止。

为简单起见,我们只讨论非负数的整数序列(即a[i]>=0, i=0, 1, …, N-1)的基数排序方法。(前面提到的扑克牌花色,完全可以按照黑红梅方的顺序编码为0, 1, 2, 3。于是,黑桃序列为: 002 < … < 010 < 011(J) < 012(Q) < 013(K) < 014(A), 其他三种花色依次类推即可)

基数排序的分类

  1. 最高位优先MSD(Most Significant Digit first)
  2. 最低位优先LSD(Least Significant Digit fisrt)

《常见排序算法导读(10)[基数排序]》

最高位优先(MSD)

  • 假设待排序列的最大整数的最高位为百位,先按照位排,再按照位排,最后按照位排。

最低位优先(LSD)

  • 假设待排序列的最大整数的最高位为百位,先按照位排,再按照位排,最后按照位排。

基于LSD实现的基数排序相对简单,本文将给出这种基数排序的代码实现。(P.S. 在观看了Radix Sort Visualization之后,我大约花了15分钟就完成了代码实现,比啃数据结构的书容易多了。)

1. 获取整数的某个特定位的数字

 1 /*
 2  * Get the digit of number by index(= 0, 1, 2, ...)
 3  *
 4  * e.g.
 5  *     num = 6543210
 6  *     ------+------
 7  *     index | digit
 8  *     ------+------
 9  *         0 | 0
10  *         1 | 1
11  *         2 | 2
12  *         3 | 3
13  *         4 | 4
14  *         5 | 5
15  *         6 | 6
16  *         7 | 0
17  *        .. | ..
18  *     ------+------
19  */
20 static unsigned char
21 get_digit_byindex(int num, unsigned int index)
22 {
23         int q = 0; /* quotient */
24         int r = 0; /* remainder */
25         for (int i = index; i >= 0; i--) {
26                 r = num % 10;
27                 q = num / 10;
28                 num = q;
29         }
30         return (unsigned char)r;
31 }

2. 获取整数的宽度

 1 /*
 2  * Get width of a number
 3  * e.g.
 4  *   for i in [  0 .. 9  ] // width = 1
 5  *   for i in [ 10 .. 99 ] // width = 2
 6  *   for i in [100 .. 999] // width = 3
 7  *   ...
 8  */
 9 static int
10 get_width_of_num(int num)
11 {
12         int w = 1;
13         for (int q = num / 10; q != 0; q /= 10)
14                 w++;
15         return w;
16 }

3. 构造10个容器bin[]bin[i]存数字i的个数,i = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

 1 /*
 2  * Build bin[] by index(=0, 1, ..., N-1) (N is the max width of num in a[])
 3  *
 4  * NOTE: This is the core function to understand radix sorting based on LSD
 5  */
 6 static void
 7 build_bin_byindex(int bin[], size_t nb, int a[], size_t na, int index)
 8 {
 9         /* 1. always reset bin[] */
10         for (int i = 0; i < nb; i++)
11                 bin[i] = 0;
12 
13         /* 2. init bin[] by walking a[] */
14         for (int i = 0; i < na; i++) {
15                 unsigned char d = get_digit_byindex(a[i], index);
16                 bin[d]++;
17         }
18 
19         /* 3. build bin[] */
20         for (int i = 1; i < nb; i++)
21                 bin[i] += bin[i-1];
22 }

4. 基于LSD的基数排序

 1 void
 2 radixsort(int a[], int n)
 3 {
 4         /* get the max width of num in a[] */
 5         int max = a[0];
 6         for (int i = 0; i < n; i++) {
 7                 if (a[i] > max)
 8                         max = a[i];
 9         }
10         int maxwidth = get_width_of_num(max);
11 
12         /* alloc bin[] to store the number of per digit */
13         int bin[10] = { 0 };
14 
15         /* alloc aux[] to save a[] while rebuilding a[] */
16         int *aux = (int *)malloc(sizeof(int) * n);
17         if (aux == NULL) /* error */
18                 return;
19 
20         /* LSD (Least Significant Digit first) */
21         for (int index = 0; index < maxwidth; index++) {
22                 /* 1. build bin[] */
23                 build_bin_byindex(bin, 10, a, n, index);
24 
25                 /* 2. copy a[] to aux[] */
26                 for (int i = 0; i < n; i++)
27                         aux[i] = a[i];
28 
29                 /* 3. rebuild a[] */
30                 for (int i = n - 1; i >= 0; i--) {
31                         unsigned char d = get_digit_byindex(aux[i], index);
32                         a[--bin[d]] = aux[i];
33                 }
34         }
35 
36         free(aux);
37 }

5. 完整的C代码实现

o radixsort.c

  1 #include <stdio.h>
  2 #include <stdlib.h>
  3 #include <string.h>
  4 
  5 typedef enum bool_s {false, true} bool_t;
  6 
  7 bool_t g_isint = true;
  8 
  9 static void
 10 show(int a[], size_t n)
 11 {
 12         if (g_isint) {
 13                 for (int i = 0; i < n; i++)
 14                         printf("%-2d ", a[i]);
 15         } else {
 16                 for (int i = 0; i < n; i++)
 17                         printf("%-2c ", a[i]);
 18         }
 19         printf("\n");
 20 }
 21 
 22 /*
 23  * Get the digit of number by index(= 0, 1, 2, ...)
 24  *
 25  * e.g.
 26  *     num = 6543210
 27  *     ------+------
 28  *     index | digit
 29  *     ------+------
 30  *         0 | 0
 31  *         1 | 1
 32  *         2 | 2
 33  *         3 | 3
 34  *         4 | 4
 35  *         5 | 5
 36  *         6 | 6
 37  *         7 | 0
 38  *        .. | ..
 39  *     ------+------
 40  */
 41 static unsigned char
 42 get_digit_byindex(int num, unsigned int index)
 43 {
 44         int q = 0; /* quotient */
 45         int r = 0; /* remainder */
 46         for (int i = index; i >= 0; i--) {
 47                 r = num % 10;
 48                 q = num / 10;
 49                 num = q;
 50         }
 51         return (unsigned char)r;
 52 }
 53 
 54 /*
 55  * Get width of a number
 56  * e.g.
 57  *   for i in [  0 .. 9  ] // width = 1
 58  *   for i in [ 10 .. 99 ] // width = 2
 59  *   for i in [100 .. 999] // width = 3
 60  *   ...
 61  */
 62 static int
 63 get_width_of_num(int num)
 64 {
 65         int w = 1;
 66         for (int q = num / 10; q != 0; q /= 10)
 67                 w++;
 68         return w;
 69 }
 70 
 71 /*
 72  * Build bin[] by index(=0, 1, ..., N-1) (N is the max width of num in a[])
 73  *
 74  * NOTE: This is the core function to understand radix sorting based on LSD
 75  */
 76 static void
 77 build_bin_byindex(int bin[], size_t nb, int a[], size_t na, int index)
 78 {
 79         /* 1. always reset bin[] */
 80         for (int i = 0; i < nb; i++)
 81                 bin[i] = 0;
 82 
 83         /* 2. init bin[] by walking a[] */
 84         for (int i = 0; i < na; i++) {
 85                 unsigned char d = get_digit_byindex(a[i], index);
 86                 bin[d]++;
 87         }
 88 
 89         /* NOTE: dump bin[] just for visual observation */
 90         printf("1#bin[]:\t"); show(bin, nb);
 91 
 92         /* 3. build bin[] */
 93         for (int i = 1; i < nb; i++)
 94                 bin[i] += bin[i-1];
 95 
 96         /* NOTE: dump bin[] just for visual observation */
 97         printf("2#bin[]:\t"); show(bin, nb);
 98 }
 99 
100 void
101 radixsort(int a[], int n)
102 {
103         /* get the max width of num in a[] */
104         int max = a[0];
105         for (int i = 0; i < n; i++) {
106                 if (a[i] > max)
107                         max = a[i];
108         }
109         int maxwidth = get_width_of_num(max);
110 
111         /* alloc bin[] to store the number of per digit */
112         int bin[10] = { 0 };
113 
114         /* alloc aux[] to save a[] while rebuilding a[] */
115         int *aux = (int *)malloc(sizeof(int) * n);
116         if (aux == NULL) /* error */
117                 return;
118 
119         /* LSD (Least Significant Digit first) */
120         for (int index = 0; index < maxwidth; index++) {
121                 /* 1. build bin[] */
122                 build_bin_byindex(bin, 10, a, n, index);
123 
124                 /* 2. copy a[] to aux[] */
125                 for (int i = 0; i < n; i++)
126                         aux[i] = a[i];
127 
128                 /* 3. rebuild a[] */
129                 for (int i = n - 1; i >= 0; i--) {
130                         unsigned char d = get_digit_byindex(aux[i], index);
131                         a[--bin[d]] = aux[i];
132                 }
133 
134                 /* NOTE: dump a[] for visual observation */
135                 printf("%dth bit(done): \t", index); show(a, n);
136         }
137 
138         free(aux);
139 }
140 
141 int
142 main(int argc, char *argv[])
143 {
144         if (argc < 2) {
145                 fprintf(stderr, "Usage: %s <C1> [C2] ...\n", argv[0]);
146                 return -1;
147         }
148 
149         argc--;
150         argv++;
151 
152         int n = argc;
153         int *a = (int *)malloc(sizeof(int) * n);
154 #define VALIDATE(p) do { if (p == NULL) return -1; } while (0)
155         VALIDATE(a);
156 
157         char *s = getenv("ISINT");
158         if (s != NULL && strncmp(s, "true", 4) == 0)
159                 g_isint = true;
160         else if (s != NULL && strncmp(s, "false", 4) == 0)
161                 g_isint = false;
162 
163         if (g_isint) {
164                 for (int i = 0; i < n; i++)
165                         *(a+i) = atoi(argv[i]);
166         } else {
167                 for (int i = 0; i < n; i++)
168                         *(a+i) = argv[i][0];
169         }
170 
171         printf("                ");
172         for (int i = 0; i < n; i++)
173                 printf("%-2x ", i);
174         printf("\n");
175 
176         printf("Before sorting: "); show(a, n);
177         radixsort(a, n);
178         printf("After  sorting: "); show(a, n);
179 
180 #define FREE(p) do { free(p); p = NULL; } while (0)
181         FREE(a);
182         return 0;
183 }

o 编译并测试

$ gcc -g -Wall -std=gnu99 -m32 -o radixsort radixsort.c

$ ./radixsort   32 51 31 52 81
                0  1  2  3  4
Before sorting: 32 51 31 52 81
1#bin[]:        0  3  2  0  0  0  0  0  0  0
2#bin[]:        0  3  5  5  5  5  5  5  5  5
0th bit(done):  51 31 81 32 52
1#bin[]:        0  0  0  2  0  2  0  0  1  0
2#bin[]:        0  0  0  2  2  4  4  4  5  5
1th bit(done):  31 32 51 52 81
After  sorting: 31 32 51 52 81

基数排序(Radix Sort)是一种稳定的排序算法。设待排对象的个数为N,关键字个数为w,则时间复杂度和空间复杂度为:

Worst-case performance O(w*N) Worst-case space complexity O(w+N)

在上面给出的C代码实现中:

  • w=10, 即有10个容器(bin),辅助存储aux[]的长度为N,所以空间复杂度为O(10+N);
  • 对每个数字(0..9)都要处理N次,所以最坏时间复杂度为O(10*N)。

参考资料:

  1. Radix Sort Visualization
  2. Radix sort
  3. https://visualgo.net/sorting

补充说明: 鉴于在本文开始的动画演示中,实现bin[]和重构a[]的过程可读性好,于是本人在阅读了Robert Sedgewick的算法第3版第10章 基数排序之后,给出如下C代码实现。该实现是基于按最低位优先(LSD)的方法,针对大小为32位的非负整数,遍历处理每一个二进制位。为简单起见,我在将a[N]的元素放置到bin[i](i=0,1)中去的时候,没有使用链式存储,而是一次性malloc大小为2*N的顺序存储空间aux[],将aux[0:N]分配给bin[0]使用,将aux[N:]分配给bin[1]使用(代码行: L108-121)。 (PS: 遍历处理每一个二进制位的思路是很酷的,读大师的书,总是会有新发现,大师就是大师,always膜拜ing…)

o radixsort2.c

  1 #include <stdio.h>
  2 #include <stdlib.h>
  3 #include <string.h>
  4 
  5 typedef enum bool_s {false, true} bool_t;
  6 
  7 bool_t g_isint = true;
  8 
  9 typedef struct bin_unit_s {
 10         size_t len;
 11         int *unit;
 12 } bin_unit_t;
 13 
 14 static void
 15 show(int a[], size_t n)
 16 {
 17         if (g_isint) {
 18                 for (int i = 0; i < n; i++)
 19                         printf("%-2d ", a[i]);
 20         } else {
 21                 for (int i = 0; i < n; i++)
 22                         printf("%-2c ", a[i]);
 23         }
 24         printf("\n");
 25 }
 26 
 27 static bool_t
 28 is_sorted(int a[], size_t n)
 29 {
 30         for (int i = 1; i < n; i++)
 31                 if (a[i] < a[i-1])
 32                         return false;
 33         return true;
 34 }
 35 
 36 /*
 37  * Get the digit of number by index(= 0, 1, 2, ...)
 38  */
 39 static unsigned char
 40 get_digit_byindex(int num, unsigned int index)
 41 {
 42         return ((unsigned int)num >> index) & 0x1;
 43 }
 44 
 45 /*
 46  * Build bin[] by index(=0, 1, ..., N-1) (N is the max width of num in a[])
 47  *
 48  * NOTE: This is the core function to understand radix sorting based on LSD
 49  */
 50 static void
 51 build_bin_byindex(bin_unit_t *bin[], size_t nb, int a[], size_t na, int index)
 52 {
 53         /* 1. always reset bin[][] */
 54         for (int i = 0; i < nb; i++) {
 55                 bin[i]->len = 0;
 56                 for (int j = 0; j < na; j++)
 57                         (bin[i]->unit)[j] = 0x0;
 58         }
 59 
 60         /* 2. init bin[][] by walking a[] */
 61         for (int i = 0; i < na; i++) {
 62                 unsigned char d = get_digit_byindex(a[i], index);
 63                 (bin[d]->unit)[bin[d]->len] = a[i];
 64                 (bin[d]->len)++;
 65 
 66                 /* NOTE: print a[i] and Xth bit just for visual observation */
 67                 printf("%dth bit of\t%d\t\tis %d\n", index, a[i], d);
 68         }
 69 
 70         /* NOTE: dump bin[] just for visual observation */
 71         for (int i = 0; i < nb; i++) {
 72                 printf("bin[%d]:\t\t", i);
 73                 show(bin[i]->unit, bin[i]->len);
 74         }
 75 }
 76 
 77 /*
 78  * Rebuild a[] by walking numbers saved in (bin[0]->unit)[] and
 79  *                                         (bin[1]->unit)[]
 80  *
 81  * e.g.
 82  *        bin[0] = { .len = 2, .unit = {4, 2}    }
 83  *        bin[1] = { .len = 3, .unit = {5, 3, 1} }
 84  *          a[5] = { 4, 2, 5, 3, 1 }
 85  *
 86  *        bin[0] bin[1] ===> a[5]
 87  *
 88  *                             1
 89  *                             3
 90  *                 1           5
 91  *          2      3           2
 92  *          4      5    ===>   4
 93  *        -----  -----       -----
 94  *          0      1
 95  */
 96 static void
 97 rebuild(int a[], size_t na, bin_unit_t *bin[], size_t nb)
 98 {
 99         int k = 0;
100         for (int i = 0; i < nb; i++)
101                 for (int j = 0; j < bin[i]->len; j++)
102                         a[k++] = (bin[i]->unit)[j];
103 }
104 
105 void
106 radixsort(int a[], int n)
107 {
108         /* alloc aux[] for puting a[] to (bin[i]->unit)[] (i= 0, 1) */
109         int *aux = (int *)malloc(sizeof(int) * n * 2);
110         if (aux == NULL) /* error */
111                 return;
112 
113         /*
114          * alloc bin[2], (hence, worst space complexity is O(n*2 + 2))
115          * if number M of a[] whose Xth digit is 0, save M to (bin[0]->unit)[]
116          * else                                     save M to (bin[1]->unit)[]
117          *                    where X = 0, 1, ..., 31
118          */
119         bin_unit_t bin0 = {.len = 0, .unit = aux + 0};
120         bin_unit_t bin1 = {.len = 0, .unit = aux + n};
121         bin_unit_t *bin[2] = {&bin0, &bin1};
122 
123         /* LSD (Least Significant Digit first) */
124         for (int index = 0; index < 32; index++) {
125                 /* 0. break if a[] is in order */
126                 if (is_sorted(a, n))
127                         break;
128 
129                 /* 1. build bin[] */
130                 build_bin_byindex(bin, 2, a, n, index);
131 
132                 /* 2. rebuild a[] by walking bin[] */
133                 rebuild(a, n, bin, 2);
134 
135                 /* NOTE: dump a[] for visual observation */
136                 printf("%dth bit(done):\t", index); show(a, n);
137         }
138 
139         free(aux);
140 }
141 
142 int
143 main(int argc, char *argv[])
144 {
145         if (argc < 2) {
146                 fprintf(stderr, "Usage: %s <C1> [C2] ...\n", argv[0]);
147                 return -1;
148         }
149 
150         argc--;
151         argv++;
152 
153         int n = argc;
154         int *a = (int *)malloc(sizeof(int) * n);
155 #define VALIDATE(p) do { if (p == NULL) return -1; } while (0)
156         VALIDATE(a);
157 
158         char *s = getenv("ISINT");
159         if (s != NULL && strncmp(s, "true", 4) == 0)
160                 g_isint = true;
161         else if (s != NULL && strncmp(s, "false", 4) == 0)
162                 g_isint = false;
163 
164         if (g_isint) {
165                 for (int i = 0; i < n; i++)
166                         *(a+i) = atoi(argv[i]);
167         } else {
168                 for (int i = 0; i < n; i++)
169                         *(a+i) = argv[i][0];
170         }
171 
172         printf("                ");
173         for (int i = 0; i < n; i++)
174                 printf("%-2x ", i);
175         printf("\n");
176 
177         printf("Before sorting: "); show(a, n);
178         radixsort(a, n);
179         printf("After  sorting: "); show(a, n);
180 
181 #define FREE(p) do { free(p); p = NULL; } while (0)
182         FREE(a);
183         return 0;
184 }

o 编译并测试

$ gcc -g -Wall -m32 -std=c99 -o radixsort2 radixsort2.c

$ ISINT=true ./radixsort2 \
>               4  2  1  8
                0  1  2  3
Before sorting: 4  2  1  8
0th bit of      4               is 0
0th bit of      2               is 0
0th bit of      1               is 1
0th bit of      8               is 0
bin[0]:         4  2  8
bin[1]:         1
0th bit(done):  4  2  8  1
1th bit of      4               is 0
1th bit of      2               is 1
1th bit of      8               is 0
1th bit of      1               is 0
bin[0]:         4  8  1
bin[1]:         2
1th bit(done):  4  8  1  2
2th bit of      4               is 1
2th bit of      8               is 0
2th bit of      1               is 0
2th bit of      2               is 0
bin[0]:         8  1  2
bin[1]:         4
2th bit(done):  8  1  2  4
3th bit of      8               is 1
3th bit of      1               is 0
3th bit of      2               is 0
3th bit of      4               is 0
bin[0]:         1  2  4
bin[1]:         8
3th bit(done):  1  2  4  8
After  sorting: 1  2  4  8

小结:

基数排序(Radix Sort)是一种非比较型整数(non-comparative integer)排序算法,也是一种分布式(distributed)排序算法,其本质是多关键字排序,既可以使用LSD(最低位优先)方法,也可以使用MSD(最高位优先)方法。 下一节将介绍另一种分布式排序算法 – 桶排序

    原文作者:veli
    原文地址: https://www.cnblogs.com/idorax/p/6653038.html
    本文转自网络文章,转载此文章仅为分享知识,如有侵权,请联系博主进行删除。
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