问题
什么是例如最好的算法，下面的转换？
0010 0000 => 0000 0100
从详细的转换MSB->LSB至LSB->MSB, 所有的Bit必须扭转，着。这并非字节顺序的交换。
最佳答案
注意: 以下的算法都用C实现。但应该能够迁移到其他语言(仅仅是不那么快的时候可别找我)。
可选方案
内存占用少(32位int。32位机器)(来源于这里)
unsigned int
reverse(register unsigned int x)
{
x = (((x & 0xaaaaaaaa) >> 1) | ((x & 0x55555555) << 1));
x = (((x & 0xcccccccc) >> 2) | ((x & 0x33333333) << 2));
x = (((x & 0xf0f0f0f0) >> 4) | ((x & 0x0f0f0f0f) << 4));
x = (((x & 0xff00ff00) >> 8) | ((x & 0x00ff00ff) << 8));
return((x >> 16) | (x << 16));
}
最快(查找表)
static const unsigned char BitReverseTable256[] =
{
0x00, 0x80, 0x40, 0xC0, 0x20, 0xA0, 0x60, 0xE0, 0x10, 0x90, 0x50, 0xD0, 0x30, 0xB0, 0x70, 0xF0,
0x08, 0x88, 0x48, 0xC8, 0x28, 0xA8, 0x68, 0xE8, 0x18, 0x98, 0x58, 0xD8, 0x38, 0xB8, 0x78, 0xF8,
0x04, 0x84, 0x44, 0xC4, 0x24, 0xA4, 0x64, 0xE4, 0x14, 0x94, 0x54, 0xD4, 0x34, 0xB4, 0x74, 0xF4,
0x0C, 0x8C, 0x4C, 0xCC, 0x2C, 0xAC, 0x6C, 0xEC, 0x1C, 0x9C, 0x5C, 0xDC, 0x3C, 0xBC, 0x7C, 0xFC,
0x02, 0x82, 0x42, 0xC2, 0x22, 0xA2, 0x62, 0xE2, 0x12, 0x92, 0x52, 0xD2, 0x32, 0xB2, 0x72, 0xF2,
0x0A, 0x8A, 0x4A, 0xCA, 0x2A, 0xAA, 0x6A, 0xEA, 0x1A, 0x9A, 0x5A, 0xDA, 0x3A, 0xBA, 0x7A, 0xFA,
0x06, 0x86, 0x46, 0xC6, 0x26, 0xA6, 0x66, 0xE6, 0x16, 0x96, 0x56, 0xD6, 0x36, 0xB6, 0x76, 0xF6,
0x0E, 0x8E, 0x4E, 0xCE, 0x2E, 0xAE, 0x6E, 0xEE, 0x1E, 0x9E, 0x5E, 0xDE, 0x3E, 0xBE, 0x7E, 0xFE,
0x01, 0x81, 0x41, 0xC1, 0x21, 0xA1, 0x61, 0xE1, 0x11, 0x91, 0x51, 0xD1, 0x31, 0xB1, 0x71, 0xF1,
0x09, 0x89, 0x49, 0xC9, 0x29, 0xA9, 0x69, 0xE9, 0x19, 0x99, 0x59, 0xD9, 0x39, 0xB9, 0x79, 0xF9,
0x05, 0x85, 0x45, 0xC5, 0x25, 0xA5, 0x65, 0xE5, 0x15, 0x95, 0x55, 0xD5, 0x35, 0xB5, 0x75, 0xF5,
0x0D, 0x8D, 0x4D, 0xCD, 0x2D, 0xAD, 0x6D, 0xED, 0x1D, 0x9D, 0x5D, 0xDD, 0x3D, 0xBD, 0x7D, 0xFD,
0x03, 0x83, 0x43, 0xC3, 0x23, 0xA3, 0x63, 0xE3, 0x13, 0x93, 0x53, 0xD3, 0x33, 0xB3, 0x73, 0xF3,
0x0B, 0x8B, 0x4B, 0xCB, 0x2B, 0xAB, 0x6B, 0xEB, 0x1B, 0x9B, 0x5B, 0xDB, 0x3B, 0xBB, 0x7B, 0xFB,
0x07, 0x87, 0x47, 0xC7, 0x27, 0xA7, 0x67, 0xE7, 0x17, 0x97, 0x57, 0xD7, 0x37, 0xB7, 0x77, 0xF7,
0x0F, 0x8F, 0x4F, 0xCF, 0x2F, 0xAF, 0x6F, 0xEF, 0x1F, 0x9F, 0x5F, 0xDF, 0x3F, 0xBF, 0x7F, 0xFF
};
unsigned int v; // reverse 32-bit value, 8 bits at time
unsigned int c; // c will get v reversed
// Option 1:
c = (BitReverseTable256[v & 0xff] << 24) |
(BitReverseTable256[(v >> 8) & 0xff] << 16) |
(BitReverseTable256[(v >> 16) & 0xff] << 8) |
(BitReverseTable256[(v >> 24) & 0xff]);
// Option 2:
unsigned char * p = (unsigned char *) &v;
unsigned char * q = (unsigned char *) &c;
q[3] = BitReverseTable256[p[0]];
q[2] = BitReverseTable256[p[1]];
q[1] = BitReverseTable256[p[2]];
q[0] = BitReverseTable256[p[3]];
来自于著名的Bit Twiddling Hacks page:
你能够扩展这个算法到64位int的场景，或者为了更快的速度而牺牲多一些的内存(如果你的L1数据缓存足够大)。有一个64K的查找表且每次反转16位。
其他方案
简单
unsigned int v; // input bits to be reversed
unsigned int r = v; // r will be reversed bits of v; first get LSB of v
int s = sizeof(v) * CHAR_BIT - 1; // extra shift needed at end
for (v >>= 1; v; v >>= 1)
{
r <<= 1;
r |= v & 1;
s--;
}
r <<= s; // shift when v's highest bits are zero
更快(32位处理器)
unsigned char b = x;
b = ((b * 0x0802LU & 0x22110LU) | (b * 0x8020LU & 0x88440LU)) * 0x10101LU >> 16;
更快(64位处理器)
unsigned char b; // reverse this (8-bit) byte b = (b * 0x0202020202ULL & 0x010884422010ULL) % 1023;
假设你想在32位int环境这样做，那么仅仅须要把每个byte反转，然后再颠倒byte的次序就可以。例如以下:
unsigned int toReverse; unsigned int reversed; unsigned char inByte0 = (toReverse & 0xFF); unsigned char inByte1 = (toReverse & 0xFF00) >> 8; unsigned char inByte2 = (toReverse & 0xFF0000) >> 16; unsigned char inByte3 = (toReverse & 0xFF000000) >> 24; reversed = (reverseBits(inByte0) << 24) | (reverseBits(inByte1) << 16) | (reverseBits(inByte2) << 8) | (reverseBits(inByte3);
结果
我測试了两个最有效的方案。查找表和按位与(第一个方案)。測试机器为一台笔记本电脑，配置为4G DDR2内存，2.4GHz的双核T7500处理器。4MB的L2缓存。我使用的是gcc 4.3.2。64位Linux。OpenMP(外加GCC绑定)被用来提高计时器的调度能力。
reverse.c
#include <stdlib.h>
#include <stdio.h>
#include <omp.h>
unsigned int
reverse(register unsigned int x)
{
x = (((x & 0xaaaaaaaa) >> 1) | ((x & 0x55555555) << 1));
x = (((x & 0xcccccccc) >> 2) | ((x & 0x33333333) << 2));
x = (((x & 0xf0f0f0f0) >> 4) | ((x & 0x0f0f0f0f) << 4));
x = (((x & 0xff00ff00) >> 8) | ((x & 0x00ff00ff) << 8));
return((x >> 16) | (x << 16));
}
int main()
{
unsigned int *ints = malloc(100000000*sizeof(unsigned int));
unsigned int *ints2 = malloc(100000000*sizeof(unsigned int));
for(unsigned int i = 0; i < 100000000; i++)
ints[i] = rand();
unsigned int *inptr = ints;
unsigned int *outptr = ints2;
unsigned int *endptr = ints + 100000000;
// Starting the time measurement
double start = omp_get_wtime();
// Computations to be measured
while(inptr != endptr)
{
(*outptr) = reverse(*inptr);
inptr++;
outptr++;
}
// Measuring the elapsed time
double end = omp_get_wtime();
// Time calculation (in seconds)
printf("Time: %f seconds
", end-start);
free(ints);
free(ints2);
return 0;
}
reverse_lookup.c
#include <stdlib.h>
#include <stdio.h>
#include <omp.h>
static const unsigned char BitReverseTable256[] =
{
0x00, 0x80, 0x40, 0xC0, 0x20, 0xA0, 0x60, 0xE0, 0x10, 0x90, 0x50, 0xD0, 0x30, 0xB0, 0x70, 0xF0,
0x08, 0x88, 0x48, 0xC8, 0x28, 0xA8, 0x68, 0xE8, 0x18, 0x98, 0x58, 0xD8, 0x38, 0xB8, 0x78, 0xF8,
0x04, 0x84, 0x44, 0xC4, 0x24, 0xA4, 0x64, 0xE4, 0x14, 0x94, 0x54, 0xD4, 0x34, 0xB4, 0x74, 0xF4,
0x0C, 0x8C, 0x4C, 0xCC, 0x2C, 0xAC, 0x6C, 0xEC, 0x1C, 0x9C, 0x5C, 0xDC, 0x3C, 0xBC, 0x7C, 0xFC,
0x02, 0x82, 0x42, 0xC2, 0x22, 0xA2, 0x62, 0xE2, 0x12, 0x92, 0x52, 0xD2, 0x32, 0xB2, 0x72, 0xF2,
0x0A, 0x8A, 0x4A, 0xCA, 0x2A, 0xAA, 0x6A, 0xEA, 0x1A, 0x9A, 0x5A, 0xDA, 0x3A, 0xBA, 0x7A, 0xFA,
0x06, 0x86, 0x46, 0xC6, 0x26, 0xA6, 0x66, 0xE6, 0x16, 0x96, 0x56, 0xD6, 0x36, 0xB6, 0x76, 0xF6,
0x0E, 0x8E, 0x4E, 0xCE, 0x2E, 0xAE, 0x6E, 0xEE, 0x1E, 0x9E, 0x5E, 0xDE, 0x3E, 0xBE, 0x7E, 0xFE,
0x01, 0x81, 0x41, 0xC1, 0x21, 0xA1, 0x61, 0xE1, 0x11, 0x91, 0x51, 0xD1, 0x31, 0xB1, 0x71, 0xF1,
0x09, 0x89, 0x49, 0xC9, 0x29, 0xA9, 0x69, 0xE9, 0x19, 0x99, 0x59, 0xD9, 0x39, 0xB9, 0x79, 0xF9,
0x05, 0x85, 0x45, 0xC5, 0x25, 0xA5, 0x65, 0xE5, 0x15, 0x95, 0x55, 0xD5, 0x35, 0xB5, 0x75, 0xF5,
0x0D, 0x8D, 0x4D, 0xCD, 0x2D, 0xAD, 0x6D, 0xED, 0x1D, 0x9D, 0x5D, 0xDD, 0x3D, 0xBD, 0x7D, 0xFD,
0x03, 0x83, 0x43, 0xC3, 0x23, 0xA3, 0x63, 0xE3, 0x13, 0x93, 0x53, 0xD3, 0x33, 0xB3, 0x73, 0xF3,
0x0B, 0x8B, 0x4B, 0xCB, 0x2B, 0xAB, 0x6B, 0xEB, 0x1B, 0x9B, 0x5B, 0xDB, 0x3B, 0xBB, 0x7B, 0xFB,
0x07, 0x87, 0x47, 0xC7, 0x27, 0xA7, 0x67, 0xE7, 0x17, 0x97, 0x57, 0xD7, 0x37, 0xB7, 0x77, 0xF7,
0x0F, 0x8F, 0x4F, 0xCF, 0x2F, 0xAF, 0x6F, 0xEF, 0x1F, 0x9F, 0x5F, 0xDF, 0x3F, 0xBF, 0x7F, 0xFF
};
int main()
{
unsigned int *ints = malloc(100000000*sizeof(unsigned int));
unsigned int *ints2 = malloc(100000000*sizeof(unsigned int));
for(unsigned int i = 0; i < 100000000; i++)
ints[i] = rand();
unsigned int *inptr = ints;
unsigned int *outptr = ints2;
unsigned int *endptr = ints + 100000000;
// Starting the time measurement
double start = omp_get_wtime();
// Computations to be measured
while(inptr != endptr)
{
unsigned int in = *inptr;
// Option 1:
//*outptr = (BitReverseTable256[in & 0xff] << 24) |
// (BitReverseTable256[(in >> 8) & 0xff] << 16) |
// (BitReverseTable256[(in >> 16) & 0xff] << 8) |
// (BitReverseTable256[(in >> 24) & 0xff]);
// Option 2:
unsigned char * p = (unsigned char *) &(*inptr);
unsigned char * q = (unsigned char *) &(*outptr);
q[3] = BitReverseTable256[p[0]];
q[2] = BitReverseTable256[p[1]];
q[1] = BitReverseTable256[p[2]];
q[0] = BitReverseTable256[p[3]];
inptr++;
outptr++;
}
// Measuring the elapsed time
double end = omp_get_wtime();
// Time calculation (in seconds)
printf("Time: %f seconds
", end-start);
free(ints);
free(ints2);
return 0;
}
在不同的优化级别(Optimizations)，两个方案我都尝试了。每一个级别跑3个案例，每一个案例反转1亿个随机的无符号整数。
对于查找表方案。bitwise hacks page上面的两种方法(Option 1 and Option 2)我都測试过。
结果例如以下:
按位与
mrj10@mjlap:~/code$ gcc -fopenmp -std=c99 -o reverse reverse.c mrj10@mjlap:~/code$ ./reverse Time: 2.000593 seconds mrj10@mjlap:~/code$ ./reverse Time: 1.938893 seconds mrj10@mjlap:~/code$ ./reverse Time: 1.936365 seconds mrj10@mjlap:~/code$ gcc -fopenmp -std=c99 -O2 -o reverse reverse.c mrj10@mjlap:~/code$ ./reverse Time: 0.942709 seconds mrj10@mjlap:~/code$ ./reverse Time: 0.991104 seconds mrj10@mjlap:~/code$ ./reverse Time: 0.947203 seconds mrj10@mjlap:~/code$ gcc -fopenmp -std=c99 -O3 -o reverse reverse.c mrj10@mjlap:~/code$ ./reverse Time: 0.922639 seconds mrj10@mjlap:~/code$ ./reverse Time: 0.892372 seconds mrj10@mjlap:~/code$ ./reverse Time: 0.891688 seconds
查找表(Option 1)
mrj10@mjlap:~/code$ gcc -fopenmp -std=c99 -o reverse_lookup reverse_lookup.c mrj10@mjlap:~/code$ ./reverse_lookup Time: 1.201127 seconds mrj10@mjlap:~/code$ ./reverse_lookup Time: 1.196129 seconds mrj10@mjlap:~/code$ ./reverse_lookup Time: 1.235972 seconds mrj10@mjlap:~/code$ gcc -fopenmp -std=c99 -O2 -o reverse_lookup reverse_lookup.c mrj10@mjlap:~/code$ ./reverse_lookup Time: 0.633042 seconds mrj10@mjlap:~/code$ ./reverse_lookup Time: 0.655880 seconds mrj10@mjlap:~/code$ ./reverse_lookup Time: 0.633390 seconds mrj10@mjlap:~/code$ gcc -fopenmp -std=c99 -O3 -o reverse_lookup reverse_lookup.c mrj10@mjlap:~/code$ ./reverse_lookup Time: 0.652322 seconds mrj10@mjlap:~/code$ ./reverse_lookup Time: 0.631739 seconds mrj10@mjlap:~/code$ ./reverse_lookup Time: 0.652431 seconds
查找表(Option 2)
mrj10@mjlap:~/code$ gcc -fopenmp -std=c99 -o reverse_lookup reverse_lookup.c mrj10@mjlap:~/code$ ./reverse_lookup Time: 1.671537 seconds mrj10@mjlap:~/code$ ./reverse_lookup Time: 1.688173 seconds mrj10@mjlap:~/code$ ./reverse_lookup Time: 1.664662 seconds mrj10@mjlap:~/code$ gcc -fopenmp -std=c99 -O2 -o reverse_lookup reverse_lookup.c mrj10@mjlap:~/code$ ./reverse_lookup Time: 1.049851 seconds mrj10@mjlap:~/code$ ./reverse_lookup Time: 1.048403 seconds mrj10@mjlap:~/code$ ./reverse_lookup Time: 1.085086 seconds mrj10@mjlap:~/code$ gcc -fopenmp -std=c99 -O3 -o reverse_lookup reverse_lookup.c mrj10@mjlap:~/code$ ./reverse_lookup Time: 1.082223 seconds mrj10@mjlap:~/code$ ./reverse_lookup Time: 1.053431 seconds mrj10@mjlap:~/code$ ./reverse_lookup Time: 1.081224 seconds
结论
假设你比較在意性能，那么使用查找表Option 1(Byte的寻址不出意外的慢)。
假设你须要尽可能的利用完每个Byte内存(且你也在意bit反转的性能)，那么优化后的按位与方案也还不赖。
附加说明
我知道上面的代码仅仅是一个粗略的版本号，很欢迎大家提供一些优化的建议。下面是我知道的几点:
- 我没有权限訪问ICC，那可能更快些(假设你能够測试请在评论中回复)。
- 在一些L1缓存比較大的现代机器上面。64K的查找表可能工作得更好。
-mtune=native对 -O2/-O3(发生符号重定义的错误)无效，所以我不相信产生的代码是为我的微架构而优化。- SSE环境下应该有一种方法处理得更快。
我不知道怎么做，但又更快的内存复制。批量的按位与。调整的指令集， 总是有一些手段的。
- 我知道只x86的指令集是危急的。以下是GCC在-O3环境产生的代码，所以比我更厉害的大牛能够检查一下。
32-bit
.L3: movl (%r12,%rsi), %ecx movzbl %cl, %eax movzbl BitReverseTable256(%rax), %edx movl %ecx, %eax shrl $24, %eax mov %eax, %eax movzbl BitReverseTable256(%rax), %eax sall $24, %edx orl %eax, %edx movzbl %ch, %eax shrl $16, %ecx movzbl BitReverseTable256(%rax), %eax movzbl %cl, %ecx sall $16, %eax orl %eax, %edx movzbl BitReverseTable256(%rcx), %eax sall $8, %eax orl %eax, %edx movl %edx, (%r13,%rsi) addq $4, %rsi cmpq $400000000, %rsi jne .L3
更改: 我也尝试在自己机器上使用uint64，看看是否性能有所提高。
相对于32-bit性能大概提高了10%。不管你是每次用64-bit类型去反转2个32-bit的int。或者实际上看作64-bit并分两次来反转。性能都大致相当。
代码例如以下(对于前者。每次反转2个32-bit的int):
.L3: movq (%r12,%rsi), %rdx movq %rdx, %rax shrq $24, %rax andl $255, %eax movzbl BitReverseTable256(%rax), %ecx movzbq %dl,%rax movzbl BitReverseTable256(%rax), %eax salq $24, %rax orq %rax, %rcx movq %rdx, %rax shrq $56, %rax movzbl BitReverseTable256(%rax), %eax salq $32, %rax orq %rax, %rcx movzbl %dh, %eax shrq $16, %rdx movzbl BitReverseTable256(%rax), %eax salq $16, %rax orq %rax, %rcx movzbq %dl,%rax shrq $16, %rdx movzbl BitReverseTable256(%rax), %eax salq $8, %rax orq %rax, %rcx movzbq %dl,%rax shrq $8, %rdx movzbl BitReverseTable256(%rax), %eax salq $56, %rax orq %rax, %rcx movzbq %dl,%rax shrq $8, %rdx movzbl BitReverseTable256(%rax), %eax andl $255, %edx salq $48, %rax orq %rax, %rcx movzbl BitReverseTable256(%rdx), %eax salq $40, %rax orq %rax, %rcx movq %rcx, (%r13,%rsi) addq $8, %rsi cmpq $400000000, %rsi jne .L3