有效地转置大（密集）二进制矩阵
[英]Efficiently transposing a large (dense) binary matrix
问题描述
我想转置一个二进制矩阵（由零和一组成的矩阵）。 矩阵的每一行都是一列32位整数的1D数组，整个矩阵是一行的一维数组。
这是一个128 x 128二进制矩阵的示例，该矩阵由128行的128 / 32 32位整数组成。 （实际上，该矩阵是一个N x N矩阵，其中N数以万计。）
// gcc example0.c -std=c99 && ./a.out
#include <stdlib.h>
#include <stdio.h>
#include <stdint.h>
typedef uint32_t uint32;
#define N (32 * 4) // Assume this to be divisible by 32. The matrix is intended to be an N x N matrix
#define INTS_PER_ROW (N / 32)
int main(int argc, char** argv){
uint32** matrix = malloc(N * sizeof(uint32*));
for(int i=0; i<N; ++i)
matrix[i] = malloc(INTS_PER_ROW * sizeof(uint32));
for(int i=0; i<N; ++i)
for(int j=0; j<INTS_PER_ROW; ++j)
matrix[i][j] = rand();
for(int i=0; i<N; ++i){
for(int j=0; j<INTS_PER_ROW; ++j){
printf(" %08x", matrix[i][j]);
}
puts("");
}
for(int i=0; i<N; ++i)
free(matrix[i]);
free(matrix);
}
转置此类矩阵的有效方法是什么？ （AVX2，CUDA，OpenCL ...一切正常。）
（ Hacker's Delight有用于转置8x8二进制矩阵的代码，它建议对较大的矩阵递归使用它们的方法，但我不知道它对大型矩阵是否可缩放。
1 个解决方案
解决方案1
3 已采纳 2017-10-07 06:37:40
这是__ballot()建议的使用__ballot()一种可能方法。
- 以合并的方式，通过图块将数据加载到共享内存中
- 在加载期间在共享内存中进行初始转置，以将列排列为行，以方便
__ballot()操作。 - 执行投票以从列数据（已存储在行中）中构建“最终”行数据
- 写回数据
至少据我所知，步骤1-3在GPU上应该相当有效。 第4步受到以下限制：位块的按位换位不会产生一组在内存中相邻的32位量。 因此，通常情况下，全局存储上的内存访问模式是分散的。
输出索引操作是最可视化/最繁琐的工作。
以下是一个包含4个测试用例的完整示例。 对于测试用例，我还借鉴了问题注释中的另一个想法，编写了一个简单的CPU函数来执行“黄金”测试。
$ cat t435.cu
#include <stdio.h>
#include <stdlib.h>
#define IDX(d,x,y,ld) d[y*ld+x]
#define cudaCheckErrors(msg) \
do { \
cudaError_t __err = cudaGetLastError(); \
if (__err != cudaSuccess) { \
fprintf(stderr, "Fatal error: %s (%s at %s:%d)\n", \
msg, cudaGetErrorString(__err), \
__FILE__, __LINE__); \
fprintf(stderr, "*** FAILED - ABORTING\n"); \
exit(1); \
} \
} while (0)
#include <time.h>
#include <sys/time.h>
#define USECPSEC 1000000ULL
unsigned long dtime_usec(unsigned long start){
timeval tv;
gettimeofday(&tv, 0);
return ((tv.tv_sec*USECPSEC)+tv.tv_usec)-start;
}
__global__ void bt(const unsigned * __restrict__ in, unsigned * __restrict__ out, const unsigned idim){
// assumes "square" binary matrix transpose
// assumes kernel is launched with 1024 threads per block, 2D, 32x32
// assumes input matrix dimension idim is evenly divisible by 32 bits
// assumes idim is specified in bits
__shared__ unsigned smem[32][33];
int idx = threadIdx.x+blockDim.x*blockIdx.x;
int idy = threadIdx.y+blockDim.y*blockIdx.y;
smem[threadIdx.x][threadIdx.y] = ((idx < idim/32)&&(idy < idim))?IDX(in,idx,idy,idim/32):0;
__syncthreads();
unsigned myval = smem[threadIdx.y][31-threadIdx.x];
__syncthreads();
smem[ 0][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<31)&myval));
smem[ 1][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<30)&myval));
smem[ 2][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<29)&myval));
smem[ 3][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<28)&myval));
smem[ 4][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<27)&myval));
smem[ 5][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<26)&myval));
smem[ 6][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<25)&myval));
smem[ 7][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<24)&myval));
smem[ 8][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<23)&myval));
smem[ 9][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<22)&myval));
smem[10][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<21)&myval));
smem[11][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<20)&myval));
smem[12][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<19)&myval));
smem[13][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<18)&myval));
smem[14][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<17)&myval));
smem[15][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<16)&myval));
smem[16][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<15)&myval));
smem[17][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<14)&myval));
smem[18][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<13)&myval));
smem[19][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<12)&myval));
smem[20][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<11)&myval));
smem[21][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<<10)&myval));
smem[22][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<< 9)&myval));
smem[23][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<< 8)&myval));
smem[24][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<< 7)&myval));
smem[25][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<< 6)&myval));
smem[26][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<< 5)&myval));
smem[27][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<< 4)&myval));
smem[28][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<< 3)&myval));
smem[29][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<< 2)&myval));
smem[30][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<< 1)&myval));
smem[31][threadIdx.y] = __ballot_sync(0xFFFFFFFFU, ((1U<< 0)&myval));
__syncthreads();
int indx = (idx*idim)+(gridDim.y*threadIdx.y)+blockIdx.y;
if ((idx < (idim/32))&&(idy < idim)) out[indx] = smem[threadIdx.y][threadIdx.x];
}
void naive_cpu_bt(const unsigned *in, unsigned *out, const unsigned idim){
memset(out, 0, idim*(idim/32)*sizeof(unsigned));
for (int i = 0; i < (idim/32); i++)
for (int j = 0; j < idim; j++){
for (int bit = 0; bit < 32; bit++){
if ((in[(j*(idim/32)) + i]>>bit)&1) out[(((i*32)+(31-bit))*(idim/32))+(j/32)] |= 1<<(31 - (j%32));
}
}
}
int main(){
unsigned *h_idata, *h_odata, *d_idata, *d_odata, *c_odata;
unsigned idim;
// test case 1, 32x32, upper triangular
idim = 32;
h_idata = (unsigned *)malloc(idim*(idim/32)*sizeof(unsigned));
h_odata = (unsigned *)malloc(idim*(idim/32)*sizeof(unsigned));
c_odata = (unsigned *)malloc(idim*(idim/32)*sizeof(unsigned));
cudaMalloc(&d_idata, idim*(idim/32)*sizeof(unsigned));
cudaMalloc(&d_odata, idim*(idim/32)*sizeof(unsigned));
unsigned data = 0x0FFFFFFFFU;
for (int i = 0; i < 32; i++){
h_idata[i] = data;
data>>=1;}
cudaMemcpy(d_idata, h_idata, idim*(idim/32)*sizeof(unsigned), cudaMemcpyHostToDevice);
bt<<<dim3(1,1),dim3(32,32)>>>(d_idata, d_odata, idim);
cudaMemcpy(h_odata, d_odata, idim*(idim/32)*sizeof(unsigned), cudaMemcpyDeviceToHost);
naive_cpu_bt(h_idata, c_odata, idim);
for (int i = 0; i < (idim/32)*idim; i++) if (c_odata[i] != h_odata[i]) {printf("mismatch at %u, was: %u, should be: %u\n", i, h_odata[i], c_odata[i]); return -1;}
for (int i = 0; i < 32; i++) printf("0x%8x\n", h_odata[i]);
free(h_idata);
free(h_odata);
free(c_odata);
cudaFree(d_idata);
cudaFree(d_odata);
// test case 2, 64x64, opposite diagonal
idim = 64;
h_idata = (unsigned *)malloc(idim*(idim/32)*sizeof(unsigned));
h_odata = (unsigned *)malloc(idim*(idim/32)*sizeof(unsigned));
c_odata = (unsigned *)malloc(idim*(idim/32)*sizeof(unsigned));
cudaMalloc(&d_idata, idim*(idim/32)*sizeof(unsigned));
cudaMalloc(&d_odata, idim*(idim/32)*sizeof(unsigned));
data = 0x01;
for (int i = 0; i < 32; i++){
h_idata[2*i] = 0; h_idata[(2*i)+1] = data;
h_idata[64+(2*i)] = data; h_idata[65+(2*i)] = 0xFFFFFFFFU;
data<<=1; data++;}
cudaMemcpy(d_idata, h_idata, idim*(idim/32)*sizeof(unsigned), cudaMemcpyHostToDevice);
bt<<<dim3(1,2),dim3(32,32)>>>(d_idata, d_odata, idim);
cudaMemcpy(h_odata, d_odata, idim*(idim/32)*sizeof(unsigned), cudaMemcpyDeviceToHost);
naive_cpu_bt(h_idata, c_odata, idim);
for (int i = 0; i < (idim/32)*idim; i++) if (c_odata[i] != h_odata[i]) { printf("mismatch at %u, was: %u, should be: %u\n", i, h_odata[i], c_odata[i]); return -1;}
for (int i = 0; i < 64; i++) printf("0x%8x 0x%8x\n", h_odata[i*2], h_odata[(i*2)+1]);
free(h_idata);
free(h_odata);
free(c_odata);
cudaFree(d_idata);
cudaFree(d_odata);
// test case 3, 96x96, ones in alternating columns
idim = 96;
h_idata = (unsigned *)malloc(idim*(idim/32)*sizeof(unsigned));
h_odata = (unsigned *)malloc(idim*(idim/32)*sizeof(unsigned));
c_odata = (unsigned *)malloc(idim*(idim/32)*sizeof(unsigned));
cudaMalloc(&d_idata, idim*(idim/32)*sizeof(unsigned));
cudaMalloc(&d_odata, idim*(idim/32)*sizeof(unsigned));
data = 0x55555555U;
for (int i = 0; i < idim*(idim/32); i++)
h_idata[i] = data;
cudaMemcpy(d_idata, h_idata, idim*(idim/32)*sizeof(unsigned), cudaMemcpyHostToDevice);
bt<<<dim3(1,3),dim3(32,32)>>>(d_idata, d_odata, idim);
cudaMemcpy(h_odata, d_odata, idim*(idim/32)*sizeof(unsigned), cudaMemcpyDeviceToHost);
naive_cpu_bt(h_idata, c_odata, idim);
for (int i = 0; i < (idim/32)*idim; i++) if (c_odata[i] != h_odata[i]) { printf("mismatch at %u, was: %u, should be: %u\n", i, h_odata[i], c_odata[i]); return -1;}
for (int i = 0; i < 96; i++) printf("0x%8x 0x%8x 0x%8x\n", h_odata[i*3], h_odata[(i*3)+1], h_odata[(i*3)+2]);
free(h_idata);
free(h_odata);
free(c_odata);
cudaFree(d_idata);
cudaFree(d_odata);
// test case 4, 8kx8k random
idim = 8192;
int xblocks = (idim/1024)+((idim%1024)?1:0);
h_idata = (unsigned *)malloc(idim*(idim/32)*sizeof(unsigned));
h_odata = (unsigned *)malloc(idim*(idim/32)*sizeof(unsigned));
c_odata = (unsigned *)malloc(idim*(idim/32)*sizeof(unsigned));
cudaMalloc(&d_idata, idim*(idim/32)*sizeof(unsigned));
cudaMalloc(&d_odata, idim*(idim/32)*sizeof(unsigned));
for (int i = 0; i < idim*(idim/32); i++)
h_idata[i] = rand();
unsigned long gt = dtime_usec(0);
cudaMemcpy(d_idata, h_idata, idim*(idim/32)*sizeof(unsigned), cudaMemcpyHostToDevice);
unsigned long gkt = dtime_usec(0);
bt<<<dim3(xblocks,(idim/32)),dim3(32,32)>>>(d_idata, d_odata, idim);
cudaDeviceSynchronize();
gkt = dtime_usec(gkt);
cudaMemcpy(h_odata, d_odata, idim*(idim/32)*sizeof(unsigned), cudaMemcpyDeviceToHost);
gt = dtime_usec(gt);
unsigned long ct = dtime_usec(0);
naive_cpu_bt(h_idata, c_odata, idim);
ct = dtime_usec(ct);
for (int i = 0; i < (idim/32)*idim; i++) if (c_odata[i] != h_odata[i]) { printf("mismatch at %u, was: %u, should be: %u\n", i, h_odata[i], c_odata[i]); return -1;}
printf("gputime: %fms, kerneltime: %fms, cputime: %fms\n", (gt*1000)/(float)USECPSEC, (gkt*1000)/(float)USECPSEC, (ct*1000)/(float)USECPSEC);
printf("kernel bandwidth = %fMB/s\n", (idim*(idim/32)*4*2)/(float)gkt);
printf("Success!\n");
free(h_idata);
free(h_odata);
free(c_odata);
cudaFree(d_idata);
cudaFree(d_odata);
return 0;
}
$ nvcc -arch=sm_61 -o t435 t435.cu
$ ./t435
0x80000000
0xc0000000
0xe0000000
0xf0000000
0xf8000000
0xfc000000
0xfe000000
0xff000000
0xff800000
0xffc00000
0xffe00000
0xfff00000
0xfff80000
0xfffc0000
0xfffe0000
0xffff0000
0xffff8000
0xffffc000
0xffffe000
0xfffff000
0xfffff800
0xfffffc00
0xfffffe00
0xffffff00
0xffffff80
0xffffffc0
0xffffffe0
0xfffffff0
0xfffffff8
0xfffffffc
0xfffffffe
0xffffffff
0x 0 0x 1
0x 0 0x 3
0x 0 0x 7
0x 0 0x f
0x 0 0x 1f
0x 0 0x 3f
0x 0 0x 7f
0x 0 0x ff
0x 0 0x 1ff
0x 0 0x 3ff
0x 0 0x 7ff
0x 0 0x fff
0x 0 0x 1fff
0x 0 0x 3fff
0x 0 0x 7fff
0x 0 0x ffff
0x 0 0x 1ffff
0x 0 0x 3ffff
0x 0 0x 7ffff
0x 0 0x fffff
0x 0 0x 1fffff
0x 0 0x 3fffff
0x 0 0x 7fffff
0x 0 0x ffffff
0x 0 0x 1ffffff
0x 0 0x 3ffffff
0x 0 0x 7ffffff
0x 0 0x fffffff
0x 0 0x1fffffff
0x 0 0x3fffffff
0x 0 0x7fffffff
0x 0 0xffffffff
0x 1 0xffffffff
0x 3 0xffffffff
0x 7 0xffffffff
0x f 0xffffffff
0x 1f 0xffffffff
0x 3f 0xffffffff
0x 7f 0xffffffff
0x ff 0xffffffff
0x 1ff 0xffffffff
0x 3ff 0xffffffff
0x 7ff 0xffffffff
0x fff 0xffffffff
0x 1fff 0xffffffff
0x 3fff 0xffffffff
0x 7fff 0xffffffff
0x ffff 0xffffffff
0x 1ffff 0xffffffff
0x 3ffff 0xffffffff
0x 7ffff 0xffffffff
0x fffff 0xffffffff
0x 1fffff 0xffffffff
0x 3fffff 0xffffffff
0x 7fffff 0xffffffff
0x ffffff 0xffffffff
0x 1ffffff 0xffffffff
0x 3ffffff 0xffffffff
0x 7ffffff 0xffffffff
0x fffffff 0xffffffff
0x1fffffff 0xffffffff
0x3fffffff 0xffffffff
0x7fffffff 0xffffffff
0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
0x 0 0x 0 0x 0
0xffffffff 0xffffffff 0xffffffff
gputime: 5.530000ms, kerneltime: 0.301000ms, cputime: 659.205994ms
kernel bandwidth = 55738.257812MB/s
Success!
$
对于前三个测试用例，除了进行cpu验证外，我还使用了3种不同的输入模式并打印出结果。 对于第四个测试用例（较大的随机模式），我跳过打印输出，但保留cpu验证。
还要注意，我使用了__ballot_sync()变体来与CUDA 9兼容 。
我还没有做任何性能分析。 像大多数转置操作一样，我希望此处的大部分性能来自有效使用全局内存。 主要的阻碍因素是上面步骤4中的分散写入。
尽管这种按位情况与实际情况大不相同，但可以在此处找到有关CUDA中矩阵转置的一些一般背景信息。
一个用于转置矩阵的程序，但在这个打印结果的程序中它有一个错误
[英]a program for transposing a matrix but in this program in printing result it has a bug
使用“follow-the-cycles”算法就地转置矩阵
[英]Transposing a matrix in-place using the "follow-the-cycles" algorithm
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