为什么这些矩阵换位时间如此反直觉？

Question

以下示例代码生成一个size的矩阵N,并将其转换SAMPLES次数.当N = 512转置操作的平均执行时间是2144 ?s(coliru链接)时.乍一看,没有什么特别的,对吗？...

好吧,这是结果

• N = 513 → 1451 ?s
• N = 519 → 600 ?s
• N = 530 → 486 ?s
• N = 540→ 492 ?s(终于!理论开始工作:).

那么为什么在实践中这些简单的计算与理论如此不同呢？此行为是否与CPU缓存一致性或缓存未命中有关？如果是这样请解释.

#include <algorithm>
#include <iostream>
#include <chrono>

constexpr int N       = 512; // Why is 512 specifically slower (as of 2016)
constexpr int SAMPLES = 1000;
using us = std::chrono::microseconds;

int A[N][N];

void transpose()
{
    for ( int i = 0 ; i < N ; i++ )
    for ( int j = 0 ; j < i ; j++ )
        std::swap(A[i][j], A[j][i]);
}

int main()
{
    // initialize matrix
    for ( int i = 0 ; i < N ; i++ )
    for ( int j = 0 ; j < N ; j++ )
        A[i][j] = i+j;

    auto t1 = std::chrono::system_clock::now();
    for ( int i = 0 ; i < SAMPLES ; i++ )
        transpose();
    auto t2 = std::chrono::system_clock::now();

    std::cout << "Average for size " << N << ": " << std::chrono::duration_cast<us>(t2 - t1).count() / SAMPLES << " (us)"; 
}

Answer 1

这是由于缓存未命中.您可以使用valgrind --tool=cachegrind查看未命中数量.使用N = 512您获得以下输出:

Average for size 512: 13052 (us)==21803== 
==21803== I   refs:      1,054,721,935
==21803== I1  misses:            1,640
==21803== LLi misses:            1,550
==21803== I1  miss rate:          0.00%
==21803== LLi miss rate:          0.00%
==21803== 
==21803== D   refs:        524,278,606  (262,185,156 rd   + 262,093,450 wr)
==21803== D1  misses:      139,388,226  (139,369,492 rd   +      18,734 wr)
==21803== LLd misses:           25,828  (      7,959 rd   +      17,869 wr)
==21803== D1  miss rate:          26.6% (       53.2%     +         0.0%  )
==21803== LLd miss rate:           0.0% (        0.0%     +         0.0%  )
==21803== 
==21803== LL refs:         139,389,866  (139,371,132 rd   +      18,734 wr)
==21803== LL misses:            27,378  (      9,509 rd   +      17,869 wr)
==21803== LL miss rate:            0.0% (        0.0%     +         0.0%  )

同时,使用N=530您获得以下输出:

Average for size 530: 13264 (us)==22783== 
==22783== I   refs:      1,129,929,859
==22783== I1  misses:            1,640
==22783== LLi misses:            1,550
==22783== I1  miss rate:          0.00%
==22783== LLi miss rate:          0.00%
==22783== 
==22783== D   refs:        561,773,362  (280,923,156 rd   + 280,850,206 wr)
==22783== D1  misses:       32,899,398  ( 32,879,492 rd   +      19,906 wr)
==22783== LLd misses:           26,999  (      7,958 rd   +      19,041 wr)
==22783== D1  miss rate:           5.9% (       11.7%     +         0.0%  )
==22783== LLd miss rate:           0.0% (        0.0%     +         0.0%  )
==22783== 
==22783== LL refs:          32,901,038  ( 32,881,132 rd   +      19,906 wr)
==22783== LL misses:            28,549  (      9,508 rd   +      19,041 wr)
==22783== LL miss rate:            0.0% (        0.0%     +         0.0%  )

如您所见,512中的D1未命中大约是530中的3.5倍