矩阵转置中的缓存利用率c

Question

This code transposes a matrix four ways. The first does sequential writes, non sequential reads. The second is the opposite. The next two are the same, but with cache skipping writes. What seems to happen is sequential writes are faster, and skipping the cache is faster. What I don't understand is, if the cache is being skipped why are sequential writes still faster?

QueryPerformanceCounter(&before);
for (i = 0; i < N; ++i)
   for (j = 0; j < N; ++j)
      tmp[i][j] = mul2[j][i];
QueryPerformanceCounter(&after);
printf("Transpose 1:\t%ld\n", after.QuadPart - before.QuadPart);

QueryPerformanceCounter(&before);
for (j = 0; j < N; ++j)
   for (i = 0; i < N; ++i)
     tmp[i][j] = mul2[j][i];
QueryPerformanceCounter(&after);
printf("Transpose 2:\t%ld\n", after.QuadPart - before.QuadPart);

QueryPerformanceCounter(&before);
for (i = 0; i < N; ++i)
   for (j = 0; j < N; ++j)
      _mm_stream_si32(&tmp[i][j], mul2[j][i]);
QueryPerformanceCounter(&after);
printf("Transpose 3:\t%ld\n", after.QuadPart - before.QuadPart);

QueryPerformanceCounter(&before);
for (j = 0; j < N; ++j)
   for (i = 0; i < N; ++i)
      _mm_stream_si32(&tmp[i][j], mul2[j][i]);
QueryPerformanceCounter(&after);
printf("Transpose 4:\t%ld\n", after.QuadPart - before.QuadPart);

EDIT: The output is

Transpose 1:    47603
Transpose 2:    92449
Transpose 3:    38340
Transpose 4:    69597

Answer 1

CPU has a write combining buffer to combine writes on a cache line to happen in one burst. In this case (cache being skipped for sequential writes), this write combining buffer acts as a one line cache which makes the results be very similar to cache not being skipped.

To be exact, in case of cache being skipped, writes are still happening in bursts to memory.

See write-combining logic behavior here.

Answer 2

You could try non linear memory layout for the matrix to improve cache utilization. With 4x4 32bit float tiles one could do transpose with only single access to each cache line. Plus as a bonus tile transposes could be done easily with _MM_TRANSPOSE4_PS.

Transposing a very large matrix is still very memory intensive operation. It will still be heavily bandwidth limited but at least cache word load is near optimal. I don't know if the performance could be still optimized. My testing shows that a few years old laptop manages to transpose 16k*16k (1G memory) in about 300ms.

I tried to use also _mm_stream_sd but it actually makes performance worse for some reason. I don't understand nontemporal memory writes enough to have any practical guess why performance would drop with _mm_stream_ps. Possible reason is of course that cache line is already in L1 cache ready for the write operation.

But actually important part with non linear matrix would possibility to avoid transpose completely and simple run the multiplication in tile friendly order. But I only have transpose code that I'm using to improve my knowledge about cache management in algorithms.

I haven't yet tried to test if prefetching would improve memory bandwidth usage. Current code runs at about 0.5 instructions per cycle (good cache friendly code runs around 2 ins per cycle on this CPU) that leaves a lot of free cycles for prefetch instructions allowing even quite complex calculation to optimize prefetching timing in runtime.

example code from my transpose benchmark test follows.

#define MATSIZE 16384
#define align(val, a) (val + (a - val % a))
#define tilewidth 4
typedef int matrix[align(MATSIZE,tilewidth)*MATSIZE] __attribute__((aligned(64)));


float &index(matrix m, unsigned i, unsigned j)
{
    /* tiled address calculation */
    /* single cache line is used for 4x4 sub matrices (64 bytes = 4*4*sizeof(int) */
    /* tiles are arranged linearly from top to bottom */
    /*
     * eg: 16x16 matrix tile positions:
     *   t1 t5 t9  t13
     *   t2 t6 t10 t14
     *   t3 t7 t11 t15
     *   t4 t8 t12 t16
     */
    const unsigned tilestride = tilewidth * MATSIZE;
    const unsigned comp0 = i % tilewidth; /* i inside tile is least significant part */
    const unsigned comp1 = j * tilewidth; /* next part is j multiplied by tile width */
    const unsigned comp2 = i / tilewidth * tilestride;
    const unsigned add = comp0 + comp1 + comp2;
    return m[add];
}

/* Get start of tile reference */
float &tile(matrix m, unsigned i, unsigned j)
{
    const unsigned tilestride = tilewidth * MATSIZE;
    const unsigned comp1 = j * tilewidth; /* next part is j multiplied by tile width */
    const unsigned comp2 = i / tilewidth * tilestride;
    return m[comp1 + comp2];

}
template<bool diagonal>
static void doswap(matrix mat, unsigned i, unsigned j)
{
        /* special path to swap whole tile at once */
        union {
            float *fs;
            __m128 *mm;
        } src, dst;
        src.fs = &tile(mat, i, j);
        dst.fs = &tile(mat, j, i);
        if (!diagonal) {
            __m128 srcrow0 = src.mm[0];
            __m128 srcrow1 = src.mm[1];
            __m128 srcrow2 = src.mm[2];
            __m128 srcrow3 = src.mm[3];
            __m128 dstrow0 = dst.mm[0];
            __m128 dstrow1 = dst.mm[1];
            __m128 dstrow2 = dst.mm[2];
            __m128 dstrow3 = dst.mm[3];

            _MM_TRANSPOSE4_PS(srcrow0, srcrow1, srcrow2, srcrow3);
            _MM_TRANSPOSE4_PS(dstrow0, dstrow1, dstrow2, dstrow3);

#if STREAMWRITE == 1
            _mm_stream_ps(src.fs +  0, dstrow0);
            _mm_stream_ps(src.fs +  4, dstrow1);
            _mm_stream_ps(src.fs +  8, dstrow2);
            _mm_stream_ps(src.fs + 12, dstrow3);
            _mm_stream_ps(dst.fs +  0, srcrow0);
            _mm_stream_ps(dst.fs +  4, srcrow1);
            _mm_stream_ps(dst.fs +  8, srcrow2);
            _mm_stream_ps(dst.fs + 12, srcrow3);
#else
            src.mm[0] = dstrow0;
            src.mm[1] = dstrow1;
            src.mm[2] = dstrow2;
            src.mm[3] = dstrow3;
            dst.mm[0] = srcrow0;
            dst.mm[1] = srcrow1;
            dst.mm[2] = srcrow2;
            dst.mm[3] = srcrow3;
#endif
        } else {
            __m128 srcrow0 = src.mm[0];
            __m128 srcrow1 = src.mm[1];
            __m128 srcrow2 = src.mm[2];
            __m128 srcrow3 = src.mm[3];

            _MM_TRANSPOSE4_PS(srcrow0, srcrow1, srcrow2, srcrow3);

#if STREAMWRITE == 1
            _mm_stream_ps(src.fs +  0, srcrow0);
            _mm_stream_ps(src.fs +  4, srcrow1);
            _mm_stream_ps(src.fs +  8, srcrow2);
            _mm_stream_ps(src.fs + 12, srcrow3);
#else
            src.mm[0] = srcrow0;
            src.mm[1] = srcrow1;
            src.mm[2] = srcrow2;
            src.mm[3] = srcrow3;
#endif
        }
    }
}

static void transpose(matrix mat)
{
    const unsigned xstep = 256;
    const unsigned ystep = 256;
    const unsigned istep = 4;
    const unsigned jstep = 4;
    unsigned x1, y1, i, j;
    /* need to increment x check for y limit to allow unrolled inner loop
     * access entries close to diagonal axis
     */
    for (x1 = 0; x1 < MATSIZE - xstep + 1 && MATSIZE > xstep && xstep; x1 += xstep)
        for (y1 = 0; y1 < std::min(MATSIZE - ystep + 1, x1 + 1); y1 += ystep)
            for ( i = x1 ; i < x1 + xstep; i += istep ) {
                for ( j = y1 ; j < std::min(y1 + ystep, i); j+= jstep )
                {
                    doswap<false>(mat, i, j);
                }
                if (i == j && j < (y1 + ystep))
                    doswap<true>(mat, i, j);
            }

    for ( i = 0 ; i < x1; i += istep ) {
        for ( j = y1 ; j < std::min(MATSIZE - jstep + 1, i); j+= jstep )
        {
            doswap<false>(mat, i, j);
        }
        if (i == j)
            doswap<true>(mat, i, j);
    }
    for ( i = x1 ; i < MATSIZE - istep + 1; i += istep ) {
        for ( j = y1 ; j < std::min(MATSIZE - jstep + 1, i); j+= jstep )
        {
            doswap<false>(mat, i, j);
        }
        if (i == j)
            doswap<true>(mat, i, j);
    }
    x1 = MATSIZE - MATSIZE % istep;
    y1 = MATSIZE - MATSIZE % jstep;

    for ( i = x1 ; i < MATSIZE; i++ )
        for ( j = 0 ; j < std::min((unsigned)MATSIZE, i); j++ )
                    std::swap(index(mat, i, j+0), index(mat, j+0, i));

    for ( i = 0; i < x1; i++ )
        for ( j = y1 ; j < std::min((unsigned)MATSIZE, i) ; j++ )
                    std::swap(index(mat, i, j+0), index(mat, j+0, i));
}