Interested Article - Разбиение цикла на блоки

Loop tiling (??????? , ????????? ????? ?? ????? ^{[

???????? ?? ?????? 2922 ???

]} ) — ?????????????? ?????????????? , ?????????? ??????? ?????????? ????????? ????? ?????? ????? ???????????.

?????? ?????? ??????????? ??????? ? ????????? ???????????? ???????????? ????????? ????? (??????? ????? ??????????? ?? ?????????? ??????????) ?? ????????? ????? ???????? ???????, ??? ????????? ??????? ???????????? ? ???? ????????? ?????? ?????? ? ???? ????????? ??? ?? ?????????????? ????????????? ? ???????? ?????????? ?????. ????????? ???????????? ???????????? ????? ???????? ? ????????? ??????? ?? ??????? ?????, ??????? ?????????? ? ???, ??? ???????? ? ????????? ????????????? ????, ???????? ?????????? ???????? ? ?????????? ?????????? ? ??????? ????.

??????: ????????? ??????? ?? ??????

 for (i = 0; i < N; i++)
     for (j = 0; j < N; j++)
         c[i] = c[i] + a[i, j] * b[j];

????? ????????? ????? ?? ????? 2 × 2

 for (i = 0; i < N; i += 2)
     for (j = 0; j < N; j += 2)
         for (ii = i; ii < min(i+2, N); ii++)
             for (jj = j; jj < min(j+2, N); jj++)
                 c[ii] = c[ii] + a[ii, jj] * b[ii];

??????????, ???????????? ???????????? ????? ??????? N × N. ???? ???????, a[i, j] ? ???????? ????????? ??????, ????? ????? ?????? N × N. ???? N ????????? ??????? ??????? ????????, ? ?????? ???? ?????????? ???????????? ??? ????, ????? ????????? ???????? ??????, ?? ????????, ??????? ???????????? ? ???????? ????? ???????? (????????, ??? i = 1, j ???????? ?? 1 ?? N), ?? ????????? ??????? ?? ????, ?????????? ????????? ????????? ????? ???????, ??? ?????? ??????? ??????????????????.

??????: ????????? ??????

?????? ???????? ????????????? ??????? ??????????????? ?????????????? ???????? ????????? ??????.

???????? ???:

 for (i = 0; i < N; i++)
     for (k = 0; k < N; k++)
         for (j = 0; j < N; j++)
             z[i, j] = z[i, j] + x[i, k] * y[k, j]

????? ????????? ?? ????? B × B:

 for (k2 = 0; k2 < N; i += B)
     for (j2 = 0; j2 < N; j += B)
         for (i = 0; i < N; i++)
             for (k1 = k2; k1 < min(k2 + B, N); k1++)
                 for (j1 = j2; k2 < min(j2 + B, N); j1++)
                     z[i, j1] = z[i, j1] + x[i, k1] * y[k1, j1];

????? ??????? ?????

?? ?????? ????? ??????????, ????? ?????? ????? ????????? ??? ??????????? ?????, ??? ??? ??? ??????? ?? ???????????? ?????????? ???????? ????????, ? ??????? ???????????? ??????. ??????? ???????? ?????? ????? ?????? ?????? ???? ? ?????????? ?????? ??????????????????.

??????????

// ??????? ???????????? ???????? ???? ???????? : ??????. — 2011. — ?. 55 . — ?. 16-22 . 4 ??????? 2017 ????.

??????????

(????.) Wolfe, M. More Iteration Space Tiling . Supercomputing'89, ???????? 655—664, 1989.
(????.) Wolf, M. E. and Lam, M. A Data Locality Optimizing Algorithm . '91, ???????? 30—44, 1991.
(????.) Irigoin, F. and Triolet, R. Supernode Partitioning . '88, ???????? 319—329, 1988.
(????.) Xue, J. Loop Tiling for Parallelism . Kluwer Academic Publishers. 2000.
(????.) M. S. Lam, E. E. Rothberg, and M. E. Wolf. The cache performance and optimizations of blocked algorithms. In Proceedings of the 4th International Conference on Architectural Support for Programming Languages and Operating Systems, pages 63–74, April 1991.