Consider the following multithreaded pseudocode for the Matrix Chain Multiplication problem (based on the single-thread example looked at in class): Input : dimensions d[0…] Output: minimum number of multiplications Let M[1…n][1..n] be an empty table for j=1 to n do M]L]+0 for s = s=1 to n – 1 do parallel for i = 1 to n – s do M[i][i+s] + M[i][i] + M[i+1][i + s] + d(i – 1).d[i] – d[i+s] for k= i +1 to its – 1 do if M[i][i+s] > M[i][b] + M[k+1][i+s] + d(i – 1] – d[ki] – d[i+s] then | Milli +s] + M[i][k] + M[k + 1][i+s] + d(i – 1] .d[k].d[i+s] return M[n] On Assignment 9, we considered why the loop on i was the only one that could be made into a parallel for loop. However, the computation being done by the loop on k can be done in parallel, but would need to be restructured to be done by a divide-and-conquer routine. (a) Redesign this algorithm to also parallelize the body of the loop on i (which in- cludes the loop on k), as described above. Your resulting algorithm should have span (n log n). (b) Prove that your redesigned algorithm has span (n logn). (c) What is the parallel slack, if there are p= log2 n processors? =
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