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Unformatted text preview: CS 140 : Matrix multiplication CS 140 : Matrix multiplication Matrix multiplication I : parallel issues Matrix multiplication II: cache issues Thanks to Jim Demmel and Kathy Yelick (UCB) for some of these slides Communication volume model Communication volume model Network of p processors Each with local memory Messagepassing Communication volume ( v ) Total size (words) of all messages passed during computation Broadcasting one word costs volume p (actually, p1 ) No explicit accounting for communication time Thus, cant really model parallel efficiency or speedup; for that, wed use the latencybandwidth model (see extra slides) MatrixMatrix Multiplication MatrixMatrix Multiplication {implements C = C + A*B} for i = 1 to n for j = 1 to n for k = 1 to n C(i,j) = C(i,j) + A(i,k) * B(k,j) = + * C(i,j) C(i,j) A(i,:) B(:,j) Algorithm has 2*n 3 = O(n 3 ) Flops and operates on 3*n 2 words of memory Parallel Matrix Multiply Parallel Matrix Multiply Compute C = C + A*B Basic sequential algorithm: C(i,j) += A(i,1)*B(1,j) + A(i,2)*B(1,j) ++ A(i,n)*B(n,j) work = t 1 = 2n 3 floating point operations (flops) Variables are: Data layout Structure of communication Schedule of communication Parallel Matrix Multiply with 1D Column Layout Parallel Matrix Multiply with 1D Column Layout Assume matrices are n x n and n is divisible by p A(i) is the nbyn/p block column that processor i owns (similarly B(i) and C(i)) B(i,j) is a n/pbyn/p sublock of B(i) in rows j*n/p through (j+1)*n/p Then: C(i) += A(0)*B(0,i) + A(1)*B(1,i) ++ A(p1)*B(p1,i) p0 p1 p2 p3 p5 p4 p6 p7 (A reasonable assumption for analysis, not for code) Matmul for 1D layout on a Processor Matmul for 1D layout on a Processor Ring Ring Proc k communicates only with procs k1 and k+1 Different pairs of processors can communicate simultaneously RoundRobin MerryGoRound algorithm Copy A(myproc) into MGR (MGR = MerryGoRound) C(myproc) = C(myproc) + MGR*B(myproc , myproc) for j = 1 to p1 send MGR to processor myproc+1 mod p (but see deadlock below) receive MGR from processor myproc1 mod p (but see below) C(myproc) = C(myproc) + MGR * B( myprocj mod p , myproc) Matmul for 1D layout on a Processor Ring Matmul for 1D layout on a Processor Ring One iteration: v = n 2 All p1 iterations: v = (p1) * n 2 ~ pn 2 Optimal for 1D data layout: Perfect speedup for arithmetic A(myproc) must meet each C(myproc) Nice communication pattern can probably overlap independent communications in the ring. In latency/bandwidth model (see extra slides), parallel efficiency e = 1  O(p/n) MatMul with 2D Layout MatMul with 2D Layout Consider processors in 2D grid (physical or logical) Processors can communicate with 4 nearest neighbors Alternative pattern: broadcast along rows and columns Assume p is square s x s grid p(0,0) p(0,1) p(0,2) p(1,0)...
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 Spring '09

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