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<p>Using a directed acyclic graph (dag) model of algorithms, we investigateprecedence-constrained multiprocessor schedules for the n/spl times/n/spl times/ndirected mesh. This cubical mesh is fundamental, representing the standard algorithm forsquare matrix product, as well as many other algorithms. Its completion requires at least3/sup n/spl minus/2/ multiprocessor steps. Time-minimal multiprocessor schedules thatuse as few processors as possible are called processor-time-minimal. For the cubicalmesh, such a schedule requires at least /spl lsqb/3n/sup 2//4/spl rsqb/ processors.Among such schedules, one with the minimum period (i.e., maximum throughput) isreferred to as a period-processor-time-minimal schedule. The period of anyprocessor-time-minimal schedule for the cubical mesh is at least 3/sup n/2/ steps. Thislower bound is shown to be exact by constructing, for n a multiple of 6, aperiod-processor-time-minimal multiprocessor schedule that can be realized on a systolicarray whose topology is a toroidally connected n/2/spl times/n/2/spl times/3 mesh.</p>
Index Termsmultiprocessor interconnection networks; directed graphs; matrix algebra; computationalcomplexity; scheduling; systolic arrays; parallel algorithms; period-processor-time-minimal schedule; cubical mesh algorithms; directed acyclic graph; precedence-constrained multiprocessor schedules; systolic array; toroidally connected mesh; computational complexity; matrix product
C. Scheiman, P. Cappello, "A Period-Processor-Time-Minimal Schedule for Cubical Mesh Algorithms", IEEE Transactions on Parallel & Distributed Systems, vol. 5, no. , pp. 274-280, March 1994, doi:10.1109/71.277790
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