A Period-Processor-Time-Minimal Schedule for Cubical Mesh Algorithms

Using a directed acyclic graph (dag) model of algorithms, we investigate precedence-constrained multiprocessor schedules for the n/spl times/n/spl times/n directed mesh. This cubical mesh is fundamental, representing the standard algorithm for square matrix product, as well as many other algorithms. Its completion requires at least 3/sup n/spl minus/2/ multiprocessor steps. Time-minimal multiprocessor schedules that use as few processors as possible are called processor-time-minimal. For the cubical mesh, 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) is referred to as a period-processor-time-minimal schedule. The period of any processor-time-minimal schedule for the cubical mesh is at least 3/sup n/2/ steps. This lower bound is shown to be exact by constructing, for n a multiple of 6, a period-processor-time-minimal multiprocessor schedule that can be realized on a systolic array whose topology is a toroidally connected n/2/spl times/n/2/spl times/3 mesh. >

[1]  Chris J. Scheiman,et al.  A Processor-Time-Minimal Systolic Array for Transitive Closure , 1992, IEEE Trans. Parallel Distributed Syst..

[2]  Yves Robert,et al.  Spacetime-minimal systolic arrays for Gaussian elimination and the algebraic path problem , 1990, Parallel Comput..

[3]  C. Thomborson,et al.  Area-time complexity for VLSI , 1979, STOC.

[4]  H. T. Kung,et al.  The Area-Time Complexity of Binary Multiplication , 1981, JACM.

[5]  Oscar H. Ibarra,et al.  VLSI algorithms for solving recurrence equations and applications , 1987, IEEE Trans. Acoust. Speech Signal Process..

[6]  Benjamin W. Wah,et al.  The Design of Optimal Systolic Arrays , 1985, IEEE Transactions on Computers.

[7]  Christos H. Papadimitriou,et al.  A Communication-Time Tradeoff , 1987, SIAM J. Comput..

[8]  Mihalis Yannakakis,et al.  Towards an Architecture-Independent Analysis of Parallel Algorithms , 1990, SIAM J. Comput..

[9]  Peter R. Cappello,et al.  A Processor-Time-Minimal Systolic Array for Cubical Mesh Algorithms , 1992, IEEE Trans. Parallel Distributed Syst..

[10]  M. Tcheunte,et al.  An optimal solution for Gauss-Jordan elimination of 2D systolic arrays , 1990 .

[11]  Yves Robert,et al.  Spacetime-minimal systolic architectures for Gaussian elimination and the algebraic path problem , 1990, [1990] Proceedings of the International Conference on Application Specific Array Processors.

[12]  Peter R. Cappello.  A spacetime-minimal systolic array for matrix product , 1990 .