This Article 
 Bibliographic References 
 Add to: 
An Analysis of Scatter Decomposition
November 1990 (vol. 39 no. 11)
pp. 1337-1345

A formal analysis of a powerful mapping technique known as scatter decomposition is provided. Scatter decomposition divides an irregular computational domain into a large number of equally sized pieces and distributes them modularly among processors. A probabilistic model of workload in one dimension is used to formally explain why and when scatter decomposition works. The first result is that if a correlation in workload is a convex function of distance, then scattering a more finely decomposed domain yields a lower average processor workload variance. The second result shows that if the workload process is a stationary Gaussian and the correlation function decreases linearly in distance until becoming zero and then remain zero, scattering a more finely decomposed domain yields a lower expected maximum processor workload. It is shown that if the correlation function decreases linearly across the entire domain, then among all mappings that assign an equal number of domain pieces to each processor, scatter decomposition minimizes the average processor workload variance. The dependence of these results on the assumption of decreasing correlation is illustrated with situations where a coarser granularity actually achieves better load balance.

[1] G. Fox, M. Johnson, G. Lyzenga, S. Otto, J. Salmon, and D. Walker,Solving Problems on Concurrent Computers. Englewood Cliffs, NJ: Prentice-Hall, 1988.
[2] G. A. Geist and M. T. Heath, "Matrix factorization on a hypercube multiprocessor," inProc. Hypercube Microprocessors Conf., Knoxville, TN, Sept. 1986, pp. 161-180.
[3] P. Heidelberger, "Discrete-event simulations and parallel processing: Statistical properties,"SIAM J. Scientif. Statistical Comput., vol. 9, no. 6, pp. 1114-1132, Nov. 1988.
[4] T. Hoshino,Pax Computer. Reading, MA: Addison-Wesley, 1989.
[5] I. Ipsen, Y. Saad, and M. H. Schultz, "Complexity of dense linear system solution on a multiprocessor ring,"Lin. Algebra Appl., vol. 77, pp. 205-239, 1986.
[6] C. Kruskal and A. Weiss, "Allocating independence subtasks on parallel processors,"IEEE Trans. Software Eng., vol. SE-11, no. 10, pp. 1001-1015, Oct. 1985.
[7] H. J. Larson and B. O. Shubert,Probabilistic Models in Engineering Sciences, Vol. 1. New York: Wiley, 1979.
[8] M. R. Leadbetter, G. Lindgren, and H. Rootzén,Extremes and Related Properties of Random Sequences and Processes. New York: Springer-Verlag, 1983.
[9] R. Morison and S. W. Otto, "The scattered decomposition for finite element problems,"J. Scientif. Comput., vol. 2, no. 1, pp. 59-76, Mar. 1987.
[10] D. M. Nicol, "Mapping a battlefield simulation onto parallel message-passing architectures," inProc. 1988 SCS Conf. Distributed Simulation, Feb. 1988, pp. 141-146.
[11] D. M. Nicol and P. F. Reynolds, Jr., "Optimal dynamic remapping of parallel computations,"IEEE Trans. Comput., vol. 39, no. 2, pp. 206-219, Feb. 1990.
[12] R. S. Pindyck and D. L. Rubinfeld,Econometric Models and Economic Forecasts. New York: McGraw-Hill, 1976.
[13] H. S. Ross,Stochastic Processes. New York: Wiley, 1983.
[14] Y. Saad, "Communication complexity of the Gaussian elimination algorithm on multiprocessors,"Lin. Algebra Appl., vol. 77, pp. 315-340, 1986.
[15] J. Salmon, "A mathematical analysis of the scattered decomposition," inProc. Third Conf. Hypercube Concurrent Comput. Appl., Vol. 1, ACM, 1988, pp. 239-240.
[16] J. Saltz, V. K. Naik, and D. Nicol, "Reduction of the effects of communication delays in scientific algorithms on message passing MIMD architectures,"SIAM J. Sci. Stat. Comput., vol. 8, no. 1, pp. s118-s134, 1987.
[17] Y. Won and S. Sahni, "Maze routing on a hypercube multiprocessor computer," inProc. 1987 Int. Conf. Parallel Processing, St. Charles, IL, Aug. 1987, pp. 630-637.

Index Terms:
formal analysis; mapping technique; scatter decomposition; computational domain; probabilistic model; convex function; stationary Gaussian; correlation function; coarser granularity; parallel processing; performance evaluation; probability.
D.M. Nicol, J.H. Saltz, "An Analysis of Scatter Decomposition," IEEE Transactions on Computers, vol. 39, no. 11, pp. 1337-1345, Nov. 1990, doi:10.1109/12.61043
Usage of this product signifies your acceptance of the Terms of Use.