Abstract
The mesh-connected array is an attractive architecture for parallel machines, and a number of existing parallel machines are based on a mesh topology. In this paper, we survey the results on three fundamental problems in parallel computation: routing, sorting, and selection, and demonstrate that randomized algorithms for these problems on the mesh are often the most natural and efficient algorithms available.
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References
T. Blank. The Maspar MP-1 architecture. In Proceedings of IEEE Compcon, 1990.
A. Borodin and J. Hopcroft. Routing, merging and sorting on parallel models of computation. Journal of Computer and System Science, 30:130–145, 1985.
H. Chernoff. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Annals of Mathematics and Statistics, 23:493–507, 1952.
A. Condon and L. Narayanan. Upper and lower bounds for selection on the mesh. In Symposium on Parallel and Distributed Processing, pages 497–504. IEEE, 1994.
I. Corporation. Paragon xp/s product overview. Technical report, Intel, 1991.
T. Dachraoui and L. Narayanan. Fast deterministic sorting on large parallel machines. In Symposium on Parallel and Distributed Processing. IEEE, 1996. To appear.
W. Dally. Wire efficient VLSI multi-processor communication networks. In Advanced Research in VLSI, pages 391–415, 1987.
R. Diekmann, J. Gehring, R. Luling, B. Monien, M. Nubel, and R. Wanka. Sorting large data sets on a massively parallel system. In Proceedings of the Symposium on Parallel and Distributed Processing, pages 2–9. IEEE, 1994.
U. Feige and P. Raghavan. Exact analysis of hot-potato routing. In Symposium on the Foundations of Computer Science, pages 553–562. IEEE, 1992.
S. Felperin, P. Raghavan, and E. Upfal. A theory of wormhole routing in parallel computers. In Symposium on the Foundations of Computer Science, pages 563–572. IEEE, 1992.
R. Floyd and R. Rivest. Expected time bounds for selection. Communications of the ACM, 18:165–172, 1975.
W. Hightower, J. Prins, and J. Reif. Implementations of randomized sorting on large parallel machines. In Symposium on Parallel Algorithms and Architecture, pages 158–167. ACM, 1992.
C. Kakalamanis and D. Krizanc. Optimal sorting on mesh-connected processor arrays. In Symposium on Parallel Algorithms and Architecture, pages 50–59, 1992.
C. Kaklamanis and D. Krizanc. Multipacket hot-potato routing on processor arrays. In Euro-Par, pages 270–277, 1996.
C. Kaklamanis, D. Krizanc, L. Narayanan, and A. Tsantilas. Randomized sorting and selection on mesh-connected processor arrays. In Symposium on Parallel Algorithms and Architecture, pages 17–28, 1991.
C. Kaklamanis, D. Krizanc, and S. Rao. Simple path selection for optimal routing on processor arrays. In Symposium on Parallel Algorithms and Architecture, pages 23–30, 1992.
C. Kaklamanis, D. Krizanc, and S. Rao. Hot-potato routing on processor arrays. In Symposium on Parallel Algorithms and Architecture, pages 273–282, 1993.
C. Kaklamanis, D. Krizanc, and A. Tsantilas. Tight bounds for oblivious routing in the hypercube. In Symposium on Parallel Algorithms and Architecture, pages 31–36, 1990.
M. Kaufmann, S. Rajasekaran, and J. Sibeyn. Matching the bisection bounds for routing and sorting on the mesh. In Proceedings of SPAA 92, pages 31–40, 1992.
M. Kaufmann, J. Sibeyn, and T. Suel. Derandomizing algorithms for routing and sorting on meshes. In Symposium on Discrete Algorithms, pages 669–679. ACM-SIAM, 1994.
M. Kaufmann, J. Sibeyn, and T. Suel. Beyond the bisection bound: ranking and counting on meshes. In Proceedings of the European Symposium on Algorithms, volume LNCS 979, 1995.
D. Krizanc and L. Narayanan.Optimal algorithms for selection on a mesh-connected processor array. In Symposium on Parallel and Distributed Processing, pages 70–76. IEEE, 1992.
D. Krizanc, L. Narayanan, and R. Raman. Fast deterministic selection on meshconnected processor arrays. Algorithmica, 15:319–332, 1996.
M. Kunde. 1-selection and related problems on grids of processors. Journal of New Generation Computer Systems, 2:129–143, 1989.
M. Kunde. Concentrated regular data streams on grids: Sorting and routing near to the bisection bound. In Symposium on the Foundations of Computer Science, pages 141–150. IEEE, 1991.
M. Kunde. Block gossiping on grids and tori: Sorting and routing match the bisection bound deterministically. In Proceedings of the 1st European Symposium on Algorithms, 1993.
F. T. Leighton. Average case analysis of greedy routing algorithms on arrays. In Symposium on Parallel Algorithms and Architecture, pages 2–10, 1990.
F. T. Leighton. Introduction to Parallel Algorithms and Architectures: Arrays, Trees and Hypercubes. Morgan Kaufmann, 1992.
F. T. Leighton. Methods for message routing in parallel computers. In Symposium on the Theory of Computation, pages 77–96, 1992.
F. Meyer auf der Heide and M. Westermann.Hot-potato routing on multidimensional tori. In Work-shop on Graph-Theoretic Concepts in Computer Science, pages 275–287, 1995.
L. Narayanan. Selection, Sorting, and Routing on Mesh-Connected Processor Arrays. PhD thesis, University of Rochester, 1992.
M. Noakes, D. Wallach, and W. Dally. The 3-machine multicomputer: an architectural evaluation. In International Symposium on Computer Architecture, 1993.
S. Rajasekaran. Mesh-connected computers with fixed and reconfigurable buses: packet routing, sorting and selection. In European Symposium on Algorithms, 1993.
S. Rajasekaran. k-k routing, k-k sorting and cut-through routing on the mesh. Journal of Algorithms, 19:361–382, 1995.
S. Rajasekaran, W. Chen, and S. Yooseph. Unifying themes for network selection. In Proceedings of ISSAC, 1995.
S. Rajasekaran and T. Tsantilas. Optimal algorithms for routing on the mesh. Algorithmica, 8:21–38, 1992.
J. Reif and L. Valiant. A logarithmic time sort for linear size networks. Journal of the ACM, 34(1):60–76, 1987.
R. Reischuk. Probabilistic parallel algorithms for sorting and selection. SIAM Journal of Computing, 14(2):396–411, May 1985.
C. Schnorr and A. Shamir. An optimal sorting algorithm for mesh-connected computers. In Symposium on the Theory of Computation, pages 255–263, 1986.
W. Seitz, C. amd Athas, F. C., A. Martin, J. Siezovic, C. Steele, and W. Su. The architecture and programming of the ametek series 20101 multicomputer. In Proceedings of the Third Conference on Hypertube Concurrent Computers and Applications, pages 33–36, 1988.
C. Thompson and H. Kung. Sorting on a mesh connected parallel computer. Communications of the ACM, 20:263–270, 1977.
L. Valiant. A scheme for fast parallel communication. SIAM Journal of Computing, 11:350–361, 1982.
L. Valiant and G. Brebner. Universal schemes for parallel communication.In Symposium on the Theory of Computation, pages 263–277, 1981.
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Narayanan, L. (1998). Randomized algorithms on the mesh. In: Rolim, J. (eds) Parallel and Distributed Processing. IPPS 1998. Lecture Notes in Computer Science, vol 1388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-64359-1_715
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DOI: https://doi.org/10.1007/3-540-64359-1_715
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