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Optimal routing algorithms for mesh-connected processor arrays

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Abstract

We show that there is a randomizedoblivious algorithm for routing any (partial) permutation on ann ×n grid in 2n +O(logn) parallel communication steps. The queues will not grow larger than Θ(logn/log logn) with high probability. We then modify this to obtain a (nonoblivious) algorithm with the same running time such that the size of the queues is bounded by a constant with high probability. For permutations withlocality, where each packet has to travel a distance at mostL, a generalization of the algorithm routes in time proportional toL with high probability. Finally, we identify a class of meshlike networks that have optimal or near-optimal diameter. These meshes have the potential of being adapted to run existing sorting and routing algorithms with corresponding reduction in their running times.

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References

  1. K. E. Batcher, Design of a Massively Parallel Processor,IEEE Trans. Comput.,29 (1980), 836–840.

    Article  Google Scholar 

  2. Y. Han, Y. Igarashi, and M. Truszczynski, Indexing Schemes and Lower Bounds for Sorting on a Mesh-Connected Computer, Technical Report TR-114-88, Department of Computer Science, University of Kentucky, Lexington.

  3. V. F. Kolchin, Random Mappings, Optimization Software, New York, 1986.

    MATH  Google Scholar 

  4. D. Krizanc, Oblivious Routing with Limited Buffer Capacity, Technical Report TR-14-87, Aiken Computation Laboratory, Harvard University, Cambridge, Mass.

  5. D. Krizanc, S. Rajasekaran, and Th. Tsantilas, Optimal Routing Algorithms for Mesh-Connected Processor Arrays In:VLSI Algorithms and Architectures, 3rd Aegean Workshop on Computing (AWOC '88), Lecture Notes in Computer Science, Vol. 319, pp. 411–422, Springer-Verlag, New York, 1988.

    Google Scholar 

  6. M. Kumar and D. Hirschberg, An Efficient Implementation of Batcher's Odd-Even Merge Algorithm and its Application in Parallel Sorting SchemesIEEE Trans. Comput.,32 (1983), 254–264.

    Article  MATH  Google Scholar 

  7. M. Kunde, Optimal Sorting On Multi-Dimensionally Mesh-Connected Computers, In:STACS 1987, Lecture Notes in Computer Science, Vol. 247, pp. 408–419, Springer-Verlag, New York, 1987.

    Google Scholar 

  8. M. Kunde, Lower Bounds for Sorting on Mesh-Connected Architectures,Acta Inform.,24 (1987), 121–130.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Kunde, Routing and Sorting on Mesh-Connected Arrays, In:VLSI Algorithms and Architectures, 3rd Aegean Workshop on Computing (A WOC '88) Lecture Notes in Computer Science, Vol. 319, pp. 423–433, Springer-Verlag, New York, 1988.

    Google Scholar 

  10. M. Kunde, Bounds for 1-Selection and Related Problems on Grids of Processors,J. New Generation Comput. Systems,2 (1989), 129–143.

    MathSciNet  Google Scholar 

  11. H.-W. Langet al., Systolic Sorting on a Mesh Connected Network,IEEE Trans. Comput.,34 (1985), 652–658.

    Article  MATH  Google Scholar 

  12. T. Leighton, F. Makedon, and I. Tollis, A2n —2 Step Algorithm for Routing in an n ×n Array with Constant Size Queues, In:Proc. of the 1989 Symp. on Parallel Algorithms and Architectures (SPAA '89), pp. 328–335.

  13. Y. Ma, S. Sen, and D. Scherson, The Distance Bound for Sorting on Mesh Connected Processor Arrays is Tight, In:Proc. 27th IEEE Symposium on Foundations of Computer Science, 1986, pp. 255–263.

  14. D. Nassimi and S. Sahni, An Optimal Routing Algorithm for Mesh-Connected Parallel Computers,J. Assoc. Comput. Mach.,27 (1980), 6–29.

    MATH  MathSciNet  Google Scholar 

  15. D. Nassimi and S. Sahni, Data Broadcasting in SIMD Computers,IEEE Trans. Comput.,30 (1981), 101–107.

    Article  MathSciNet  Google Scholar 

  16. S. E. Orcutt, Implementation of Permutation Functions in Illiac IV-Type Computers,IEEE Trans. Comput.,25 (1976), 929–936.

    Article  MATH  MathSciNet  Google Scholar 

  17. C. S. Raghavendra and V. K. P. Kumar, Permutations on Illiac IV-Type Networks,IEEE Trans. Comput.,35 (1986), 662–669.

    Article  Google Scholar 

  18. S. Rajasekaran and Th. Tsantilas, An Optimal Randomized Routing Algorithm for the Mesh and a Class of Efficient Mesh-like Routing Networks, In:7th Conference on Foundations of Software Technology and Theoretical Computer Science, Pune, India, Lecture Notes in Computer Science, Vol. 287, pp. 226–241, Spinger-Verlag, New York, 1987.

    Google Scholar 

  19. C. P. Schnorr and A. Shamir, An Optimal Sorting Algorithm for Mesh-Connected Computers, In:Proc. 18th ACM Symposium on Theory of Computing, 1986, pp. 255–263.

  20. J.T. Schwartz, Ultracomputers,ACM Trans. Prog. Lang. Systems,2 (1980), 484–521.

    Article  MATH  Google Scholar 

  21. L. Snyder, Supercomputers and VLSI: The Effect of Large-Scale Integration on Computer Architecture, inAdvances in Computers, vol. 23 (1984) (M. C. Yovits, ed.), Academic Press, New York, pp. 1–33.

    Google Scholar 

  22. C. D. Thompson and H. T. Kung, Sorting on a Mesh-Connected Parallel Computer,Comm. ACM,20 (1977), 263–270.

    Article  MATH  MathSciNet  Google Scholar 

  23. L. G. Valiant, A Scheme for Fast Parallel Communication,SIAM J. Comput. 11 (1982), 350–361.

    Article  MATH  MathSciNet  Google Scholar 

  24. L. G. Valiant, Parallel Computation, In:Proc. 7th IBM Symposium on Mathematical Foundations of Computer Science, 1982, pp. 171–189. Also, Technical Report TR-16-82, Aiken Computation Laboratory, Harvard University, Cambridge, Mass.

    Google Scholar 

  25. L. G. Valiant, General Purpose Parallel Architectures, In:Handbook of Theoretical Computer Science (J. van Leeuwen, ed.), North-Holland, 1990 (to appear).

  26. L. G. Valiant and G. J. Brebner, Universal Schemes for Parallel Communication, In:Proc. 13th ACM Symposium on Theory of Computing, 1981, pp. 263–277.

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Author notes

  1. Sanguthevar Rajasekaran

    Present address: Department of Computer Science, University of Pennsylvania, 19104, Philadelphia, PA, USA

  2. Thanasis Tsantilas

    Present address: Department of Computer Science, Columbia University, 10027, New York, NY, USA

Authors and Affiliations

  1. Aiken Computation Laboratory, Harvard University, 02138, Cambridge, MA, USA

    Sanguthevar Rajasekaran & Thanasis Tsantilas

Authors
  1. Sanguthevar Rajasekaran
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  2. Thanasis Tsantilas
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Additional information

Communicated by F. Thomson Leighton.

Preliminary reports of portions of the results contained in this paper have appeared in theProceedings of the 1988 Aegean Workshop on Computing [5], and in theProceedings of the 1987 Conference on Foundations of Software Technology and Theoretical Computer Science [18]. The work of the first author was supported in part by NSF Grant NSF-DCR-85-03251 and ONR contract N00014-80-C-0647. The work of the second author was supported in part by NSF Grant NSF-DCR-86-00379.

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Rajasekaran, S., Tsantilas, T. Optimal routing algorithms for mesh-connected processor arrays. Algorithmica 8, 21–38 (1992). https://doi.org/10.1007/BF01758834

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  • DOI: https://doi.org/10.1007/BF01758834

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