Optimal average case sorting on arrays

  • Manfred Kunde
  • Rolf Niedermeier
  • Klaus Reinhardt
  • Peter Rossmanith
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 900)


We present algorithms for sorting and routing on two-dimensional mesh-connected parallel architectures that are optimal on average. If one processor has many packets then we asymptotically halve the up to now best running times. For a load of one optimal algorithms are known for the mesh. We improve this to a load of eight without increasing the running time. For tori no optimal algorithms were known even for a load of one. Our algorithm is optimal for every load. Other architectures we consider include meshes with diagonals and reconfigurable meshes. Furthermore, the method applies to meshes of arbitrary higher dimensions and also enables optimal solutions for the routing problem.


Buffer Size Average Case Indexing Scheme Adjacent Pair Processor Array 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    B. S. Chlebus. Sorting within distance bound on a mesh-connected processor array. In H. Djidjev, editor, Proc. of International Symposium on Optimal Algorithms, Number 401 in Lecture Notes in Computer Science, pages 232–238, Varna, Bulgaria, May/June 1989. Springer.Google Scholar
  2. 2.
    W. Feller. An Introduction to Probability Theory and its Applications, volume I. Wiley, 3d edition, 1968.Google Scholar
  3. 3.
    Q. P. Gu and J. Gu. Algorithms and average time bounds of sorting on a meshconnected computer. IEEE Transactions on Parallel and Distributed Systems, 5(3):308–315, March 1994.Google Scholar
  4. 4.
    T. Hagerup and C. Rüb. A guided tour of Chernoff bounds. Information Processing Letters, 33:305–308, 1990.Google Scholar
  5. 5.
    M. Hofri. Probabilistic analysis of algorithms. Texts and Monographs in Computer Science. Springer-Verlag, 1987.Google Scholar
  6. 6.
    M. Kaufmann, H. Schröder, and J. F. Sibeyn. Routing and sorting on reconfigurable meshes. 1994. To appear in Parallel Processing Letters.Google Scholar
  7. 7.
    M. Kaufmann, J. F. Sibeyn, and T. Suel. Derandomizing algorithms for routing and sorting on meshes. In Proceedings of the 5th ACM-SIAM Symposium on Discrete Algorithms, pages 669–679, 1994.Google Scholar
  8. 8.
    D. E. Knuth. Seminumerical Algorithms, volume 2 of The Art of Computer Programming. Addison-Wesley, 2nd edition, 1969.Google Scholar
  9. 9.
    D. E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, 1973.Google Scholar
  10. 10.
    M. Kunde. Block gossiping on grids and tori: Sorting and routing match the bisection bound deterministically. In T. Lengauer, editor, Proceedings of the 1st European Symposium on Algorithms, Number 726 in Lecture Notes in Computer Science, pages 272–283, Bad Honnef, Federal Republic of Germany, September 1993. Springer.Google Scholar
  11. 11.
    M. Kunde, R. Niedermeier, and P. Rossmanith. Faster sorting and routing on grids with diagonals. In P. Enjalbert, E. W. Mayr, and K. W. Wagner, editors, Proceedings of the 11th Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, pages 225–236. Springer, 1994.Google Scholar
  12. 12.
    T. Leighton. Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes. Morgan Kaufmann, 1992.Google Scholar
  13. 13.
    T. Leighton. Methods for message routing in parallel machines. In Proceedings of the 24th ACM Symposium on Theory of Computing, pages 77–96, 1992.Google Scholar
  14. 14.
    R. Miller, V. K. Prasanna-Kumar, D. I. Reisis, and Q. F. Stout. Parallel computation on reconfigurable meshes. IEEE Transactions on Computers, 42(6):678–692, June 1993.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Manfred Kunde
    • 2
  • Rolf Niedermeier
    • 1
  • Klaus Reinhardt
    • 1
  • Peter Rossmanith
    • 2
  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenFed. Rep. of Germany
  2. 2.Fakultät für InformatikTechnische Universität MünchenMünchenFed. Rep. of Germany

Personalised recommendations