Towards optimal locality in mesh-indexings

  • Rolf Niedermeier
  • Klaus Reinhardt
  • Peter Sanders
Technical Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1279)


The efficiency of many algorithms in parallel processing, computational geometry, image processing, and several other fields relies on “locality-preserving” indexing schemes for meshes. We concentrate on the case where the maximum distance between two mesh nodes indexed i and j shall be a slow-growing function of i — j (using the Euclidean, the maximum, and the Manhattan metric). In this respect, space-filling, self-similar curves like the Hilbert curve are superior to simple indexing schemes like “row-major.” We present new tight results on 2-D and 3-D Hilbert indexings which are easy to generalize to a quite large class of curves. We then present a new indexing scheme we call H- indexing, which has superior locality. For example, with respect to the Euclidean metric the H-indexing provides locality approximately 50% better than the usually used Hilbert indexing. This answers an open question of Gotsman and Lindenbaum. In addition, H-indexings have the useful property to form a Hamiltonian cycle and they are optimally locality-preserving among all cyclic indexings.


Hamiltonian Cycle Hamiltonian Path Mesh Node Indexing Scheme Manhattan Distance 
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.


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  1. 1.
    T. Asano, D. Ranjan, T. Roos, E. WeIzI, and P. Widmayer. Space filling curves and their use in the design of geometric data structures. In LATIN '95: theoretical informatics: second Latin American Symposium, number 911 in LNCS, page 36ff, Valparaiso, Chile, April 1995.Google Scholar
  2. 2.
    G. Chochia and M. Cole. Recursive 3D mesh indexings with improved locality. Technical report, University of Edinburgh, 1997. short version to be published in HPCN'97.Google Scholar
  3. 3.
    G. Chochia, M. Cole, and T. Heywood. Implementing the hierarchical PRAM on the 2D mesh: Analyses and experiments. In Symposium on Parallel and Distributed Processing, pages 587–595, Los Alamitos, Ca., USA, Oct. 1995. IEEE Computer Society Press.Google Scholar
  4. 4.
    C. Gotsman and M. Lindenbaum. On the metric properties of discrete space-filling curves. IEEE Transactions on Image Processing, 5(5):794–797, May 1996.CrossRefGoogle Scholar
  5. 5.
    C. Kaklamanis and G. Persiano. Branch-and-bound and backtrack search on mesh-connected arrays of processors. Mathematical Systems Theory, 27:471–489, 1994.CrossRefGoogle Scholar
  6. 6.
    R. Miller and Q. F. Stout. Mesh computer algorithms for computational geometry. IEEE Transactions on Computers, 38(3):321–340, March 1989.CrossRefGoogle Scholar
  7. 7.
    G. Mitchison and R. Durbin. Optimal numberings of an N x N array. SIAM J. Alg. Disc. Meth., 7(4):571–582, October 1986.Google Scholar
  8. 8.
    R. Niedermeier, K. Reinhard, and P. Sanders. Towards optimal locality in mesh-indexings. Technical Report IB 12/97, University of Karlsruhe, 1997.Google Scholar
  9. 9.
    H. Sagan. Space-Filling Curves. Universitext. Springer-Verlag, 1994.Google Scholar
  10. 10.
    P. Sanders and T. Hansch. On the efficient implementation of massively parallel quicksort. In 4th International Symposium on Solving Irregularly Structured Problems in Parallel, LNCS. Springer, 1997. to appear. *** DIRECT SUPPORT *** A0008123 00010Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Rolf Niedermeier
    • 1
  • Klaus Reinhardt
    • 1
  • Peter Sanders
    • 2
  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingen
  2. 2.Fakultät für InformatikUniversität KarlsruheKarlsruhe

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