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Automatic Mesh Partitioning

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Graph Theory and Sparse Matrix Computation

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 56))

Abstract

This paper describes an efficient approach to partitioning unstructured meshes that occur naturally in the finite element and finite difference methods. This approach makes use of the underlying geometric structure of a given mesh and finds a provably good partition in random O(n) time. It applies to meshes in both two and three dimensions. The new method has applications in efficient sequential and parallel algorithms for large-scale problems in scientific computing. This is an overview paper written with emphasis on the algorithmic aspects of the approach. Many detailed proofs can be found in companion papers.

School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213. Supported in part by National Science Foundation grant CCR-9016641.

Xerox Corporation, Palo Alto Research Center, Palo Alto, CA 94304. Part of the work was done while the author was at Carnegie Mellon University. Current address: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139.

Department of Computer Science, Cornell University, Ithaca, NY 14853. Supported by an NSF Presidential Young Investigator award. Revision work on this paper was supported by the Applied Mathematical Sciences program of the U.S. Department of Energy under contract DEAC04-76DP00789 while the author was visiting Sandia National Laboratories.

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References

  1. A. Agrawal and P. Klein. Cutting down on fill using nested dissection: Provably good elimination orderings. In Sparse Matrix Computations: Graph Theory Issues and Algorithms, IMA Volumes in Mathematics and its Applications, (this book), A. George, J. Gilbert and J. Liu, Springer-Verlag, New York. 1992.

    Google Scholar 

  2. N. Alon, P. Seymour, and R. Thomas. A separator theorem for non-planar graphs. In Proceedings of the 22th Annual ACM Symposium on Theory of Computing,Maryland, May 1990. ACM.

    Google Scholar 

  3. E. M. Andreev. On convex polyhedra in Lohacevskii space. Math. USR Sbornik, 10 (3): 413–440, 1970.

    Article  MATH  Google Scholar 

  4. E. M. Andreev. On convex polyhedra of finite volume in Lobacevskii space. Math. USR Sbornik, 12 (2): 270–259, 1970.

    Google Scholar 

  5. M. J. Berger and S. Bokhari. A partitioning strategy for nonuniform problems on multiprocessors. IEEE Trans. Comp., C-36: 570–580, 1987.

    Google Scholar 

  6. M. Bern, D. Eppstein and J. R. Gilbert. Provably good mesh generation. In 31st Annual Symposium on Foundations of Computer Science, IEEE,231–241, 1990, (to appear JCSS).

    Google Scholar 

  7. M. Bern and D. Eppstein. Mesh generation and optimal triangulation. In Computing in Euclidean Geometry, F. K. Hwang and D.-Z. Du editors, World Scientific, 1992.

    Google Scholar 

  8. G. Birkhoff and A. George. Elimination by nested dissection. Complexity of Sequential and Parallel Numerical Algorithms, J. F. Traub, Academic Press, 1973.

    Google Scholar 

  9. P. E. Bjßrstad and O. B. Widlund. Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J. Numer. Anal., 23: 1097–1120, 1986.

    Article  MathSciNet  Google Scholar 

  10. G. E. Blelloch. Vector Models for Data-Parallel Computing. MIT-Press, Cambridge MA, 1990.

    Google Scholar 

  11. J. H. Bramble, J. E. Pasciak, and A. H. Schatz. An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp. 46: 361–9, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  12. D. Calahan. Parallel solution of sparse simultaneous linear equations. in Proceedings of the 11th Annual Allerton Conference on Circuits and Systems Theory, 729–735, 1973.

    Google Scholar 

  13. T. F. Chan and D. C. Resasco. A framework for the analysis and construction of domain decomposition preconditioners. UCLA-CAM-87-09, 1987.

    Google Scholar 

  14. L. P. Chew. Guaranteed quality triangular meshes, Department of Computer Science, Cornell University TR 89–893, 1989.

    Google Scholar 

  15. K. Clarkson. Fast algorithm for the all-nearest-neighbors problem. In the 24th Annual Symposium on Foundations of Computer Science, 226–232, 1983.

    Chapter  Google Scholar 

  16. R. Cole, M. Sharir and C. K. Yap. On k-hulls and related problems. SIAM J. Computing, 61, 1987.

    Google Scholar 

  17. J. H. Conway, and N. J. A. Sloane. Sphere Packings, Lattices and Groups. Springer-Verlag, 1988.

    Google Scholar 

  18. L. Danzer, J. Fonlupt and V. Klee. Helly’s theorem and its relatives. Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 7: 101–180, 1963.

    Google Scholar 

  19. H. N. Djidjev. On the problem of partitioning planar graphs. SIAM J. Alg. Disc. Math., 3 (2): 229–240, June 1982.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. L. Dulmage and N. S. Mendelsohn. Coverings of bipartite graphs. Canadian J. Math. 10, pp 517–534, 1958.

    Article  MATH  MathSciNet  Google Scholar 

  21. I. S. Duff. Parallel implementation of multifrontal schemes. Parallel Computing, 3, 193–204, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  22. M. E. Dyer. On a multidimensional search procedure and its application to the Euclidean one-centre problem. SIAM Journal on Computing 13, pp 31–45, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  23. D. Eppstein, G. L. Miller, C. Sturtivant and S.-H. Teng. Approximating center points with and without linear programming. Manuscript, Massachusetts Institute of Technology, 1992.

    Google Scholar 

  24. D. Eppstein, G. L. Miller and S.-H. Teng. A deterministic linear time algorithm for geometric separators and its applications. Manuscript, Xerox Palo Alto Research Center, 1991.

    Google Scholar 

  25. I. Fkry. On straight line representing of planar graphs. Acta. Sci. Math. 24: 229–233, 1948.

    Google Scholar 

  26. R. W. Floyd and R. L. Rivest. Expected time bounds for selection. CACM 18 (3): 165–173, March, 1975.

    MATH  Google Scholar 

  27. H. de Fraysseix, J. Pach, and R. Pollack. Small sets supporting Fâry embeddings of planar graphs. In Proceedings of the 20th Annual ACM Symposium on Theory of Computing, 426–433, 1988.

    Google Scholar 

  28. G. N. Fredrickson and R. Janardan. Separator-based strategies for efficient message routing. In 27st Annual Symposium on Foundation of Computation Science, IEEE, 428–237, 1986.

    Google Scholar 

  29. I. Fried. Condition of finite element matrices generated from nonuniform meshes. AIAA J. 10, pp 219–221, 1972.

    Article  MATH  Google Scholar 

  30. A. M. Frieze, G. L. Miller and S.-H. Teng. Separator based divide and conquer in computational geometry. Proceedings of the 1992 ACM Symposium on Parallel Algorithms and Architectures, 1992.

    Google Scholar 

  31. H. Gazit. An improved algorithm for separating a planar graph. Manuscript, Department of Computer Science, University of Southern California, 1986.

    Google Scholar 

  32. H. Gazit and G. L. Miller. A parallel algorithm for finding a separator in planar graphs. In 28st Annual Symposium on Foundation of Computation Science, IEEE, 238–248, Los Angeles, October 1987.

    Google Scholar 

  33. H. Gazit. A deterministic parallel algorithm for planar graph isomorphism. In 32nd Annual Symposium on Foundations of Computer Science, IEEE, to appear, 1991.

    Google Scholar 

  34. J. A. George. Nested dissection of a regular finite element mesh. SIAM J. Numerical Analysis, 10: 345–363, 1973.

    Article  MATH  Google Scholar 

  35. A. George, M. T. Heath, J. Liu, E. Ng. Sparse Cholesky factorization on a local-memory multiprocessor. SIAM J. on Scientific and Statistical Computing, 9, 327–340, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  36. J. A. George and J. W. H. Liu. An automatic nested dissection algorithm for irregular finite element problems. SIAM J. on Numerical Analysis, 15, 1053–1069, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  37. J. A. George and J. W. H. Liu. Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, 1981.

    Google Scholar 

  38. J. R. Gilbert, J. P. Hutchinson, and R. E. Tarjan. A separator theorem for graphs of bounded genus. J. Algorithms, 5 pp 391–407, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  39. J.R. Gilbert, G.L. Miller, and S.-H. Teng. Geometric mesh partitioning: Implementation and experiments. Technical Report, Xerox Palo Alto Research Center, to appear, 1992.

    Google Scholar 

  40. J. R. Gilbert and R. E. Tarjan. The analysis of a nested dissection algorithm. Numerische Mathematik, 50 (4): 377–404, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  41. G. Hardy, J. E. Littlewood and G. Pólya. Inequalities. Second edition, Cambridge University Press, 1952.

    MATH  Google Scholar 

  42. D. Haussier and E. Welzl. e-net and simplex range queries. Discrete e..4 Computational Geometry, 2: 127–151, 1987.

    Article  Google Scholar 

  43. J. P. Hutchinson and G. L. Miller. On deleting vertices to make a graph of positive genus planar. In Discrete Algorithms and Complexity Theory - Proceedings of the Japan-US Joint Seminar, Kyoto,Japan, pages 81–98, Boston, 1986. Academic Press.

    Google Scholar 

  44. C. Jordan. Sur les assemblages de lignes. Journal Reine Angew. Math, 70: 185–190, 1869.

    Article  Google Scholar 

  45. F. T. Leighton. Complexity Issues in VLSI. Foundations of Computing. MIT Press, Cambridge, MA, 1983.

    Google Scholar 

  46. F. T. Leighton and S. Rao. An approximate max-flow min-cut theorem for uniform multi-commodity flow problems with applications to approximation algorithms. In 29th Annual Symposium on Foundations of Computer Science, pp 422–431, 1988.

    Google Scholar 

  47. C. E. Leiserson. Area Efficient VLSI Computation. Foundations of Computing. MIT Press, Cambridge, MA, 1983.

    Google Scholar 

  48. C. E. Leiserson and J. G. Lewis. Orderings for parallel sparse symmetric factorization. in 3rd SIAM Conference on Parallel Processing for Scientific Computing, 1987.

    Google Scholar 

  49. R. J. Lipton, D. J. Rose, and R. E. Tarjan. Generalized nested dissection. SIAM J. on Numerical Analysis, 16: 346–358, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  50. R. J. Lipton and R. E. Tarjan. A separator theorem for planar graphs. SIAM J. of Appl. Math., 36: 177–189, April 1979.

    Article  MATH  MathSciNet  Google Scholar 

  51. R. J. Lipton and R. E. Tarjan. Applications of planar separator theorem. SIAM J. Comput, 9 (3): 615–627, August 1981.

    Article  MathSciNet  Google Scholar 

  52. J. W. II. Liu. The solution of mesh equations on a parallel computer. in 2nd Langley Conference on Scientific Computing, 1974.

    Google Scholar 

  53. N. Megiddo. Linear programming in linear time when the dimension is fixed. SIAM Journal on Computing 12, pp 759–776, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  54. G. L. Miller. Finding small simple cycle separators for 2-connected planar graphs. Journal of Computer and System Sciences, 32 (3): 265–279, June 1986.

    Article  MATH  MathSciNet  Google Scholar 

  55. G. L. Miller and S.-H. Teng. Centerpoints and point divisions. Manuscript, School of Computer Science, Carnegie Mellon University, 1990.

    Google Scholar 

  56. G. L. Miller, S.-H. Teng, and S. A. Vavasis. A unified geometric approach to graph separators. In 32nd Annual Symposium on Foundations of Computer Science, IEEE, pp 538–547, 1991.

    Google Scholar 

  57. G. L. Miller, S.-H. Teng, W. Thurston and S. A. Vavasis. Separators for sphere-packings and nearest neighborhood graphs. in progress 1992.

    Google Scholar 

  58. G. L. Miller, S.-H. Teng, W. Thurston and S. A. Vavasis. Finite element meshes and geometric separators. in progress 1992.

    Google Scholar 

  59. G. L Miller and W. Thurston. Separators in two and three dimensions. In Proceedings of the 22M Annual ACM Symposium on Theory of Computing,pages 300–309, Maryland, May 1990. ACM.

    Google Scholar 

  60. G. L. Miller and S. A. Vavasis. Density graphs and separators. In Second Annual ACM-SIAM Symposium on Discrete Algorithms,pages 331–336, San Francisco, January 1991. ACM-SIAM.

    Google Scholar 

  61. S. A. Mitchell and S. A. Vavasis. Quality mesh generation in three dimensions. Proc. ACM Symposium on Computational Geometry, pp 212–221, 1992.

    Google Scholar 

  62. M. S. Paterson. Tape bounds for time-bounded Turing machines. J. Comp. Syst. Sci., 6: 116124, 1972.

    Google Scholar 

  63. V. Pan and J. Reif. Efficient parallel solution of linear systems. In Proceedings of the 17th Annual ACM Symposium on Theory of Computing,pages 143–152, Providence, RI, May 1985. ACM.

    Google Scholar 

  64. A. Pothen and C.-J. Fan. Computing the block triangular form of a sparse matrix. ACM Transactions on Mathematical Software 16 (4), pp 303–324, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  65. A. Pothen, H. D. Simon, K.-P. Liou. Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal. Appl. 11 (3), pp 430–452, July, 1990.

    MATH  MathSciNet  Google Scholar 

  66. J. H. Reif and S. Sen. Polling: A new randomized sampling technique for computational geometry. In Proceedings of the 21st annual ACM Symposium on Theory of Computing. 394–404, 1989.

    Google Scholar 

  67. E. Schwabe, G. Bielloch, A. Feldmann, O. Ghattas, J. Gilbert, G. Miller, D. O’Hallaron, J. Schewchuk and S.-H. Teng. A separator-based framework for automated partitioning and mapping of parallel algorithms in scientific computing. In First Annual Dartmouth Summer Institute on Issues and Obstacles in the Practical Implementation of Parallel Algorithms and the use of Parallel Machines, 1992.

    Google Scholar 

  68. H. D. Simon. Partitioning of unstructured problems for parallel processing. Computing Systems in Engineering 2:(2/3), pp135–148.

    Google Scholar 

  69. G. Strang and G. J. Fix. An Analysis of the Finite Element Method, Prentice-Hall, 1973.

    Google Scholar 

  70. S.-H. Teng. Points, Spheres, and Separators: A Unified Geometric Approach to Graph Partitioning. PhD thesis, School of Computer Science, Carnegie Mellon University, Pittsburgh, 1991. CMU-CS-91–184.

    Google Scholar 

  71. C. Thomassen. Planarity and duality of finite and infinite graphs. Journal of Combinatorial Theory, Series B, 29: 244–271, 1980.

    MATH  MathSciNet  Google Scholar 

  72. J. F. Thompson, Z. U. A. Warsi and C. W. Mastin. Numerical Grid Generation: Foundations and Applications. New York, North Holland, 1985.

    MATH  Google Scholar 

  73. W. P. Thurston. The geometry and topology of 3-manifolds. Princeton University Notes, 1988.

    Google Scholar 

  74. W. T. Tutte. Convex representations of graphs. Proc. London Math. Soc. 10 (3): 304–320, 1960.

    Article  MATH  MathSciNet  Google Scholar 

  75. W. T. Tutte. How to draw a graph. Proc. London Math. Soc. 13 (3): 743–768, 1963.

    Article  MATH  MathSciNet  Google Scholar 

  76. J. D. Ullman. Computational Aspects of VLSI. Computer Science Press, Rockville MD, 1984.

    MATH  Google Scholar 

  77. P. Ungar. A theorem on planar graphs. Journal London Math Soc. 26: 256–262, 1951.

    Article  MATH  MathSciNet  Google Scholar 

  78. P. M. Vaidya. Constructing provably good cheap preconditioners for certain symmetric positive definite matrices. IMA Workshop on Sparse Matrix Computation: Graph Theory Issues and Algorithms, Minneapolis, Minnesota, October 1991.

    Google Scholar 

  79. L. G. Valiant. Universality consideration in VLSI circuits. IEEE Transaction on Computers, 30 (2): 135–140, February, 1981.

    MATH  MathSciNet  Google Scholar 

  80. V. N. Vapnik and A. Ya. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl., 16: 264–280, 1971.

    Article  MATH  Google Scholar 

  81. S. A. Vavasis. Automatic domain partitioning in three dimensions. SIAM J. Sci. Stat. Comp., 12 (1991) 950–970.

    Article  MATH  MathSciNet  Google Scholar 

  82. R. D. Williams. Performance of dynamic load balancing algorithms for unstructured mesh calculations. Technical Report, California Institute of Technology, 1990.

    Google Scholar 

  83. F.-F. Yao. A 3-space partition and its application. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, ACM, 258–263, 1983.

    Google Scholar 

  84. E. E. Zmijewski. Sparse Cholesky Factorization on a Multiprocessor. PhD thesis, Department of Computer Science, Cornell University, 1987.

    Google Scholar 

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Author information

Author notes

  1. Shang-Hua Teng

    Present address: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

Authors and Affiliations

  1. School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Gary L. Miller

  2. Palo Alto Research Center, Xerox Corporation, Palo Alto, CA, 94304, USA

    Shang-Hua Teng

  3. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    William Thurston

  4. Department of Computer Science, Cornell University, Ithaca, NY, 14853, USA

    Stephen A. Vavasis

Authors
  1. Gary L. Miller
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  2. Shang-Hua Teng
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  3. William Thurston
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  4. Stephen A. Vavasis
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Editor information

Editors and Affiliations

  1. University of Waterloo, Needles Hall, Waterloo, Ontario, N2L 3G1, Canada

    Alan George

  2. Xerox Palo Alto Research Center, 3333 Coyote Hill Road, Palo Alto, CA, 94304-1314, USA

    John R. Gilbert

  3. Department of Computer Science, York University, North York, Ontario, M3J 1P3, Canada

    Joseph W. H. Liu

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Miller, G.L., Teng, SH., Thurston, W., Vavasis, S.A. (1993). Automatic Mesh Partitioning. In: George, A., Gilbert, J.R., Liu, J.W.H. (eds) Graph Theory and Sparse Matrix Computation. The IMA Volumes in Mathematics and its Applications, vol 56. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8369-7_3

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  • DOI: https://doi.org/10.1007/978-1-4613-8369-7_3

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