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Min-Max-Boundary Domain Decomposition

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Computing and Combinatorics (COCOON 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1449))

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Abstract

Domain decomposition is one of the most effective and popular parallel computing techniques for solving large scale numerical systems. In the special case when the amount of computation in a subdomain is proportional to the volume of the subdomain, domain decomposition amounts to minimizing the surface area of each subdomain while dividing the volume evenly. Motivated by this fact, we study the following min-max boundary multi-way partitioning problem: Given a graph G and an integer k > 1, we would like to divide G into k subgraphs G 1, . . . , G k (by removing edges) such that (i) |G i| = Θ(|G|/k) for all i ∈ 1, . . . , k; and (ii) the maximum boundary size of any subgraph (the set of edges connecting it with other subgraphs) is minimized.

We provide an algorithm that given G, a well-shaped mesh in d dimensions, finds a partition of G into k subgraphs G 1, . . . , G k, such that for all i, G i has Θ(|G|/k) vertices and the number of edges connecting G i with the other subgraphs is O((|G|/k)1−1/d). Our algorithm can find such a partition in O(|G| log k) time. Finally, we extend our results to vertex-weighted and vertex-based graph decomposition. Our results can be used to simultaneously balance the computational and memory requirement on a distributed-memory parallel computer without sacrificing the communication overhead.

Part of this work was done while Daniel Spielman and Shang-Hua Teng were visiting Universidad de Chile.

Supported in part by Fondecyt No. 1981182, and Fondap in Applied Mathematics 1998.

Supported in part by an Alfred P. Sloan Research Fellowship

Supported in part by an NSF CAREER award (CCR-9502540), an Alfred P. Sloan Research Fellowship, and an Intel research grant.

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Author information

Authors and Affiliations

  1. Dept. de Ingeniería Matemática, Fac. de Ciencias Físicas y Matemáticas, U. de Chile, Casilla 170/3, Correo 3, Santiago, Chile

    Marcos Kiwi

  2. Department of Mathematics, M.I.T., Cambridge, MA, 02139, USA

    Daniel A. Spielman

  3. Department of Computer Science, University of Illinois, Urbana, IL, 61801, USA

    Shang-Hua Teng

Authors
  1. Marcos Kiwi
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  2. Daniel A. Spielman
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  3. Shang-Hua Teng
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Editor information

Editors and Affiliations

  1. Institute of Information Science, Academia Sinica, 11529, Nankang, Taipei, Taiwan

    Wen-Lian Hsu

  2. Department of Computer Science, Yale University, 51 Prospect Street, New Haven, CT, 06520, USA

    Ming-Yang Kao

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© 1998 Springer-Verlag Berlin Heidelberg

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Kiwi, M., Spielman, D.A., Teng, SH. (1998). Min-Max-Boundary Domain Decomposition. In: Hsu, WL., Kao, MY. (eds) Computing and Combinatorics. COCOON 1998. Lecture Notes in Computer Science, vol 1449. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-68535-9_17

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  • DOI: https://doi.org/10.1007/3-540-68535-9_17

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64824-6

  • Online ISBN: 978-3-540-68535-7

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