Low diameter graph decompositions

Abstract

Adecomposition of a graphG=(V,E) is a partition of the vertex set into subsets (calledblocks). Thediameter of a decomposition is the leastd such that any two vertices belonging to the same connected component of a block are at distance ≤d. In this paper we prove (nearly best possible) statements, of the form: Anyn-vertex graph has a decomposition into a small number of blocks each having small diameter. Such decompositions provide a tool for efficiently decentralizing distributed computations. In [4] it was shown that every graph has a decomposition into at mosts(n) blocks of diameter at mosts(n) for\(s(n) = n^{O(\sqrt {\log \log n/\log n)} }\). Using a technique of Awerbuch [3] and Awerbuch and Peleg [5], we improve this result by showing that every graph has a decomposition of diameterO (logn) intoO(logn) blocks. In addition, we give a randomized distributed algorithm that produces such a decomposition and runs in timeO(log2 n). The construction can be parameterized to provide decompositions that trade-off between the number of blocks and the diameter. We show that this trade-off is nearly best possible, for two families of graphs: the first consists of skeletons of certain triangulations of a simplex and the second consists of grid graphs with added diagonals. The proofs in both cases rely on basic results in combinatorial topology, Sperner's lemma for the first class and Tucker's lemma for the second.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    Y. Afek andM. Ricklin: Sparser: A paradigm for running distributed algorithms, 6th International Workshop, on Distributed Algorithms, Haifa, Israel November 1992, Springer-Verlag. (J. of Algorithms, in press).

  2. [2]

    N. Alon, L. Babai, andA. Itai: A fast and simple randomized parallel algorithm for the maximal independent set problem,J. of Algorithms 7 (1986), 567–583.

    Google Scholar 

  3. [3]

    B. Awerbuch: Complexity of network synchronization,J. ACM 32 (1985), 804–823.

    Google Scholar 

  4. [4]

    B. Awerbuch, A. Goldberg. M. Luby, andS. Plotkin: Network decomposition and locality in distributed computation,Proc. 30th IEEE Symp. on Foundations of Comp. Sci. (1989) 364–369.

  5. [5]

    B. Awerbuch andD. Peleg: Sparse partitions,FOCS 31 (1990), 503–513.

    Google Scholar 

  6. [6]

    I. F. Blake andR. C. Mullin:An Introduction to Algebraic and Combinatorial Coding Theory, Academic Press, New York, 1976.

    Google Scholar 

  7. [7]

    B. Bollobás:Extremal graph theory, Academic Press, New York, 1987.

    Google Scholar 

  8. [8]

    R. Cole andU. Vishkin: Deterministic coin tossing and accelerating cascades: micro and macro techniques for designing parallel algorithms,Proc. 18th ACM Sump. on Theory of Computing (1986) 206–219.

  9. [9]

    R. M. Freund andM. J. Todd: A constructive proof of Tucker's combinatorial lemma.J. Comb. Theory A 30 (1981) 321–325.

    Google Scholar 

  10. [10]

    D. Gale: The game of hex and the Brouwer, fixed-point theorem,Amer. Math. Month. 86 (1979) 818–827.

    Google Scholar 

  11. [11]

    A. V. Goldberg., S. V. Plotkin andG. E. Shannon: Parallel symmetry-breaking in sparsed graphs,SIAM J. Disc. Math. 1 (1989) 434–446.

    Google Scholar 

  12. [12]

    C. Kaklamanis, A. R. Karlin, F. T. Leighton, V. Milenkovic, P. Raghavan, S. Rao, C., Thomborson, A. Tsantilas: Asymptotically tight bounds for computing with faulty arrays of processors,FOCS 31 (1990), 285–296.

    Google Scholar 

  13. [13]

    N. Linial: Locality in distributed graph alorithms,SIAM Journal on Computing,21 (1992) 193–201., Preliminary version: N. Linial, Distributive algorithmsglobal solutions from local data,FOCS 28 (1987), 331–335.

    Google Scholar 

  14. [14]

    M. Luby: A simple parallel algorithm for the maximal independent set problem,SIAM J. on Computing,15 (1986) 1036–1053.

    Google Scholar 

  15. [15]

    E. H. Spanier:Algebraic Topology, McGraw-Hill, New York, 1966.

    Google Scholar 

  16. [16]

    M. Todd:The computation of fixed points and applications, Lecture Notes in Econnomics and Mathematical Systems, 124, Springer-Verlag, 1976.

  17. [17]

    B. Weiss: A combinatorial proof of the Borsuk-Ulam antipodal point theorem,Israel J. Math. 66 (1989) 364–368.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

A preliminary version of this paper appeared as “Decomposing Graphs into Regions of Small Diameter” in Proc. 2nd ACM-SIAM Symposium on Discrete Algorithms (1991) 321-330.

This work was supported in part by NSF grant DMS87-03541 and by a grant from the Israel Academy of Science.

This work was supported in part by NSF grant DMS87-03541 and CCR89-11388.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Linial, N., Saks, M. Low diameter graph decompositions. Combinatorica 13, 441–454 (1993). https://doi.org/10.1007/BF01303516

Download citation

AMS subject classification codes (1991)

  • 05 C 12
  • 05 C 15
  • 05 C 35
  • 05 C 70
  • 05 C 85
  • 68 Q 22, 68 R 10