## Abstract

A*decomposition* of a graph*G=(V,E)* is a partition of the vertex set into subsets (called*blocks*). The*diameter* of a decomposition is the least*d* 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: Any*n*-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 most*s(n)* blocks of diameter at most*s(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 diameter*O* (log*n*) into*O*(log*n*) blocks. In addition, we give a randomized distributed algorithm that produces such a decomposition and runs in time*O*(log^{2}*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.

## 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## Preview

Unable to display preview. Download preview PDF.

## References

- [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).Google Scholar - [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]
- [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.Google Scholar - [5]
- [6]I. F. Blake andR. C. Mullin:
*An Introduction to Algebraic and Combinatorial Coding Theory*, Academic Press, New York, 1976.Google Scholar - [7]
- [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.Google Scholar - [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]D. Gale: The game of hex and the Brouwer, fixed-point theorem,
*Amer. Math. Month.***86**(1979) 818–827.Google Scholar - [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]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]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]M. Luby: A simple parallel algorithm for the maximal independent set problem,
*SIAM J. on Computing*,**15**(1986) 1036–1053.Google Scholar - [15]
- [16]M. Todd:
*The computation of fixed points and applications*, Lecture Notes in Econnomics and Mathematical Systems, 124, Springer-Verlag, 1976.Google Scholar - [17]B. Weiss: A combinatorial proof of the Borsuk-Ulam antipodal point theorem,
*Israel J. Math.***66**(1989) 364–368.Google Scholar