ABSTRACT
A maximal independent set in a hypergraph is a subset of vertices that is maximal with respect to the property of not containing any edge of the hypergraph. We show that an algorithm proposed by Beame and Luby is in randomized NC for hypergraphs in which the maximum edge size is bounded by a constant. To prove this, we bound the upper tail of sums of dependent random variables defined on the edges of a hypergraph. These bounds may be viewed as extensions of bounds on the tail of the binomial distribution. We derandomize this algorithm to obtain the first sublinear time deterministic algorithm for hypergraphs with edges of size O(1). The algorithm exhibits the following time-processor tradeoff: it can be made to run in time O(nε) with nO(1/ε) processors for a hypergraph on n vertices, for any ε ≥ 2d+1• (log log n)/(log n); here d = O(1) denotes the maximum size of an edge in H. In particular, for any constant ε > O, we have an algorithm running in time O(nε) on a polynomial number of processors, and we have an algorithm running in time (log n)O(1) on nO(log n/log log n) processors.
- 1.N. Alon, L. Babai, A. itai, A fast randomized parallel algorithm for the maximal independent set problem, J. Algorithms 7, pp. 567-583, 1986. Google ScholarDigital Library
- 2.P. Beame, Personal Communication, 1991.Google Scholar
- 3.P. Beame, M. Luby, Parallel Search for Maximal independence Given Minimal Dependence, Proceedings of the First SODA Conference, 1990, pp. 212- 218. Google ScholarDigital Library
- 4.B. Berger, J. Rompel, Simulating (log# n)-wise Independence in NC, JACM, vol. 38, 1991, pp. 1026- 1046. Google ScholarDigital Library
- 5.E. Dahlhaus, M. Karpinski, An efficient algorithm for the 3MIS problem, Technical report TR-89-052, September 1989, International Computer Science institute, Berkeley, CA.Google Scholar
- 6.E. Dahlhaus, M. Karpinski, P. Kelsen, An Efficient Parallel Algorithm for Computing a Maximal Independent Set in a Hypergraph of Dimension 3, information Processing Letters, to appear. Google ScholarDigital Library
- 7.M. Goldberg, T. Spencer, A new parallel algorithm for the maximal independent set problem, SIAM J. Computing, vol. 18, 1989, pp.419-427. Google ScholarDigital Library
- 8.M. Goldberg, T. Spencer, Constructing a Maximal Independent Set in Parallel, SIAM J. Disc. Math., vol. 2, 1989, pp. 322-328. Google ScholarDigital Library
- 9.R. Karp, V. Ramachandran, Parallel algorithms for shared memory machines, in Handbook of Theoretical Computer Science, J. Van Leeuwen, ed., North Holland, 1990, pp.869-941. Google ScholarDigital Library
- 10.R. Karp, A. Wigderson, A fast parallel algorithm for the maximal independent set problem, JACM 32 (1985), pp. 762-773. Google ScholarDigital Library
- 11.R. Karp, E. Upfal, A. Wigderson, The complexity of parallel search, JCSS, vol. 36, 1988, pp.225-253. Google ScholarDigital Library
- 12.P. Kelsen, An efficient parallel algorithm for finding an mis in hypergraphs of dimension 3, manuscript, Department of Computer Sciences, University of Texas, Austin, TX, January 1990.Google Scholar
- 13.M. Luby, A simple parallel algorithm for the maximal independent set problem, SIAM J. Computing, vol. 15, 1986, pp. 1036-1053. Google ScholarDigital Library
- 14.M. Luby, Removing randomness in parallel computation without a processor penalty, 29th FOCS, 1988, pp. 162-173.Google ScholarDigital Library
- 15.R. Motwani, J. Naor, M. Naor, The probabilistic method yields deterministic parallel algorithms, 30th FOCS, 1989, pp. 8-13. Google ScholarDigital Library
- 16.P. Raghavan, Probabilistic construction of deterministic algorithms: Approximating packing integer programs, JCSS, vol. 37, 1988, pp. 130-143. Google ScholarDigital Library
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- On the parallel complexity of computing a maximal independent set in a hypergraph
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