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Expected parallel time and sequential space complexity of graph and digraph problems

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Abstract

This paper determines upper bounds on the expected time complexity for a variety of parallel algorithms for undirected and directed random graph problems. For connectivity, biconnectivity, transitive closure, minimum spanning trees, and all pairs minimum cost paths, we prove the expected time to beO(log logn) for the CRCW PRAM (this parallel RAM machine allows resolution of write conflicts) andO(logn · log logn) for the CREW PRAM (which allows simultaneous reads but not simultaneous writes). We also show that the problem of graph isomorphism has expected parallel timeO(log logn) for the CRCW PRAM andO(logn) for the CREW PRAM. Most of these results follow because of upper bounds on the mean depth of a graph, derived in this paper, for more general graphs than was known before.

For undirected connectivity especially, we present a new probabilistic algorithm which runs on a randomized input and has an expected running time ofO(log logn) on the CRCW PRAM, withO(n) expected number of processors only.

Our results also improve known upper bounds on the expected space required for sequential graph algorithms. For example, we show that the problems of finding connected components, transitive closure, minimum spanning trees, and minimum cost paths have expected sequential spaceO(logn · log logn) on a deterministic Turing Machine. We use a simulation of the CRCW PRAM to get these expected sequential space bounds.

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Authors and Affiliations

  1. Department of Computer Science, Duke University, 27706, Durham, NC, USA

    John Reif

  2. Computer Technology Institute, P.O. Box 1122, 26110, Patras, Greece

    Paul Spirakis

  3. Computer Science Department, Courant Institute of Mathematical Sciences, New York University, 10012, New York, NY, USA

    Paul Spirakis

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  1. John Reif
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  2. Paul Spirakis
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Additional information

Communicated by Greg N. Frederickson.

This research was supported by National Science Foundation Grant DCR-85-03251 and Office of Naval Research Contract N00014-80-C-0647.

This research was partially supported by the National Science Foundation Grants MCS-83-00630, DCR-8503497, by the Greek Ministry of Research and Technology, and by the ESPRIT Basic Research Actions Project ALCOM.

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Reif, J., Spirakis, P. Expected parallel time and sequential space complexity of graph and digraph problems. Algorithmica 7, 597–630 (1992). https://doi.org/10.1007/BF01758779

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  • DOI: https://doi.org/10.1007/BF01758779

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