Advertisement

Fast parallel algorithms for coloring random graphs

  • Zvi M. Kedem
  • Krishna V. Palem
  • Grammati E. Pantziou
  • Paul G. Spirakis
  • Christos D. Zaroliagis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 570)

Abstract

We improve here the expected performance of parallel algorithms for graph coloring. This is achieved through new adaptive techniques that may be useful for the average-case analysis of many graph algorithms. We apply our techniques to:
  1. (a)

    the class G n,p of random graphs. We present a parallel algorithm which colors the graph with a number of colors at most twice its chromatic number and runs in time O(log4n/ log log n) almost surely, for p = Ω((log(3)n)2/ log(2)n). The number of processors used is O(m) where m is the number of edges of the graph.

     
  2. (b)

    the class of all k-colorable graphs, uniformly chosen. We present a parallel algorithm which actually constructs the coloring in expected parallel time O(log2n), for constant k, by using O(m) processors on the average. This problem is not known to have a polynomial time algorithm in the worst case.

     

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    B. Bollobas, “Random Graphs”, Academic Press, London, 1985.Google Scholar
  2. [2]
    H. Chernoff, “A measure of asymptotic efficiency for tests based on the sum of observations”, Ann. Math. Statist. 23 (1952), 493–509.Google Scholar
  3. [3]
    A. Calkin, A. Frieze, “Probabilistic Analysis of a Parallel Algorithm for Finding Maximal Independent Sets”, Random Structures & Algorithms, Vol.1, No.l, 39–50, 1990.Google Scholar
  4. [4]
    D. Coppersmith, P. Raghavan, M. Tompa, “Parallel Graph Algorithms that are Efficient on Average”, Proc. of the 28th Annual IEEE FOCS, 1987, pp.260–269.Google Scholar
  5. [5]
    M. E. Dyer, A. M. Frieze, “The Solution of Some Random NP-hard Problems in Polynomial Expected Time”, Journal of Algorithms, 10, 451–489, 1989.Google Scholar
  6. [6]
    P. Erdos, A. Renyi, “On random graphs I”, Publ. Math. Debrecen, 6 (1959), 290–297.Google Scholar
  7. [7]
    Z. Kedem, K. Palem, P. Spirakis, “Adaptive average case analysis”, unpublished manuscript, 1990.Google Scholar
  8. [8]
    L. Kucera, “Expected behaviour of graph coloring algorithms”, Proc. of Fundamentals in Computation Theory LNCS, Vol.56, pp.447–451, Springer-Verlag, 1977.Google Scholar
  9. [9]
    J.E. Littlewood, “On the probability in the tail of a binomial distribution”, Adv. Appl. Probab., 1(1969), 43–72.Google Scholar
  10. [10]
    G. Pantziou, P. Spirakis, C. Zaroliagis, “Coloring Random Graphs Efficiently in Parallel, through Adaptive Techniques”, CTI TR-90.10.25, Computer Technology Institute, Patras. Also presented in the ALCOM Workshop on Graph Algorithms, Data Structures and Computational Geometry, Berlin 3–5 October, 1990.Google Scholar
  11. [11]
    J. Spencer, “Ten Lectures on the Probabilistic Method”, SIAM, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Zvi M. Kedem
    • 1
  • Krishna V. Palem
    • 2
  • Grammati E. Pantziou
    • 3
  • Paul G. Spirakis
    • 1
    • 3
    • 4
  • Christos D. Zaroliagis
    • 3
    • 4
  1. 1.Dept of Computer ScienceCourant Institute of Math. Sciences NYUNew YorkUSA
  2. 2.IBM Research DivisionT.J. Watson Research CenterYorktown HeightsUSA
  3. 3.Computer Technology InstitutePatrasGreece
  4. 4.Computer Sc and Eng DeptUniversity of PatrasGreece

Personalised recommendations