Abstract
A parallel (CRCW PRAM) algorithm is given to find a k-coloring of a graph randomly drawn from the family of k-colorable graphs with nvertices, where k = logO(1)n. The average running time of the algorithm is constant, and the number of processors is equal to ¦V¦ + ¦E¦, where ¦V¦, ¦E¦, resp. is the number of vertices, edges, resp. of the input graph.
This research was partially supported by EC Cooperative Action IC-1000 (project ALTEC: Algorithms for Future Technologies)
Preview
Unable to display preview. Download preview PDF.
References
M. Ajtai, and M. Ben-Or, A theorem on probabilistic constant depth computation, Proceedings of the 16th Symposium on Theory of Computing, (1984), 471–474.
N. Alon, J. Spencer, and P. Erdős, The probabilistic Method, J. Wiley and Sons, New York, 1992.
B. Bollobás, The chromatic number of random graphs, Combinatorica 8, 49–56.
B. Brustmann, and I. Wegener, The complexity of symmetric functions in bounded depth circuits, Information Processing Letters 25 (1987), 217–219.
H. Chernoff, A measure of asymptotic efficiency for tests based on the sum of observations, Ann. Math. Statist. 23 (1952), 493–509.
B. Chlebus, K. Diks, T. Hagerup, and T. Radzik, Efficient simulations between CRCW PRAMs, Proc. 13th Symp. on the Mathematical Foundations of Computer Science, 1988, 230–239.
D. Coppersmith, P. Raghavan, and M. Tompa, Parallel graph algorithms that are efficient on average, Proceedings of the 28th Annual IEEE Conference on Foundations of Computer Science, (1987), 260–269.
M. Dyer, and A. Frieze, The solution of some random NP-hard problems in polynomial expected time, J Algorithms 10 (1989), 451–489.
A. Frieze, and L. Kučera, Parallel colouring of random graphs, in Random Graphs 87, M. Karonski, J. Jaworski, A. Rucinski, eds, J. Wiley 1990, 41–52
M. R. Garey, and D. S. Johnson, The complexity of near optimal graph coloring, JACM 23 (1976), 43–49.
G.R. Grimmett, and C. J. H. McDiarmid, On colouring random graphs, Math. Proc. Cambridge Phil. Soc., 77 (1975), 313–324.
J. Håstad, I. Wegener, N. Wurm, and S.-Z. Yi, Optimal depth, very small size circuits for symmetric functions in AC0, Tech. Rep. 384, Univ. Dortmund (1991).
L. Kučera, Expected behavior of graph coloring algorithms, in FCT'77, M. Karpinski, ed., Lecture Notes in Computer Science 56 (Springer, Berlin, 1977), 447–451.
L. Kučera, Parallel computation and conflicts in memory access, Information Processing Letters 14 (1982), 93–96.
L. Kučera, Graphs with small chromatic numbers are easy to color, Information Processing Letters 30 (1989), 233–236.
L. Kučera, The greedy coloring is a bad probabilistic algorithm, J Algorithms, 12 (1991), 674–684.
L. Levin, Average case complete problems, SIAM J. Computing 15 (1986), 285–286.
D. Matula, and L. Kučera, An expose-and-merge algorithm and the chromatic number of a random graph, in Random Graphs'87, M.Karonski, J.Jaworski, A.Rucinski, eds., (J.Wiley and Sons, 1990), 175–187.
S. Moran, Generalized lower bound derived from Håstad's main lemma, Information Processing Letters 25 (1987), 383–388.
P. Ragde, The parallel simplicity of compaction and chaining, ICALP'90, Lecture Notes in Computer Science 443, 744–751, 1990.
L. Stockmayer, The complexity of approximate computing, Proceedings of the 15th Symposium on Theory of Computing (1983), 118–126.
J. Turner, Almost all k-colorable graphs are easy to color, J. Algorithms 9 (1988), 63–82.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kučera, L. (1994). Coloring k-colorable graphs in constant expected parallel time. In: van Leeuwen, J. (eds) Graph-Theoretic Concepts in Computer Science. WG 1993. Lecture Notes in Computer Science, vol 790. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57899-4_50
Download citation
DOI: https://doi.org/10.1007/3-540-57899-4_50
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-57899-4
Online ISBN: 978-3-540-48385-4
eBook Packages: Springer Book Archive