# Optimal parallel 3-colouring algorithm for rooted trees and its application

Part III Communications

First Online:

## Abstract

A new optimal parallel algorithm for 3-colouring rooted trees with maximum degree Δ is presented. The algorithm runs in O(Δ log *n*/log log *n*) time on a CRCW PRAM using O(Δ *n* log log *n*/log *n*) processors. This technique is used to develop optimal algorithms for several graph problems including (Δ+1)-colouring of constant degree graphs, 7-colouring of planar graphs or finding a maximal independent set in a planar graph. The technique can be applied to expression tree evaluation as well and yields an optimal logarithmic time algorithm.

## Preview

Unable to display preview. Download preview PDF.

## References

- [ADKP]K. Abrahamson, N. Dadoun, D.G. Kirkpatrick and T. Przytycka, "A simple parallel tree contraction algorithm", J. Algorithms, 10 (1989), pp.287–302.CrossRefGoogle Scholar
- [CV1]R. Cole and U. Vishkin, "The accelerated centroid decomposition technique for optimal parallel tree evaluation in logarithmic time", Algorithmica, 3 (1988a), pp. 329–346.CrossRefGoogle Scholar
- [CV2]R. Cole and U. Vishkin, "Faster optimal parallel prefix sums and list ranking", Inform. and Comput., 81 (1989), pp.334–352.Google Scholar
- [GMT]H. Gazit, G.L. Miller and S.H. Teng, "Optimal tree contraction in an EREW model", in S.K. Tewkesbury, B.W. Dickinson and S.C. Schwartz, editors, Concurrent Computations: Algorithms, Architecture and Technology, Plenum Press, New York, 1988.Google Scholar
- [GR1]A. Gibbons and W. Rytter, "Efficient parallel algorithms", Cambridge University Press, Cambridge, 1988.Google Scholar
- [GR2]A. Gibbons and W. Rytter, "Optimal parallel algorithms for dynamic expression evaluation and context-free recognition", Inform. and Comput., 81 (1989), pp.32–45.Google Scholar
- [GPS]A. V. Goldberg, S. A. Plotkin and G. E. Shannon, "Parallel symmetry — breaking in sparse graphs", SIAM J. Disc. Math., 1 (1988), pp.434–446.Google Scholar
- [GS]M. Goldberg and T. Spencer, "Constructing a maximal independent set in parallel", SIAM J. Disc. Math., 2 (1989), pp.322–328.Google Scholar
- [Hag]T. Hagerup, "Optimal parallel algorithms on planar graphs", in J.H. Reif, editor, VLSI Algorithms and Architectures, Lecture Notes in Computer Science 319, Springer-Verlag, New York, Berlin, 1988, pp. 24–32.Google Scholar
- [Har]F. Harary, "Graph theory", Addison Wesley, Reading, Mass., 1969.Google Scholar
- [KD]S.R. Kosaraju and A.L. Delcher, "Optimal parallel evaluation of tree — structured computations by raking", in J.H. Reif, editor, VLSI Algorithms and Architectures, Lecture Notes in Computer Science, 319, Springer-Verlag, New York, Berlin, 1988, pp.101–110.Google Scholar
- [Lu]M. Luby, "Removing randomness in parallel computation without a processor penalty", in Proc. 29th Annual IEEE Symp. on Foundations of Computer Science, 1988, pp.162–173.Google Scholar
- [MR]G.L.Miller and J.H. Reif, "Parallel tree contraction and its application", in Proc. 26th Annual IEEE Symp. on Foundations of Computer Science, 1985, pp.478–489.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1990