Abstract
The problem of coloring a set of n intervals (from the real line) with a set of k colors is studied. In such a coloring, two intervals may have the same color if and only if those intervals do not overlap. Two versions of the problem are considered. For the first, we provide an O(k+n) time algorithm for k-coloring a maximum cardinality subset of the intervals. The best previous algorithm for this problem required time O(kn). In the second version, we assume that each interval has a weight, and provide an O(knlogn) algorithm for k-coloring a set of intervals of maximum total weight. The best previous algorithm for this problem required time O(n2logn). These results provide improved solutions to problems of local register allocation, task scheduling, and the routing of nets on a chip.
(Extended Abstract)
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
Arkin, E.M. and E.B. Silverberg, "Scheduling jobs with fixed start and end times," Discrete Applied Mathematics 18 (1987), 1–8.
Belady, L.A., "A study of replacement algorithms for a virtual-storage computer," IBM Systems Journal 5 (1966).
Booth, K.S. and G.S. Leuker, "Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms," J. Comp. and Sys. Sci. 13 (1976), 335–379.
Edmonds, J. and R.M. Karp, "Theoretical improvements in algorithmic efficiency for network flow problems," JACM 19 (1972), 248–264.
Fredman, M.L. and R.E. Tarjan, "Fibonacci heaps and their uses in improved network optimization algorithms," JACM 34 (1987), 596–615.
Golumbic, M.C., Algorithmic Graph Theory and Perfect Graphs, Academic Press, 1980.
Gabow, H.N. and R.E. Tarjan, "A linear-time algorithm for a special case of disjoint set union," J. Comp. and Sys. Sci. 30 (1985), 209–221.
Hashimoto, A. and J. Stevens, "Wire routing by optimizing channel assignment within large apertures," Proc. 8th IEEE Design Automation Workshop (1971), 155–169.
Karp, R.M., "Reducibility among combinatorial problems," in Complexity of Computer Computations, R.E. Miller and J.W. Thatcher, editors, Plenum Press, 1972.
Tarjan, R.E., "Efficiency of a good but not linear set union algorithm," JACM 22 (1975), 215–225.
Yannakakis, M. and F. Gavril, "The maximum k-colorable subgraph problem for chordal graphs," Information Processing Letters 24 (1987), 133–137.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Carlisle, M.C., Lloyd, E.L. (1991). On the k-coloring of intervals. In: Dehne, F., Fiala, F., Koczkodaj, W.W. (eds) Advances in Computing and Information — ICCI '91. ICCI 1991. Lecture Notes in Computer Science, vol 497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54029-6_157
Download citation
DOI: https://doi.org/10.1007/3-540-54029-6_157
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-54029-8
Online ISBN: 978-3-540-47359-6
eBook Packages: Springer Book Archive