Abstract
We give the first linear-time algorithm for computing single-source shortest paths in a weighted interval or circular-arc graph, when we are given the model of that graph, i.e., the actual weighted intervals or circular-arcs and the sorted list of the interval endpoints. Our algorithm solves this problem optimally in O(n) time, where n is the number of intervals or circular-arcs in a graph. An immediate consequence of our result is an O(qn + n log n) time algorithm for the minimum-weight circle-cover problem, where q is the minimum number of arcs crossing any point on the circle; the n log n term in this time complexity is from a preprocessing sorting step when the sorted list of endpoints is not given as part of the input. The previously best time bounds were O(n log n) for this shortest paths problem, and O(qn log n) for the minimum-weight circle-cover problem. Thus we improve the bounds of both problems. More importantly, the techniques we give hold the promise of achieving similar log n-factor improvements in other problems on such graphs.
This research was supported in part by the Leonardo Fibonacci Institute in Trento, Italy.
Research supported in part by the Air Force Office of Scientific Research under Contract AFOSR-90-0107 and by the National Science Foundation under Grant CCR-9202807.
Research supported in part by the National Science Foundation under Grant CCR-8901815.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
A. V. Aho, J. E. Hopcroft, and J. D. Ullman. The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, Massachusetts, 1974.
M. J. Atallah and D. Z. Chen. “An optimal parallel algorithm for the minimum circle-cover problem,” Information Processing Letters, 32 (1989), pp. 159–165.
A. A. Bertossi. “Parallel circle-cover algorithms,” Information Processing Letters, 27 (1988), pp. 133–139.
K. S. Booth and G. S. Lukeher. “Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms,” Journal of Computer and System Sciences, 13 (1976), pp. 335–379.
D. Z. Chen and D. T. Lee. “Solving the all-pair shortest path problem on interval and circulararc graphs,” Technical Report No. 93-3, Department of Computer Science and Engineering, University of Notre Dame, May 1993.
H. N. Gabow and R. E. Tarjan. “A linear-time algorithm for a special case of disjoint set union,” Journal of Computer and System Sciences, 30 (1985), pp. 209–221.
M. C. Golumbic. Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.
U. I. Gupta, D.T. Lee, and J. Y.-T. Leung. “Efficient algorithms for interval graphs and circular-arc graphs,” Networks, Vol. 12 (1982), pp. 459–467.
O. H. Ibarra, H. Wang, and Q. Zheng. “Minimum cover and single source shortest path problems for weighted interval graphs and circular-arc graphs,” Proc. of 30th Annual Allerton Conf. on Commun., Contr., and Comput., 1992, Univ. of Illinois, Urbana, pp. 575–584.
C. C. Lee and D. T. Lee. “On a circle-cover minimization problem,” Information Processing Letters, 18 (1984), pp. 109–115.
R. Ravi, M.V. Marathe, and C.P. Rangan. “An optimal algorithm to solve the all-pair shortest path problem on interval graphs,” Networks, Vol. 22 (1992), pp. 21–35.
M. Sarrafzadeh and D. T. Lee, “Restricted track assignment with applications,” Int'l Journal Computational Geometry & Applications, to appear.
R. E. Tarjan. “A class of algorithms which require nonlinear time to maintain disjoint sets,” set union algorithm,” Journal of Computer and System Sciences, 18 (2) (1979), pp. 110–127.
P. Van Emde Boas. “Preserving order in a forest in less than logarithmic time and linear space,” Information Processing Letters, 6 (3) (1977), pp. 80–82.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Atallah, M.J., Chen, D.Z., Lee, D.T. (1993). An optimal algorithm for shortest paths on weighted interval and circular-arc graphs, with applications. In: Lengauer, T. (eds) Algorithms—ESA '93. ESA 1993. Lecture Notes in Computer Science, vol 726. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57273-2_40
Download citation
DOI: https://doi.org/10.1007/3-540-57273-2_40
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-57273-2
Online ISBN: 978-3-540-48032-7
eBook Packages: Springer Book Archive