Abstract
We prove super-linear lower bounds for some shortest path problems in directed graphs, where no such bounds were previously known. The central problem in our study is the replacement paths problem: Given a directed graph G with non-negative edge weights, and a shortest path P = e 1, e 2, . . . , e p between two nodes s and t, compute the shortest path distances from s to t in each of the p graphs obtained from G by deleting one of the edges e i . We show that the replacement paths problem requires ω(m√n) time in the worst case whenever m = O(n√n). Our construction also implies a similar lower bound for the k shortest paths problem for a broad class of algorithms that includes all known algorithms for the problem. To put our lower bound in perspective, we note that both these problems (replacement paths and k shortest paths) can be solved in near linear time for undirected graphs.
Supported in part by NSF grants IIS-0121562 and CCR-9901958.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
N. Alon, Z. Galil, and O. Margalit. On the exponent of the all pairs shortest path problem. In 32nd Symposium on Foundations of Computer Science, 569–575, 1991.
A. Archer. Private communication, 2001.
A. Archer and E. Tardos. Frugal path mechanisms. In Proc. 13th Annual ACMSIAM Symposium on Discrete Algorithms, pages 991–999, 2002.
M. O. Ball, B. L. Golden, and R. V. Vohra. Finding the most vital arcs in a network. Oper. Res. Letters, 8:73–76, 1989.
A. Bar-Noy, S. Khuller, and B. Schieber. The complexity of finding most vital arcs and nodes. Technical Report CS-TR-3539, Institute for Advanced Studies, University of Maryland, College Park, MD, 1995.
A. Brander and M. Sinclair. A comparative study of K-shortest path algorithms. In Proc. of 11th UK Performance Engineering Workshop, pages 370–379, 1995.
D. Eppstein. Finding the k shortest paths. SIAM J. Computing, 28(2):652–673, 1998.
B. Fortz and M. Thorup. Internet traffic engineering by optimizing OSPF weights. In INFOCOM, pages 519–528, 2000.
M. Fredman. New bounds on the complexity of the shortest path problem. SIAM J. of Computing, 5:83–89, 1976.
E. Hadjiconstantinou and N. Christofides. An efficient implementation of an algorithm for finding K shortest simple paths. Networks, 34(2):88–101, September 1999.
J. Hershberger and S. Suri. Vickrey prices and shortest paths: What is an edge worth? In Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science, pages 252–259, 2001.
J. Hershberger, M. Maxel, and S. Suri. Finding the k Shortest Simple Paths: A New Algorithm and its Implementation. To appear in Proc. of ALENEX, 2003.
D. R. Karger, D. Koller, and S. J. Phillips. Finding the hidden path: Time bounds for all-pairs shortest paths. SIAM J. Comput., 22:1199–1217, 1993.
N. Katoh, T. Ibaraki, and H. Mine. An efficient algorithm for k shortest simple paths. Networks, 12:411–427, 1982.
D. Kirkpatrick and S. Reisch. Upper bounds for sorting integers on random access machines. Theoretical Computer Science, 28(3):263–276, 1984.
E. L. Lawler. A procedure for computing the K best solutions to discrete optimization problems and its application to the shortest path problem. Management Science, pages 401–405, 1972.
K. Malik, A. K. Mittal, and S. K. Gupta. The k most vital arcs in the shortest path problem. Oper. Res. Letters, 8:223–227, 1989.
E. Martins and M. Pascoal. A new implementation of Yen’s ranking loopless paths algorithm. Submited for publication, Universidade de Coimbra, Portugal, 2000.
E. Martins, M. Pascoal, and J. Santos. A new algorithm for ranking loopless paths. Technical report, Universidade de Coimbra, Portugal, 1997.
E. Nardelli, G. Proietti, and P. Widmayer. Finding the most vital node of a shortest path. In Proc. COCOON, 2001.
N. Nisan and A. Ronen. Algorithmic mechanism design. In Proc. 31st Annu. ACM Sympos. Theory Comput., 1999.
W. J. Paul and J. Simon. Decision trees and random access machines. In Proc. International Symp. on Logic and Algorithmic, pages 331–340, 1980.
A. Perko. Implementation of algorithms for K shortest loopless paths. Networks, 16:149–160, 1986.
J. Y. Yen. Finding the K shortest loopless paths in a network. Management Science, 17:712–716, 1971.
J. Y. Yen. Another algorithm for finding the K shortest loopless network paths. In Proc. of 41st Mtg. Operations Research Society of America, volume 20, 1972.
U. Zwick. All Pairs Shortest Paths inWeighted Directed Graphs-exact and almost exact algorithms. In Proc. IEEE Symposium on Foundations of Computer Science, 310–319, 1998.
U. Zwick. All Pairs Shortest Paths using Bridging Sets and Rectangular Matrix Multiplication. In Electronic Colloquium on Computational Complexity, 2000.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hershberger, J., Suri, S., Bhosle, A. (2003). On the Difficulty of Some Shortest Path Problems. In: Alt, H., Habib, M. (eds) STACS 2003. STACS 2003. Lecture Notes in Computer Science, vol 2607. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36494-3_31
Download citation
DOI: https://doi.org/10.1007/3-540-36494-3_31
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-00623-7
Online ISBN: 978-3-540-36494-8
eBook Packages: Springer Book Archive