Abstract
Let P G (s,t) denote a shortest path between two nodes s and t in an undirected graph G with nonnegative edge weights. A detour at a node u∈P G (s,t)=(s,…,u,v,…,t) is defined as a shortest path P G−e (u,t) from u to t which does not make use of (u,v). In this paper, we focus on the problem of finding an edge e=(u,v)∈P G (s,t) whose removal produces a detour at node u such that the ratio of the length of P G−e (u,t) to the length of P G (u,t) is maximum. We define such an edge as an anti-block vital edge (AVE for short), and show that this problem can be solved in O(mn) time, where n and m denote the number of nodes and edges in the graph, respectively. Some applications of the AVE for two special traffic networks are shown.
Similar content being viewed by others
References
Bar-Noy A, Schieber B (1991) The canadian traveler problem. In: Proceedings of the second annual ACM-SIAM symposium on discrete algorithms, pp 261–270
Bhosle AM (2005) Improved algorithms for replacement paths problems in restricted graphs. Oper Res Lett 33:459–466
Corley HW, Sha DY (1982) Most vital links and nodes in weighted networks. Oper Res Lett 1:157–161
Dijkstra EW (1959) A note on two problems in connexion with graphs. Numer Math 1:269–271
Hershberger J, Suri S (2001) Vickrey prices and shortest paths: What is an edge worth? In: Proceedings of the 42nd annual IEEE symposium on foundations of computer science, pp 252–259
Hershberger J, Suri S, Bhosle A (2003) On the difficulty of some shortest path problems. In: Proceedings of the 20th symposium on theoretical aspects of computer science, pp 343–354
Li Y, Guo Y (2004) Study on vital edges of shortest paths in traffic and transportation networks. Chin J Manag Sci 12(4):69–73
Malik K, Mittal AK, Gupta SK (1989) The k most vital arcs in the shortest path problem. Oper Res Lett 8:223–227
Nardelli E, Proietti G, Widmayer P (1998) Finding the detour critical edge of a shortest path between two nodes. Inf Process Lett 67(1):51–54
Nardelli E, Proietti G, Widmayer P (2001) A faster computation of the most vital edge of a shortest path between two nodes. Inf Process Lett 79(2):81–85
Nardelli E, Proietti G, Widmayer P (2003) Finding the most vital node of a shortest path. Theor Comput Sci 296:167–177
Papadimitriou CH, Yannakakis M (1989) Shortest paths without a map. In: Ronchi Della Rocca S, Ausiello G, Dezani-Ciancaglini M (eds) Automata, languages and programming. Lecture notes in computer science, vol 372. Springer, Heidelberg, pp 610–620
Su B (2005) Research on strategy for sequential unexpected blockages during the transportation process. PhD dissertation
Author information
Authors and Affiliations
Corresponding author
Additional information
This research is supported by NSF of China under Grants 70471035, 70525004, 701210001 and 60736027, and PSF of China under Grant 20060401003.
Rights and permissions
About this article
Cite this article
Su, B., Xu, Q. & Xiao, P. Finding the anti-block vital edge of a shortest path between two nodes. J Comb Optim 16, 173–181 (2008). https://doi.org/10.1007/s10878-007-9120-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-007-9120-2