Abstract
Given an undirected graph with a source node s and a sink node t. The anti-risk path problem is defined as the problem of finding a path between node s to node t with the least risk under the assumption that at most one edge of each path may be blocked. Xiao et al. (J Comb Optim 17:235–246, 2009) defined the problem and presented an \(O(mn+n^2 \log n)\) time algorithm to find an anti-risk path, where n and m are the number of nodes and edges, respectively. Recently, Mahadeokar and Saxena (J Comb Optim 27:798–807, 2014) solved the problem in \(O(m+n \log n)\) time. In this paper, first, a new version of the anti-risk path (called contra-risk path) is defined, which is more effective than an anti-risk path in many networks. Then, an algorithm to find a contra-risk path is presented, which runs in \(O(m+n \log n)\) time.
Similar content being viewed by others
References
Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows: theory, algorithms, and applications. Prentice-Hall, Englewood Cliffs
Mahadeokar J, Saxena S (2014) Faster algorithm to find anti-risk path between two nodes of an undirected graph. J Comb Optim 27:798–807
Malik K, Mittal AK, Gupta SK (1989) The k most vital arcs in the shortest path problem. Oper Res Lett 8(4):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) Faster computation of the most vital edge of a shortest path. Inf Process Lett 79:81–85
Nardelli E, Proietti G, Widmayer P (2001) Finding the most vital node of a shortest path. In: Wang Jie (ed) Computing and combinatorics, volume 2108 of lecture notes in computer science. Springer, Berlin, pp 278–287
Nisan N, Ronen A (1999) Algorithmic mechanism design (extended abstract). STOC’99, New York, NY, USA
Xiao P, Xu Y, Su B (2009) Finding an anti-risk path between two nodes in undirected graphs. J Comb Optim 17:235–246
Zhang H, Xu Y, Wen X (2014) Optimal shortest path set problem in undirected graphs. J Comb Optim 29:511–530
Acknowledgments
We would like to express our very great appreciation to reviewers for their valuable comments and suggestions, which have helped to improve the quality and presentation of this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ghiyasvand, M., Keshtkar, I. Finding a contra-risk path between two nodes in undirected graphs. J Comb Optim 32, 917–926 (2016). https://doi.org/10.1007/s10878-015-9912-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-015-9912-8