High probability parallel transitive-closure algorithms
SIAM Journal on Computing
Finding the hidden path: time bounds for all-pairs shortest paths
SIAM Journal on Computing
SIAM Journal on Computing
Undirected single-source shortest paths with positive integer weights in linear time
Journal of the ACM (JACM)
A faster computation of the most vital edge of a shortest path
Information Processing Letters
Improved decremental algorithms for maintaining transitive closure and all-pairs shortest paths
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Oracles for distances avoiding a link-failure
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
All pairs shortest paths using bridging sets and rectangular matrix multiplication
Journal of the ACM (JACM)
Maintaining all-pairs approximate shortest paths under deletion of edges
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Erratum to "Vickrey Pricing and Shortest Paths: What is an Edge Worth?"
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
On the Difficulty of Some Shortest Path Problems
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
Fully dynamic biconnectivity and transitive closure
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Vickrey Prices and Shortest Paths: What is an Edge Worth?
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Dynamic Approximate All-Pairs Shortest Paths in Undirected Graphs
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
On the K-simple shortest paths problem in weighted directed graphs
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Oracles for Distances Avoiding a Failed Node or Link
SIAM Journal on Computing
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
The k most vital arcs in the shortest path problem
Operations Research Letters
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Let G = (V,E) be a directed graph and let P be a shortest path from s to t in G. In the replacement paths problem, we are required to find, for every edge e on P, a shortest path from s to t in G that avoids e. The only known algorithm for solving the problem, even for unweighted directed graphs, is the trivial algorithm in which each edge on the path, in its turn, is excluded from the graph and a shortest paths tree is computed from s. The running time is O(mn + n2 log n). The replacement paths problem is strongly motivated by two different applications: (1) The fastest algorithm to compute the k simple shortest paths between s and t in directed graphs [Yen 1971; Lawler 1972] computes the replacement paths between s and t. Its running time is Õ(mnk). (2) The replacement paths problem is used to compute the Vickrey pricing of edges in a distributed network. It was raised as an open problem by Nisan and Ronen [2001] whether it is possible to compute the Vickrey pricing faster than n computations of a shortest paths tree. In this article we present the first nontrivial algorithm for computing replacement paths in unweighted directed graphs (and in graphs with small integer weights). Our algorithm is Monte-Carlo and its running time is Õ(m√n). This result immediately improves the running time of the two applications mentioned above in a factor of √n. We also show how to reduce the problem of computing k simple shortest paths between s and t to O(k) computations of a second simple shortest path from s to t each time in a different subgraph of G. The importance of this result is that computing a second simple shortest path may turn out to be an easier problem than computing the replacement paths, thus, we can focus our efforts to improve the k simple shortest paths algorithm in obtaining a faster algorithm for the second shortest path problem.