On the exponent of the all pairs shortest path problem
Journal of Computer and System Sciences - Special issue: papers from the 32nd and 34th annual symposia on foundations of computer science, Oct. 2–4, 1991 and Nov. 3–5, 1993
Fast rectangular matrix multiplication and applications
Journal of Complexity
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
All pairs shortest paths using bridging sets and rectangular matrix multiplication
Journal of the ACM (JACM)
On the Difficulty of Some Shortest Path Problems
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
Finding the most vital node of a shortest path
Theoretical Computer Science - Computing and combinatorics
Vickrey Prices and Shortest Paths: What is an Edge Worth?
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Experimental analysis of dynamic all pairs shortest path algorithms
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
A fully dynamic reachability algorithm for directed graphs with an almost linear update time
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Worst-case update times for fully-dynamic all-pairs shortest paths
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Answering distance queries in directed graphs using fast matrix multiplication
FOCS '05 Proceedings of the 46th 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
Improved distance sensitivity oracles via random sampling
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
A near-linear time algorithm for computing replacement paths in planar directed graphs
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Oracles for Distances Avoiding a Failed Node or Link
SIAM Journal on Computing
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Dual-failure distance and connectivity oracles
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Improved algorithms for the k simple shortest paths and the replacement paths problems
Information Processing Letters
A nearly optimal oracle for avoiding failed vertices and edges
Proceedings of the forty-first annual ACM symposium on Theory of computing
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Solving the replacement paths problem for planar directed graphs in O(n log n) time
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
f-sensitivity distance Oracles and routing schemes
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Subcubic Equivalences between Path, Matrix and Triangle Problems
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Replacement Paths via Fast Matrix Multiplication
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Replacement paths and k simple shortest paths in unweighted directed graphs
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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A distance sensitivity oracle of an n-vertex graph G = (V,E) is a data structure that can report shortest paths when edges of the graph fail. A query (u ∈ V, v ∈ V, S ⊆ E) to this oracle returns a shortest u-to-v path in the graph G′ = (V,E ∖ S). We present randomized (Monte Carlo) algorithms for constructing a distance sensitivity oracle of size Õ(n3−α) for |S| = O(lg n/lg lg n) and any choice of 0 α O(n4−α) time and a query to this oracle takes Õ(n2−2(1−α)/|S|) time. For integral edge-lengths in {−M,..., M}, using the current ω O(Mn3.376−α) time with Õ(n2−(1−α)/|S|) query, or alternatively in O(M0.681n3.575−α) time with Õ(n2−2(1−α)/|S|) query. Distance sensitivity oracles generalize the replacement paths problem in which u and v are known in advance and |S| = 1. In other words, if P is a shortest path from u to v in G, then the replacement paths problem asks to compute, for every edge e on P, a shortest u-to-v path that avoids e. Our new technique for constructing distance sensitivity oracles using fast matrix multiplication also yields the first subcubic-time algorithm for the replacement paths problem when the edge-lengths are small integers. In particular, it yields a randomized (Monte Carlo) Õ(Mn2.376 + M2 3 n2.584)-time algorithm for the replacement paths problem assuming M ≤ n0.624. Finally, we mention that both our replacement paths algorithm and our distance sensitivity oracle can be made to work, in the same time and space bounds, for the case of failed vertices rather than edges, that is, when S is a set of vertices and we seek a shortest u-to-v path in the graph obtained from G by removing all vertices in S and their adjacent edges.