Fibonacci heaps and their uses in improved network optimization algorithms
Journal of the ACM (JACM)
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Finding the hidden path: time bounds for all-pairs shortest paths
SIAM Journal on Computing
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
Rectangular matrix multiplication revisited
Journal of Complexity
Fast rectangular matrix multiplication and applications
Journal of Complexity
Scaling algorithms for the shortest paths problem
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
Efficient Algorithms for Shortest Paths in Sparse Networks
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
Vickrey Prices and Shortest Paths: What is an Edge Worth?
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Answering distance queries in directed graphs using fast matrix multiplication
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
More algorithms for all-pairs shortest paths in weighted graphs
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
On the K-simple shortest paths problem in weighted directed graphs
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Improved algorithms for the k simple shortest paths and the replacement paths problems
Information Processing Letters
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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
An experimental study on approximating K shortest simple paths
ESA'11 Proceedings of the 19th European conference on Algorithms
Replacement Paths and Distance Sensitivity Oracles via Fast Matrix Multiplication
ACM Transactions on Algorithms (TALG)
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The replacement paths problem for directed graphs is to find for given nodes s and t and every edge e on the shortest path between them, the shortest path between s and t which avoids e. For unweighted directed graphs on n vertices, the best known algorithm runtime was Õ(n2.5) by Roditty and Zwick. For graphs with integer weights in {− M,..., M}, Weimann and Yuster showed that one can use fast matrix multiplication and solve the problem in O(Mn2.584) time, a runtime which would be O(Mn2.33) if the exponent ω of matrix multiplication is 2. We improve on both of these algorithms. Our new algorithm also relies on fast matrix multiplication and runs in Mnω+o(1) time. Our result shows that, at least for small integer weights, the replacement paths problem in directed graphs may be easier than the related all pairs shortest paths problem in directed graphs, as the current best runtime for the latter is Ω(n2.5) time even if ω = 2.