Fibonacci heaps and their uses in improved network optimization algorithms
Journal of the ACM (JACM)
SIAM Journal on Computing
A new approach to all-pairs shortest paths on real-weighted graphs
Theoretical Computer Science - Special issue on automata, languages and programming
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
Replacement paths and k simple shortest paths in unweighted directed graphs
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
The k most vital arcs in the shortest path problem
Operations Research Letters
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
Applications of graph algorithms in GIS
SIGSPATIAL Special
An experimental study on approximating K shortest simple paths
ESA'11 Proceedings of the 19th European conference on Algorithms
Efficient associative algorithm for finding the second simple shortest paths in a digraph
PaCT'11 Proceedings of the 11th international conference on Parallel computing technologies
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Computing replacement paths in surface embedded graphs
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Replacement Paths and Distance Sensitivity Oracles via Fast Matrix Multiplication
ACM Transactions on Algorithms (TALG)
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Given a directed, non-negatively weighted graph G=(V,E) and s,t@?V, we consider two problems. In the k simple shortest paths problem, we want to find the k simple paths from s to t with the k smallest weights. In the replacement paths problem, we want the shortest path from s to t that avoids e, for every edge e in the original shortest path from s to t. The best known algorithm for the k simple shortest paths problem has a running of O(k(mn+n^2logn)). For the replacement paths problem the best known result is the trivial one running in time O(mn+n^2logn). In this paper we present two simple algorithms for the replacement paths problem and the k simple shortest paths problem in weighted directed graphs (using a solution of the All Pairs Shortest Paths problem). The running time of our algorithm for the replacement paths problem is O(mn+n^2loglogn). For the k simple shortest paths we will perform O(k) iterations of the second simple shortest path (each in O(mn+n^2loglogn) running time) using a useful property of Roditty and Zwick [L. Roditty, U. Zwick, Replacement paths and k simple shortest paths in unweighted directed graphs, in: Proc. of International Conference on Automata, Languages and Programming (ICALP), 2005, pp. 249-260]. These running times immediately improve the best known results for both problems over sparse graphs. Moreover, we prove that both the replacement paths and the k simple shortest paths (for constant k) problems are not harder than APSP (All Pairs Shortest Paths) in weighted directed graphs.