Small sets supporting fary embeddings of planar graphs
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Introduction to algorithms
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
Oracles for distances avoiding a link-failure
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
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
Planar Graphs, Negative Weight Edges, Shortest Paths, and Near Linear Time
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Compact oracles for reachability and approximate distances in planar digraphs
Journal of the ACM (JACM)
Multiple-source shortest paths in planar graphs
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
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
Informational overhead of incentive compatibility
Proceedings of the 9th ACM conference on Electronic commerce
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
A nearly optimal oracle for avoiding failed vertices and edges
Proceedings of the forty-first annual ACM symposium on Theory of computing
ACM Transactions on Algorithms (TALG)
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
Single source distance oracle for planar digraphs avoiding a failed node or link
Proceedings of the twenty-third 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
Finding Alternative Shortest Paths in Spatial Networks
ACM Transactions on Database Systems (TODS)
Replacement Paths and Distance Sensitivity Oracles via Fast Matrix Multiplication
ACM Transactions on Algorithms (TALG)
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Let G = (V(G), E(G)) be a weighted directed graph and let P be a shortest path from s to t in G. In the replacement paths problem we are required to compute for every edge e in P, the length of a shortest path from s to t that avoids e. The fastest known algorithm for solving the problem in weighted directed graphs is the trivial one: each edge in P is removed from the graph in its turn and the distance from s to t in the modified graph is computed. The running time of this algorithm is O (mn + n2 log n), where n = |V(G)| and m = |E(G)|. The replacement paths problem is strongly motivated by two different applications. First, the fastest algorithm to compute the k simple shortest paths from s to t in directed graphs [21, 13] repeatedly computes the replacement paths from s to t. Its running time is O(kn(m + n log n)). Second, the computation of Vickrey pricing of edges in distributed networks can be reduced to the replacement paths problem. An open question raised by Nisan and Ronen [16] asks whether it is possible to compute the Vickrey pricing faster than the trivial algorithm described in the previous paragraph. In this paper we present a near-linear time algorithm for computing replacement paths in weighted planar directed graphs. In particular, the algorithm computes the lengths of the replacement paths in O(n log3 n) time. This result immediately improves the running time of the two applications mentioned above by almost a linear factor. Our algorithm is obtained by combining several new ideas with a data structure of Klein [12] that supports multi-source shortest paths queries in planar directed graphs in logarithmic time. Our algorithm can be adapted to address the variant of the problem in which one is interested in the replacement path itself (rather than the length of the path). In that case the algorithm is executed in a preprocessing stage constructing a data structure that supports replacement path queries in time Õ(h), where h is the number of hops in the replacement path. In addition, we can handle the variant in which vertices should be avoided instead of edges.