An On-Line Edge-Deletion Problem
Journal of the ACM (JACM)
All-Pairs Almost Shortest Paths
SIAM Journal on Computing
Journal of Algorithms
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
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
Fully dynamic biconnectivity and transitive closure
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Fully Dynamic Algorithms for Maintaining All-Pairs Shortest Paths and Transitive Closure in Digraphs
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Proceedings of the forty-second ACM symposium on Theory of computing
Replacement paths and k simple shortest paths in unweighted directed graphs
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Improved dynamic algorithms for maintaining approximate shortest paths under deletions
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Constant-time all-pairs geodesic distance query on triangle meshes
I3D '12 Proceedings of the ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games
Fully dynamic approximate distance oracles for planar graphs via forbidden-set distance labels
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Replacement paths and k simple shortest paths in unweighted directed graphs
ACM Transactions on Algorithms (TALG)
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We present a hierarchical scheme for efficiently maintaining all-pairs approximate shortest paths in undirected unweighted graphs under deletions of edges.An α-approximate shortest-path between two vertices is a path of length at-most α times the length of the shortest path. For maintaining α-approximate shortest paths for all pairs of vertices separated by distance ≤ d in a graph of n vertices, we present the first o(nd) update time algorithm based on our hierarchical scheme. In particular, the update time per edge deletion achieved by our algorithm is Õ(min{√nd,(nd)2/3}) for 3-approximate shortest-paths, and Õ(min{√nd,(nd)4/7}) for 7-approximate shortest-paths. For graphs with θ(n2) edges, we achieve even further improvement in update time : Õ(√nd) for 3-approximate shortest-paths, and Õ(3√nd2) for 5-approximate shortest-paths.For maintaining all-pairs approximate shortest-paths, weimprove the previous Õ(n3/2)bound on the update time per edge deletion for approximation factor ≥ 3. In particular, update time achieved by our algorithm is Õ(n10/9) for 3-approximate shortest-paths, Õ(n14/13) for 5-approximate shortest-paths, and Õ(n28/27) for 7-approximate shortest-paths.All our algorithms achieve optimal query time and are simple to implement.