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Maintaining all-pairs approximate shortest paths under deletion of edges
S. Baswana, R. Hariharan,
Published in
Pages: 394 - 403
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{3√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, we improve 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.
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