Maintaining all-pairs approximate shortest paths under deletion of edges

Abstract

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 O(min{√nd,(nd)2/3}) for 3-approximate shortest-paths, and O(min{√nd,(nd)4/7}) for 7-approximate shortest-paths. For graphs with θ(n2) edges, we achieve even further improvement in update time : O(√nd) for 3-approximate shortest-paths, and O(3√nd2) for 5-approximate shortest-paths.For maintaining all-pairs approximate shortest-paths, weimprove the previous O(n3/2)bound on the update time per edge deletion for approximation factor ≥ 3. In particular, update time achieved by our algorithm is O(n10/9) for 3-approximate shortest-paths, O(n14/13) for 5-approximate shortest-paths, and O(n28/27) for 7-approximate shortest-paths.All our algorithms achieve optimal query time and are simple to implement.