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 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, we improve the previous Õ (n^3/2)bound on the update time per edge deletion for approximation factor ≥ 3. In particular, update time achieved by our algorithm is Õ (n^10/9) for 3-approximate shortest-paths, Õ (n^14/13) for 5-approximate shortest-paths, and Õ (n^28/27) for 7-approximate shortest-paths. All our algorithms achieve optimal query time and are simple to implement.