Maintaining all-pairs approximate shortest paths under deletion of edges

We present a hierarchical scheme for efficiently maintaining all-pairs approximate shortest-paths in undirected unweighted graphs under deletions of edges. An alpha-approximate shortest-path between two vertices is a path of length at-most alpha times the length of the shortest path. For maintaining alpha-approximate shortest paths for all pairs of vertices separated by distance less than or equal to 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) over tilde (min{rootnd, (nd)(2/3)}) for 3-approximate shortest-paths, and (O) over tilde (min{(3)rootnd, (nd)(4/7)}) for 7-approximate shortest-paths. For graphs with theta(n(2)) edges, we achieve even further improvement in update time : (O) over tilde(rootnd) for 3-approximate shortest-paths, and (O) over tilde((3)rootnd(2)) for 5-approximate shortest-paths. For maintaining all-pairs approximate shortest-paths, we improve the previous (O) over tilde (n(3/2)) bound on the update time per edge deletion for approximation factor greater than or equal to 3. In particular, update time achieved by our algorithm is (O) over tilde (n(10/9)) for 3-approximate shortest-paths, (O) over tilde (n(14/13)) for 5-approximate shortest-paths, and (O) over tilde (n(28/27)) for 7-approximate shortest-paths. All our algorithms achieve optimal query time and are simple to implement.