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 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 ≤ <i>d</i> in a graph of n vertices, we present the first <i>o</i>(<i>nd</i>) 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 <i>θ</i>(<i>n</i>²) 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(<i>n</i>^3/2)bound on the update time per edge deletion for approximation factor ≥ 3. In particular, update time achieved by our algorithm is Õ(<i>n</i>^10/9) for 3-approximate shortest-paths, Õ(<i>n</i>^14/13) for 5-approximate shortest-paths, and Õ(<i>n</i>^28/27) for 7-approximate shortest-paths.All our algorithms achieve optimal query time and are simple to implement.