Incremental distance products via faulty shortest paths

Given a constant number d of n×n matrices with total m non-infinity entries, we show how to construct in (essentially optimal) O˜(mn+n²) time a data structure that can compute in (essentially optimal) O˜(n²) time the distance product of these matrices after incrementing the value (possibly to infinity) of a constant number k of entries. Our result is obtained by designing an oracle for single source replacement paths that is suited for short distances: Given a graph G=(V,E), a source vertex s, and a shortest paths tree T of depth d rooted at s, the oracle can be constructed in O˜(md^k) time for any constant k. Then, given an arbitrary set S of at most k=O(1) edges and an arbitrary vertex t, the oracle in O˜(1) time either reports the length of the shortest s-to-t path in (V,E∖S) or otherwise reports that any such shortest path must use more than d edges.