(1 + ϵ)-Approximate f-sensitive distance oracles

An f-Sensitive Distance Oracle with stretch α preprocesses a graph G(V;E) and produces a small data structure that is used to answer subsequent queries. A query is a triple consisting of a set F E of at most f edges, and vertices s and t. The oracle answers a query (F; s; t) by returning a value d which is equal to the length of some path between s and t in the graph G n F (the graph obtained from G by discarding all edges in F). Moreover, d is at most α times the length of the shortest path between s and t in G n F. The oracle can also construct a path between s and t in Gn F of length d. To the best of our knowledge we give the first nontrivial f-sensitive distance oracle with fast query time and small stretch capable of handling multiple edge failures. Specifically, for any f = o( log n log log n ) and a fixed ϵ > 0 our oracle answers queries (F; s; t) in time e O(1) with (1 + ϵ) stretch using a data structure of size n2+o(1). For comparison, the naive alternative requires mfn2 space for sublinear query time.