An f-Sensitive Distance Oracle with stretch 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 valued 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 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 G n F of lengthd. To the best of our knowledge we give the rst nontrivial f-sensitive distance oracle with fast query time and small stretch capable of handling multiple edge failures. Specically, for any f = o( log n log log n ) and a xed > 0 our oracle answers queries (F; s; t) in time e O(1) with (1 + ) stretch using a data structure of size n 2+o(1) . For comparison, the na ve alternative requires m f n 2 space for sublinear query time.
The Single Source Replacement Paths (SSRP) problem is as follows; Given a graph G = (V, E), a source vertex s and a shortest paths tree T s rooted in s, output for every vertex t ∈ V and for every edge e in T s the length of the shortest path from s to t avoiding e.We present near optimal upper bounds, by providing O(m √ n + n 2 ) time randomized combinatorial algorithm 1 for unweighted undirected graphs, and matching conditional lower bounds for the SSRP problem.
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