Compact and fast sensitivity oracles for single-source distances

Let s denote a distinguished source vertex of a non-negatively real weighted and undirected graph G with n vertices and m edges. In this paper we present two efficient single-source approximate-distance sensitivity oracles, namely compact data structures which are able to quickly report an approximate (by a multiplicative stretch factor) distance from s to any node of G following the failure of any edge in G. More precisely, we first present a sensitivity oracle of size O(n) which is able to report 2-approximate distances from the source in O(1) time. Then, we further develop our construction by building, for any 0 < ϵ < 1, another sensitivity oracle having size O(n · 1/ϵ log 1/ϵ), and is able to report a (1 + ϵ)-approximate distance from s to any vertex of G in O(log n · 1/ϵ log 1/ϵ) time. Thus, this latter oracle is essentially optimal as far as size and stretch are concerned, and it only asks for a logarithmic query time. Finally, our results are complemented with a space lower bound for the related class of single-source additively-stretched sensitivity oracles, which is helpful to realize the hardness of designing compact oracles of this type.