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 which 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.
For any single edge failure in weighted graphs, Bilò et al. REF showed that we can construct an oracle of O(n) size which can report 2-approximate distances from the source in O(1) time.
Compact and Fast Sensitivity Oracles for Single-Source Distances
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Mathematics Computer Science