Fault-Tolerant ST-Diameter Oracles

Abstract

Given two vertex sets S and T in a graph, the ST-diameter is the maximum s-t-distance between vertices \(s \in S\) and \(t \in T\). We study the problem of estimating the ST-diameter of graphs that are subject to a small number of transient edge failures. An f-edge fault-tolerant ST-diameter oracle (f-FDO-ST) is a data structure that preprocesses a graph G, sets S, T, and a positive integer f. When queried with a set F of at most f failing edges, the oracle returns an estimate \(\widehat{D}\) of the ST-diameter in \(G\,{-}\,F\). The oracle is said to have stretch \(\sigma \geqslant 1\) if \({{\,\textrm{diam}\,}}(G{-}F,S,T) \leqslant \widehat{D} \leqslant \sigma \cdot {{\,\textrm{diam}\,}}(G{-}F,S,T)\). We design new f-FDO-STs by reducing their construction to that of all-pairs and single-source distance sensitivity oracles (f-DSOs). These are data structures that estimate the pairwise graph distances, or respectively the distances from a distinguished source, under up to f failures. We obtain several new trade-offs between the size of the ST-diameter oracles, their stretch guarantees, query and preprocessing times by combining our black-box reductions with f-DSO results from the literature. We further provide a lower bound on the space requirement of approximate ST-diameter oracles. We prove that there exists a family of graphs for which any f-FDO-ST with sensitivity \(f \geqslant 2\) and stretch better than 5/3 requires \(\Omega (n^{3/2})\) bits of space, regardless of the query time.