Space-Efficient Fault-Tolerant Diameter Oracles

We design f-edge fault-tolerant diameter oracles (f-FDO, or simply FDO if f = 1). For a given directed or undirected and possibly edge-weighted graph G with n vertices and m edges and a positive integer f, we preprocess the graph and construct a data structure that, when queried with a set F of edges, where |F|f, returns the diameter of GF. An f-FDO has stretch σ1 if the returned value bD satisfies diam(GF)bDσ diam(GF). For the case of a single edge failure (f = 1) in an unweighted directed graph, there exists an approximate FDO by Henzinger et al. [ITCS 2017] with stretch (1 + ϵ), constant query time, space O(m), and a combinatorial preprocessing time of eO(mn + n1.5p Dm/ϵ ), where D is the diameter. We present an FDO for directed graphs with the same stretch, query time, and space. It has a preprocessing time of eO(mn + n2/ϵ), which is better for constant ϵ > 0. The preprocessing time nearly matches a conditional lower bound for combinatorial algorithms, also by Henzinger et al. With fast matrix multiplication, we achieve a preprocessing time of eO(n2.5794 + n2/ϵ). We further prove an information-theoretic lower bound showing that any FDO with stretch better than 3/2 requires Ω(m) bits of space. Thus, for constant 0 < ϵ < 3/2, our combinatorial (1 + ϵ)-approximate FDO is near-optimal in all parameters. In the case of multiple edge failures (f > 1) in undirected graphs with non-negative edge weights, we give an f-FDO with stretch (f + 2), query time O(f2 log2 n), eO(fn) space, and preprocessing time eO(fm). We complement this with a lower bound excluding any finite stretch in o(fn) space. Many real-world networks have polylogarithmic diameter. We show that for those graphs and up to f = o(log n/ log log n) failures one can swap approximation for query time and space. We present an exact combinatorial f-FDO with preprocessing time mn1+o(1), query time no(1), and space n2+o(1). When using fast matrix multiplication instead, the preprocessing time can be improved to nω+o(1), where ω < 2.373 is the matrix multiplication exponent.