Space-Efficient Fault-Tolerant Diameter Oracles.

Abstract

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 G-F. An f-FDO has stretch σ ⩾ 1 if the returned value D^ satisfies diam(G-F) ⩽ D^ ⩽ σ diam(G-F). 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+e), constant query time, space O(m), and a combinatorial preprocessing time of O(mn + n^{1.5} √{Dm/e}), 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 O(mn + n²/e), which is better for constant e > 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 O(n^{2.5794} + n²/e). 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 < e < 3/2, our combinatorial (1+e)-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(f²log²{n}), O(fn) space, and preprocessing time O(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 mn^{1+o(1)}, query time n^o(1), and space n^{2+o(1)}. When using fast matrix multiplication instead, the preprocessing time can be improved to n^{ω+o(1)}, where ω < 2.373 is the matrix multiplication exponent.