Approximate Distance Sensitivity Oracles in Subquadratic Space

An f-edge fault-tolerant distance sensitive oracle (f-DSO) with stretch σ ≥ 1 is a data structure that preprocesses a given undirected, unweighted graph G with n vertices and m edges, and a positive integer f. When queried with a pair of vertices s, t and a set F of at most f edges, it returns a σ-approximation of the s-t-distance in G-F. We study f-DSOs that take subquadratic space. Thorup and Zwick [JACM2015] showed that this is only possible for σ ≥ 3. We present, for any constant f ≥ 1 and α (0, 1/2), and any ϵ > 0, an f-DSO with stretch 3 + that takes O(n2-α/f+1/ϵ) · O(logn/ϵ)f+1 space and has an O(nα/ϵ2) query time. We also give an improved construction for graphs with diameter at most D. For any constant k, we devise an f-DSO with stretch 2k-1 that takes O(Df+o(1) n1+1/k) space and has O(Do(1)) query time, with a preprocessing time of O(Df+o(1) mn1/k). Chechik, Cohen, Fiat, and Kaplan [SODA 2017] presented an f-DSO with stretch 1+ and preprocessing time O(n5) · O(logn/ϵ)f, albeit with a super-quadratic space requirement. We show how to reduce their preprocessing time to O(mn2) · O(logn/ϵ)f.