Approximate Distance Sensitivity Oracles in Subquadratic Space

An f-edge fault-tolerant distance sensitive oracle (f-DSO) with stretch sigma >= 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 sigma-approximation of the s-t-distance in G-F. We study f-DSOs that take subquadratic space. Thorup and Zwick [JACM 2015] showed that this is only possible for sigma >= 3. We present, for any constant f >= 1 and alpha is an element of (0, 1/2), and any epsilon > 0, an f-DSO with stretch 3 + epsilon that takes (O) over tilde (n(2-) alpha/f+1/epsilon) center dot O(log n/epsilon)(f+1) space and has an O(n(alpha)/epsilon(2)) 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+0(1) n(1+1/k)) space and has (O) over tilde (D-o(1)) query time, with a preprocessing time of O(Df+0(1) mn(1/k)). Chechik, Cohen, Fiat, and Kaplan [SODA 2017] presented an f-DSO with stretch 1+epsilon and preprocessing time (O) over tilde (epsilon)(n(5)), albeit with a super-quadratic space requirement. We show how to reduce their preprocessing time to O(mn(2)) center dot O(log n/epsilon)(f).