Compact Distance Oracles with Large Sensitivity and Low Stretch
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An f-edge fault-tolerant distance sensitive oracle (f-DSO) with stretch σ≥ 1 is a data structure that preprocesses an input graph G. When queried with the triple (s,t,F), where s, t ∈ V and F ⊆ E contains at most f edges of G, the oracle returns an estimate d_G-F(s,t) of the distance d_G-F(s,t) between s and t in the graph G-F such that d_G-F(s,t) ≤d_G-F(s,t) ≤σ d_G-F(s,t). For any positive integer k ≥ 2 and any 0 < α < 1, we present an f-DSO with sensitivity f = o(log n/loglog n), stretch 2k-1, space O(n^1+1/k+α+o(1)), and an O(n^1+1/k - α/k(f+1)) query time. Prior to our work, there were only three known f-DSOs with subquadratic space. The first one by Chechik et al. [Algorithmica 2012] has a stretch of (8k-2)(f+1), depending on f. Another approach is storing an f-edge fault-tolerant (2k-1)-spanner of G. The bottleneck is the large query time due to the size of any such spanner, which is Ω(n^1+1/k) under the Erdős girth conjecture. Bilò et al. [STOC 2023] gave a solution with stretch 3+ε, query time O(n^α) but space O(n^2-α/f+1), approaching the quadratic barrier for large sensitivity. In the realm of subquadratic space, our f-DSOs are the first ones that guarantee, at the same time, large sensitivity, low stretch, and non-trivial query time. To obtain our results, we use the approximate distance oracles of Thorup and Zwick [JACM 2005], and the derandomization of the f-DSO of Weimann and Yuster [TALG 2013], that was recently given by Karthik and Parter [SODA 2021].
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