Approximate Distance Sensitivity Oracles in Subquadratic Space
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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 [JACM 2015] 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 + ε that takes O(n^2-α/f+1/ε) · O(log n/ε)^f+1 space and has an O(n^α/ε^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(D^f+o(1) n^1+1/k) space and has O(D^o(1)) query time, with a preprocessing time of O(D^f+o(1) mn^1/k). Chechik, Cohen, Fiat, and Kaplan [SODA 2017] presented an f-DSO with stretch 1+ε and preprocessing time O_ε(n^5+o(1)), albeit with a super-quadratic space requirement. We show how to reduce their preprocessing time to O_ε(mn^2+o(1)).
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