Near-optimal deterministic single-source distance sensitivity oracles

Given a graph with a distinguished source vertex s, the Single Source Replacement Paths (SSRP) problem is to compute and output, for any target vertex t and edge e, the length d(s, t, e) of a shortest path from s to t that avoids a failing edge e. A Single-Source Distance Sensitivity Oracle (Single-Source DSO) is a compact data structure that answers queries of the form (t, e) by returning the distance d(s, t, e). We show how to deterministically compress the output of the SSRP problem on n-vertex, m-edge graphs with integer edge weights in the range [1, M] into a Single-Source DSO that has size O(M1/2n3/2) and query time Oe(1). We prove that the space requirement is optimal (up to the word size). Our techniques can also handle vertex failures within the same bounds. Chechik and Cohen [SODA 2019] presented a combinatorial, randomized Oe(m√n + n2) time SSRP algorithm for undirected and unweighted graphs. We derandomize their algorithm with the same asymptotic running time and apply our compression to obtain a deterministic Single-Source DSO with Oe(m√n +n2) preprocessing time, O(n3/2) space, and Oe(1) query time. Our combinatorial Single-Source DSO has near-optimal space, preprocessing and query time for unweighted graphs, improving the preprocessing time by a √n -factor compared to previous results with o(n2) space. Grandoni and Vassilevska Williams [FOCS 2012, TALG 2020] gave an algebraic, randomized Oe(Mnω) time SSRP algorithm for (undirected and directed) graphs with integer edge weights in the range [1, M], where ω < 2.373 is the matrix multiplication exponent. We derandomize it for undirected graphs and apply our compression to obtain an algebraic Single-Source DSO with Oe(Mnω) preprocessing time, O(M1/2 n3/2) space, and Oe(1) query time. This improves the preprocessing time of algebraic Single-Source DSOs by polynomial factors compared to previous o(n2)-space oracles. We also present further improvements of our Single-Source DSOs. We show that the query time can be reduced to a constant at the cost of increasing the size of the oracle to O(M1/3 n5/3) and that all our oracles can be made path-reporting. On sparse graphs with m = O(nM5/74/−4ε) edges, for any constant ε > 0, we reduce the preprocessing to randomized Oe(M7/8 m1/2 n11/8) = O(n2−ε/2) time. To the best of our knowledge, this is the first truly subquadratic time algorithm for building Single-Source DSOs on sparse graphs.