# Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles

Given a graph with a source vertex s, the Single Source Replacement Paths (SSRP) problem is to compute, for every vertex t and edge e, the length d(s,t,e) of a shortest path from s to t that avoids e. A Single-Source Distance Sensitivity Oracle (Single-Source DSO) is a 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 of size O(M^1/2n^3/2) with query time O(1). The space requirement is optimal (up to the word size) and our techniques can also handle vertex failures. Chechik and Cohen [SODA 2019] presented a combinatorial, randomized O(m√(n)+n^2) time SSRP algorithm for undirected and unweighted graphs. Grandoni and Vassilevska Williams [FOCS 2012, TALG 2020] gave an algebraic, randomized O(Mn^ω) time SSRP algorithm for graphs with integer edge weights in the range [1,M], where ω<2.373 is the matrix multiplication exponent. We derandomize both algorithms for undirected graphs in the same asymptotic running time and apply our compression to obtain deterministic Single-Source DSOs. The O(m√(n)+n^2) and O(Mn^ω) preprocessing times are polynomial improvements over previous o(n^2)-space oracles. On sparse graphs with m=O(n^5/4-ε/M^7/4) edges, for any constant ε > 0, we reduce the preprocessing to randomized O(M^7/8m^1/2n^11/8)=O(n^2-ε/2) time. This is the first truly subquadratic time algorithm for building Single-Source DSOs on sparse graphs.

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