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• W3173922011 abstract "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 $O(M^{1/2}n^{3/2})$ with query time $widetilde{O}(1)$. 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 $widetilde{O}(msqrt{n}+n^2)$ time SSRP algorithm for undirected and unweighted graphs. Grandoni and Vassilevska Williams [FOCS 2012, TALG 2020] gave an algebraic, randomized $widetilde{O}(Mn^omega)$ time SSRP algorithm for graphs with integer edge weights in the range $[1,M]$, where $omega<2.373$ 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 $widetilde{O}(msqrt{n}+n^2)$ and $widetilde{O}(Mn^omega)$ preprocessing times are polynomial improvements over previous $o(n^2)$-space oracles. On sparse graphs with $m=O(n^{5/4-varepsilon}/M^{7/4})$ edges, for any constant $varepsilon > 0$, we reduce the preprocessing to randomized $widetilde{O}(M^{7/8}m^{1/2}n^{11/8})=O(n^{2-varepsilon/2})$ time. This is the first truly subquadratic time algorithm for building Single-Source DSOs on sparse graphs." @default.
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• W3173922011 date "2021-06-29" @default.
• W3173922011 modified "2023-09-27" @default.
• W3173922011 title "Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles" @default.
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