Deterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler
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In the decremental (1+ϵ)-approximate Single-Source Shortest Path (SSSP) problem, we are given a graph G=(V,E) with n = |V|, m = |E|, undergoing edge deletions, and a distinguished source s ∈ V, and we are asked to process edge deletions efficiently and answer queries for distance estimates dist_G(s,v) for each v ∈ V, at any stage, such that dist_G(s,v) ≤dist_G(s,v) ≤ (1+ ϵ)dist_G(s,v). In the decremental (1+ϵ)-approximate All-Pairs Shortest Path (APSP) problem, we are asked to answer queries for distance estimates dist_G(u,v) for every u,v ∈ V. In this article, we consider the problems for undirected, unweighted graphs. We present a new deterministic algorithm for the decremental (1+ϵ)-approximate SSSP problem that takes total update time O(mn^0.5 + o(1)). Our algorithm improves on the currently best algorithm for dense graphs by Chechik and Bernstein [STOC 2016] with total update time Õ(n^2) and the best existing algorithm for sparse graphs with running time Õ(n^1.25√(m)) [SODA 2017] whenever m = O(n^1.5 - o(1)). In order to obtain this new algorithm, we develop several new techniques including improved decremental cover data structures for graphs, a more efficient notion of the heavy/light decomposition framework introduced by Chechik and Bernstein and the first clustering technique to maintain a dynamic sparse emulator in the deterministic setting. As a by-product, we also obtain a new simple deterministic algorithm for the decremental (1+ϵ)-approximate APSP problem with near-optimal total running time Õ(mn /ϵ) matching the time complexity of the sophisticated but rather involved algorithm by Henzinger, Forster and Nanongkai [FOCS 2013].
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