Deterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler - Maximilian Probst Gutenberg; Wulff-Nilsen, Christian (arXiv.org, 2020)
In the decremental \((1+\epsilon)\)-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 \(s \in V\), and we are asked to process edge deletions efficiently and answer queries for distance estimates \(\widetilde{\mathbf{dist}}_G(s,v)\) for each \(v \in V\), at any stage, such that \(\mathbf{dist}_G(s,v) \leq \widetilde{\mathbf{dist}}_G(s,v) \leq (1+ \epsilon)\mathbf{dist}_G(s,v)\). In the decremental \((1+\epsilon)\)-approximate All-Pairs Shortest Path (APSP) problem, we are asked to answer queries for distance estimates \(\widetilde{\mathbf{dist}}_G(u,v)\) for every \(u,v \in V\). In this article, we consider the problems for undirected, unweighted graphs. We present a new \emph{deterministic} algorithm for the decremental \((1+\epsilon)\)-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 \(\tilde{O}(n^2)\) and the best existing algorithm for sparse graphs with running time \(\tilde{O}(n^{1.25}\sqrt{m})\) [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 \emph{sparse} emulator in the deterministic setting. As a by-product, we also obtain a new simple deterministic algorithm for the decremental \((1+\epsilon)\)-approximate APSP problem with near-optimal total running time \(\tilde{O}(mn /\epsilon)\) matching the time complexity of the sophisticated but rather involved algorithm by Henzinger, Forster and Nanongkai [FOCS 2013].