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].
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