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تسجيل مستخدم جديدDeterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler
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Maximilian Probst Gutenberg
تاريخ النشر
2020
مجال البحث
الهندسة المعلوماتية
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English
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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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اقرأ أيضاً
In the decremental single-source shortest paths (SSSP) problem, the input is an undirected graph $G=(V,E)$ with $n$ vertices and $m$ edges undergoing edge deletions, together with a fixed source vertex $sin V$. The goal is to maintain a data structur
e that supports shortest-path queries: given a vertex $vin V$, quickly return an (approximate) shortest path from $s$ to $v$. The decremental all-pairs shortest paths (APSP) problem is defined similarly, but now the shortest-path queries are allowed between any pair of vertices of $V$. Both problems have been studied extensively since the 80s, and algorithms with near-optimal total update time and query time have been discovered for them. Unfortunately, all these algorithms are randomized and, more importantly, they need to assume an oblivious adversary. Our first result is a deterministic algorithm for the decremental SSSP problem on weighted graphs with $O(n^{2+o(1)})$ total update time, that supports $(1+epsilon)$-approximate shortest-path queries, with query time $O(|P|cdot n^{o(1)})$, where $P$ is the returned path. This is the first $(1+epsilon)$-approximation algorithm against an adaptive adversary that supports shortest-path queries in time below $O(n)$, that breaks the $O(mn)$ total update time bound of the classical algorithm of Even and Shiloah from 1981. Our second result is a deterministic algorithm for the decremental APSP problem on unweighted graphs that achieves total update time $O(n^{2.5+delta})$, for any constant $delta>0$, supports approximate distance queries in $O(loglog n)$ time; the algorithm achieves an $O(1)$-multiplicative and $n^{o(1)}$-additive approximation on the path length. All previous algorithms for APSP either assume an oblivious adversary or have an $Omega(n^{3})$ total update time when $m=Omega(n^{2})$.
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