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Decremental All-Pairs Shortest Paths in Deterministic Near-Linear Time

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 نشر من قبل Julia Chuzhoy
 تاريخ النشر 2021
  مجال البحث الهندسة المعلوماتية
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 تأليف Julia Chuzhoy




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We study the decremental All-Pairs Shortest Paths (APSP) problem in undirected edge-weighted graphs. The input to the problem is an $n$-vertex $m$-edge graph $G$ with non-negative edge lengths, that undergoes a sequence of edge deletions. The goal is to support approximate shortest-path queries: given a pair $x,y$ of vertices of $G$, return a path $P$ connecting $x$ to $y$, whose length is within factor $alpha$ of the length of the shortest $x$-$y$ path, in time $tilde O(|E(P)|)$, where $alpha$ is the approximation factor of the algorithm. APSP is one of the most basic and extensively studied dynamic graph problems. A long line of work culminated in the algorithm of [Chechik, FOCS 2018] with near optimal guarantees for the oblivious-adversary setting. Unfortunately, adaptive-adversary setting is still poorly understood. For unweighted graphs, the algorithm of [Henzinger, Krinninger and Nanongkai, FOCS 13, SICOMP 16] achieves a $(1+epsilon)$-approximation with total update time $tilde O(mn/epsilon)$; the best current total update time of $n^{2.5+O(epsilon)}$ is achieved by the deterministic algorithm of [Chuzhoy, Saranurak, SODA21], with $2^{O(1/epsilon)}$-multiplicative and $2^{O(log^{3/4}n/epsilon)}$-additive approximation. To the best of our knowledge, for arbitrary non-negative edge weights, the fastest current adaptive-update algorithm has total update time $O(n^{3}log L/epsilon)$, achieving a $(1+epsilon)$-approximation. Here, L is the ratio of longest to shortest edge lengths. Our main result is a deterministic algorithm for decremental APSP in undirected edge-weighted graphs, that, for any $Omega(1/loglog m)leq epsilon< 1$, achieves approximation factor $(log m)^{2^{O(1/epsilon)}}$, with total update time $Oleft (m^{1+O(epsilon)}cdot (log m)^{O(1/epsilon^2)}cdot log Lright )$.



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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 deletion s 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].
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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Given a directed weighted graph $G=(V,E)$ undergoing vertex insertions emph{and} deletions, the All-Pairs Shortest Paths (APSP) problem asks to maintain a data structure that processes updates efficiently and returns after each update the distance ma trix to the current version of $G$. In two breakthrough results, Italiano and Demetrescu [STOC 03] presented an algorithm that requires $tilde{O}(n^2)$ emph{amortized} update time, and Thorup showed in [STOC 05] that emph{worst-case} update time $tilde{O}(n^{2+3/4})$ can be achieved. In this article, we make substantial progress on the problem. We present the following new results: (1) We present the first deterministic data structure that breaks the $tilde{O}(n^{2+3/4})$ worst-case update time bound by Thorup which has been standing for almost 15 years. We improve the worst-case update time to $tilde{O}(n^{2+5/7}) = tilde{O}(n^{2.71..})$ and to $tilde{O}(n^{2+3/5}) = tilde{O}(n^{2.6})$ for unweighted graphs. (2) We present a simple deterministic algorithm with $tilde{O}(n^{2+3/4})$ worst-case update time ($tilde{O}(n^{2+2/3})$ for unweighted graphs), and a simple Las-Vegas algorithm with worst-case update time $tilde{O}(n^{2+2/3})$ ($tilde{O}(n^{2 + 1/2})$ for unweighted graphs) that works against a non-oblivious adversary. Both data structures require space $tilde{O}(n^2)$. These are the first exact dynamic algorithms with truly-subcubic update time emph{and} space usage. This makes significant progress on an open question posed in multiple articles [COCOON01, STOC 03, ICALP04, Encyclopedia of Algorithms 08] and is critical to algorithms in practice [TALG 06] where large space usage is prohibitive. Moreover, they match the worst-case update time of the best previous algorithms and the second algorithm improves upon a Monte-Carlo algorithm in a weaker adversary model with the same running time [SODA 17].
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