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تسجيل مستخدم جديدDynamic Approximate Shortest Paths and Beyond: Subquadratic and Worst-Case Update Time
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Consider the following distance query for an $n$-node graph $G$ undergoing edge insertions and deletions: given two sets of nodes $I$ and $J$, return the distances between every pair of nodes in $Itimes J$. This query is rather general and captures sever
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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].
In the dynamic minimum set cover problem, a challenge is to minimize the update time while guaranteeing close to the optimal $min(O(log n), f)$ approximation factor. (Throughout, $m$, $n$, $f$, and $C$ are parameters denoting the maximum number of se
ts, number of elements, frequency, and the cost range.) In the high-frequency range, when $f=Omega(log n)$, this was achieved by a deterministic $O(log n)$-approximation algorithm with $O(f log n)$ amortized update time [Gupta et al. STOC17]. In the low-frequency range, the line of work by Gupta et al. [STOC17], Abboud et al. [STOC19], and Bhattacharya et al. [ICALP15, IPCO17, FOCS19] led to a deterministic $(1+epsilon)f$-approximation algorithm with $O(f log (Cn)/epsilon^2)$ amortized update time. In this paper we improve the latter update time and provide the first bounds that subsume (and sometimes improve) the state-of-the-art dynamic vertex cover algorithms. We obtain: 1. $(1+epsilon)f$-approximation ratio in $O(flog^2 (Cn)/epsilon^3)$ worst-case update time: No non-trivial worst-case update time was previously known for dynamic set cover. Our bound subsumes and improves by a logarithmic factor the $O(log^3 n/text{poly}(epsilon))$ worst-case update time for unweighted dynamic vertex cover (i.e., when $f=2$ and $C=1$) by Bhattacharya et al. [SODA17]. 2. $(1+epsilon)f$-approximation ratio in $Oleft((f^2/epsilon^3)+(f/epsilon^2) log Cright)$ amortized update time: This result improves the previous $O(f log (Cn)/epsilon^2)$ update time bound for most values of $f$ in the low-frequency range, i.e. whenever $f=o(log n)$. It is the first that is independent of $m$ and $n$. It subsumes the constant amortized update time of Bhattacharya and Kulkarni [SODA19] for unweighted dynamic vertex cover (i.e., when $f = 2$ and $C = 1$).
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].
One of the primary goals of the mathematical analysis of algorithms is to provide guidance about which algorithm is the best for solving a given computational problem. Worst-case analysis summarizes the performance profile of an algorithm by its wors
t performance on any input of a given size, implicitly advocating for the algorithm with the best-possible worst-case performance. Strong worst-case guarantees are the holy grail of algorithm design, providing an application-agnostic certification of an algorithms robustly good performance. However, for many fundamental problems and performance measures, such guarantees are impossible and a more nuanced analysis approach is called for. This chapter surveys several alternatives to worst-case analysis that are discussed in detail later in the book.
In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph $G$ and a source node $s$ the goal is to maintain shortest distances between $s$ and all other nodes in $G$ under a sequence of online adversar
ial edge deletions. In their seminal work, Even and Shiloach [JACM 1981] presented an exact solution to the problem in unweighted graphs with only $O(mn)$ total update time over all edge deletions. Their classic algorithm was the state of the art for the decremental SSSP problem for three decades, even when approximate shortest paths are allowed. A series of results showed how to improve upon $O(mn)$ if approximation is allowed, culminating in a recent breakthrough of Henzinger, Krinninger and Nanongkai [FOCS 14], who presented a $(1+epsilon)$-approximate algorithm for undirected weighted graphs whose total update time is near linear: $O(m^{1+o(1)}log(W))$, where $W$ is the ratio of the heaviest to the lightest edge weight in the graph. In this paper they posed as a major open problem the question of derandomizing their result. Until very recently, all known improvements over the Even-Shiloach algorithm were randomized and required the assumption of a non-adaptive adversary. In STOC 2016, Bernstein and Chechik showed the first emph{deterministic} algorithm to go beyond $O(mn)$ total update time: the algorithm is also $(1+epsilon)$-approximate, and has total update time $tilde{O}(n^2)$. In SODA 2017, the same authors presented an algorithm with total update time $tilde{O}(mn^{3/4})$. However, both algorithms are restricted to undirected, unweighted graphs. We present the emph{first} deterministic algorithm for emph{weighted} undirected graphs to go beyond the $O(mn)$ bound. The total update time is $tilde{O}(n^2 log(W))$.
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