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Register a new userFully-Dynamic All-Pairs Shortest Paths: Improved Worst-Case Time and Space Bounds
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Maximilian Probst Gutenberg
Publication date
2020
fields
Informatics Engineering
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English
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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 matrix 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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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
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 $Oleft (m^{1+O(epsilon)}cdot (log m)^{O(1/epsilon^2)}cdot log Lright )$.
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 sets, 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 $C = 1$).
This paper gives a new deterministic algorithm for the dynamic Minimum Spanning Forest (MSF) problem in the EREW PRAM model, where the goal is to maintain a MSF of a weighted graph with $n$ vertices and $m$ edges while supporting edge insertions and deletions. We show that one can solve the dynamic MSF problem using $O(sqrt n)$ processors and $O(log n)$ worst-case update time, for a total of $O(sqrt n log n)$ work. This improves on the work of Ferragina [IPPS 1995] which costs $O(log n)$ worst-case update time and $O(n^{2/3} log{frac{m}{n}})$ work.
The shortest secure path (routing) problem in communication networks has to deal with multiple attack layers e.g., man-in-the-middle, eavesdropping, packet injection, packet insertion, etc. Consider different probabilities for each such attack over an edge, probabilities that can differ across edges. Furthermore, usage of a single shortest path (for routing) implies possible traffic bottleneck, which should be avoided if possible, which we term pathneck security avoidance. Finding all Pareto-optimal solutions for the multi-criteria single-source single-destination shortest secure path problem with non-negative edge lengths might yield a solution with an exponential number of paths. In the first part of this paper, we study specific settings of the multi-criteria shortest secure path problem, which are based on prioritized multi-criteria and on $k$-shortest secure paths. In the second part, we show a polynomial-time algorithm that, given an undirected graph $G$ and a pair of vertices $(s,t)$, finds prioritized multi-criteria $2$-disjoint (vertex/edge) shortest secure paths between $s$ and $t$. In the third part of the paper, we introduce the $k$-disjoint all-criteria-shortest secure paths problem, which is solved in time $O(min(k|E|, |E|^{3/2}))$.
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