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Register a new userNear-Optimal Decremental Hopsets with Applications
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Yasamin Nazari
Publication date
2020
fields
Informatics Engineering
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English
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Given a weighted undirected graph $G=(V,E,w)$, a hopset $H$ of hopbound $beta$ and stretch $(1+epsilon)$ is a set of edges such that for any pair of nodes $u, v in V$, there is a path in $G cup H$ of at most $beta$ hops, whose length is within a $(1+epsilon)$ factor from the distance between $u$ and $v$ in $G$. We show the first efficient decremental algorithm for maintaining hopsets with a polylogarithmic hopbound. The update time of our algorithm matches the best known static algorithm up to polylogarithmic factors. All the previous decremental hopset constructions had a superpolylogarithmic (but subpolynomial) hopbound of $2^{log^{Omega(1)} n}$ [Bernstein, FOCS09; HKN, FOCS14; Chechik, FOCS18]. By applying our decremental hopset construction, we get improved or near optimal bounds for several distance problems. Most importantly, we show how to decrementally maintain $(2k-1)(1+epsilon)$-approximate all-pairs shortest paths (for any constant $k geq 2)$, in $tilde{O}(n^{1/k})$ amortized update time and $O(k)$ query time. This significantly improves (by a polynomial factor) over the update-time of the best previously known decremental algorithm in the constant query time regime. Moreover, it improves over the result of [Chechik, FOCS18] that has a query time of $O(log log(nW))$, where $W$ is the aspect ratio, and the amortized update time is $n^{1/k}cdot(frac{1}{epsilon})^{tilde{O}(sqrt{log n})}$. For sparse graphs our construction nearly matches the best known static running time/ query time tradeoff.
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In the decremental Single-Source Shortest Path problem (SSSP), we are given a weighted directed graph $G=(V,E,w)$ undergoing edge deletions and a source vertex $r in V$; let $n = |V|, m = |E|$ and $W$ be the aspect ratio of the graph. The goal is to obtain a data structure that maintains shortest paths from $r$ to all vertices in $V$ and can answer distance queries in $O(1)$ time, as well as return the corresponding path $P$ in $O(|P|)$ time. This problem was first considered by Even and Shiloach [JACM81], who provided an algorithm with total update time $O(mn)$ for unweighted undirected graphs; this was later extended to directed weighted graphs [FOCS95, STOC99]. There are conditional lower bounds showing that $O(mn)$ is in fact near-optimal [ESA04, FOCS14, STOC15, STOC20]. In a breakthrough result, Forster et al. showed that it is possible to achieve total update time $mn^{0.9+o(1)}log W$ if the algorithm is allowed to return $(1+{epsilon})$-approximate paths, instead of exact ones [STOC14, ICALP15]. No further progress was made until Probst Gutenberg and Wulff-Nilsen [SODA20] provided a new approach for the problem, which yields total time $tilde{O}(min{m^{2/3}n^{4/3}log W, (mn)^{7/8} log W})$. Our result builds on this recent approach, but overcomes its limitations by introducing a significantly more powerful abstraction, as well as a different core subroutine. Our new framework yields a decremental $(1+{epsilon})$-approximate SSSP data structure with total update time $tilde{O}(n^2 log^4 W)$. Our algorithm is thus near-optimal for dense graphs with polynomial edge-weights. Our framework can also be applied to sparse graphs to obtain total update time $tilde{O}(mn^{2/3} log^3 W)$. Our main technique allows us to convert SSSP algorithms for DAGs to ones for general graphs, which we believe has significant potential to influence future work.
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 $Oleft (m^{1+O(epsilon)}cdot (log m)^{O(1/epsilon^2)}cdot log Lright )$.
We give the first Congested Clique algorithm that computes a sparse hopset with polylogarithmic hopbound in polylogarithmic time. Given a graph $G=(V,E)$, a $(beta,epsilon)$-hopset $H$ with hopbound $beta$, is a set of edges added to $G$ such that for any pair of nodes $u$ and $v$ in $G$ there is a path with at most $beta$ hops in $G cup H$ with length within $(1+epsilon)$ of the shortest path between $u$ and $v$ in $G$. Our hopsets are significantly sparser than the recent construction of Censor-Hillel et al. [6], that constructs a hopset of size $tilde{O}(n^{3/2})$, but with a smaller polylogarithmic hopbound. On the other hand, the previously known constructions of sparse hopsets with polylogarithmic hopbound in the Congested Clique model, proposed by Elkin and Neiman [10],[11],[12], all require polynomial rounds. One tool that we use is an efficient algorithm that constructs an $ell$-limited neighborhood cover, that may be of independent interest. Finally, as a side result, we also give a hopset construction in a variant of the low-memory Massively Parallel Computation model, with improved running time over existing algorithms.
We study the vertex-decremental Single-Source Shortest Paths (SSSP) problem: given an undirected graph $G=(V,E)$ with lengths $ell(e)geq 1$ on its edges and a source vertex $s$, we need to support (approximate) shortest-path queries in $G$, as $G$ undergoes vertex deletions. In a shortest-path query, given a vertex $v$, we need to return a path connecting $s$ to $v$, whose length is at most $(1+epsilon)$ times the length of the shortest such path, where $epsilon$ is a given accuracy parameter. The problem has many applications, for example to flow and cut problems in vertex-capacitated graphs. Our main result is a randomized algorithm for vertex-decremental SSSP with total expected update time $O(n^{2+o(1)}log L)$, that responds to each shortest-path query in $O(nlog L)$ time in expectation, returning a $(1+epsilon)$-approximate shortest path. The algorithm works against an adaptive adversary. The main technical ingredient of our algorithm is an $tilde O(|E(G)|+ n^{1+o(1)})$-time algorithm to compute a emph{core decomposition} of a given dense graph $G$, which allows us to compute short paths between pairs of query vertices in $G$ efficiently. We believe that this core decomposition algorithm may be of independent interest. We use our result for vertex-decremental SSSP to obtain $(1+epsilon)$-approximation algorithms for maximum $s$-$t$ flow and minimum $s$-$t$ cut in vertex-capacitated graphs, in expected time $n^{2+o(1)}$, and an $O(log^4n)$-approximation algorithm for the vertex version of the sparsest cut problem with expected running time $n^{2+o(1)}$. These results improve upon the previous best known results for these problems in the regime where $m= omega(n^{1.5 + o(1)})$.
In the model of online caching with machine learned advice, introduced by Lykouris and Vassilvitskii, the goal is to solve the caching problem with an online algorithm that has access to next-arrival predictions: when each input element arrives, the algorithm is given a prediction of the next time when the element will reappear. The traditional model for online caching suffers from an $Omega(log k)$ competitive ratio lower bound (on a cache of size $k$). In contrast, the augmented model admits algorithms which beat this lower bound when the predictions have low error, and asymptotically match the lower bound when the predictions have high error, even if the algorithms are oblivious to the prediction error. In particular, Lykouris and Vassilvitskii showed that there is a prediction-augmented caching algorithm with a competitive ratio of $O(1+min(sqrt{eta/OPT}, log k))$ when the overall $ell_1$ prediction error is bounded by $eta$, and $OPT$ is the cost of the optimal offline algorithm. The dependence on $k$ in the competitive ratio is optimal, but the dependence on $eta/OPT$ may be far from optimal. In this work, we make progress towards closing this gap. Our contributions are twofold. First, we provide an improved algorithm with a competitive ratio of $O(1 + min((eta/OPT)/k, 1) log k)$. Second, we provide a lower bound of $Omega(log min((eta/OPT)/(k log k), k))$.
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