Spanners are fundamental graph structures that sparsify graphs at the cost of small stretch. In particular, in recent years, many sequential algorithms constructing additive all-pairs spanners were designed, providing...
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Spanners are fundamental graph structures that sparsify graphs at the cost of small stretch. In particular, in recent years, many sequential algorithms constructing additive all-pairs spanners were designed, providing very sparse small-stretch subgraphs. Remarkably, it was then shown that the known (+6)-spanner constructions are essentially the sparsest possible, that is, larger additive stretch cannot guarantee a sparser spanner, which brought the stretch-sparsity trade-off to its limit. distributed constructions of spanners are also abundant. However, for additive spanners, while there were algorithms constructing (+2) and (+4)-all-pairs spanners, the sparsest case of (+6)-spanners remained elusive. We remedy this by designing a new sequential algorithm for constructing a (+6)-spanner with the essentially-optimal sparsity of (O) over tilde (n(4/3)) edges. We then show a distributed implementation of our algorithm, answering an open problem in [12]. A main ingredient in our distributed algorithm is an efficient construction of multiple weighted BFS trees. A weighted BFS tree is a BFS tree in a weighted graph, that consists of the lightest among all shortest paths from the root to each node. We present a distributed algorithm in the CONGEST model, that constructs multiple weighted BFS trees in vertical bar S vertical bar + D - 1 rounds, where S is the set of sources and D is the diameter of the network graph. (C) 2020 Elsevier B.V. All rights reserved.
Many modern systems are built on top of large-scale networks like the Internet. This article provides an overview of a dissertation [29] that addresses the complexity of classic graph problems like the vertex coloring...
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Many modern systems are built on top of large-scale networks like the Internet. This article provides an overview of a dissertation [29] that addresses the complexity of classic graph problems like the vertex coloring problem in such networks. It has been known for a long time that randomization helps significantly in solving many of these problems, whereas the best known deterministic algorithms have been exponentially slower. In the first part of the dissertation we use a complexity theoretic approach to show that several problems are complete in the following sense: An efficient deterministic algorithm for any complete problem would imply an efficient algorithm for all problems that can be solved efficiently with a randomized algorithm. Among the complete problems is a rudimentary looking graph coloring problem that can be solved by a randomized algorithm without any communication. In further parts of the dissertation we develop efficient distributedalgorithms for several problems where the most important problems are distributed versions of integer linear programs, the vertex coloring problem and the edge coloring problem. We also prove a lower bound on the runtime of any deterministic algorithm that solves the vertex coloring problem in a weak variant of the standard model of the area.
Given a graph G = (V, E), an (alpha, beta)-ruling set is a subset S subset of V such that the distance between any two vertices in S is at least alpha, and the distance between any vertex in V and the closest vertex i...
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ISBN:
(纸本)9781728196213
Given a graph G = (V, E), an (alpha, beta)-ruling set is a subset S subset of V such that the distance between any two vertices in S is at least alpha, and the distance between any vertex in V and the closest vertex in S is at most beta. We present lower bounds for distributedly computing ruling sets. More precisely, for the problem of computing a (2, beta)-ruling set (and hence also any (alpha, beta)-ruling set with alpha > 2) in the LOCAL model of distributed computing, we show the following, where n denotes the number of vertices, Delta the maximum degree, and c is some universal constant independent of n and Delta. Any deterministic algorithm requires Omega(min{log Delta/beta log log Delta, log(Delta) n}) rounds, for all beta <= c . min{root log Delta/log log Delta, log(Delta) n}. By optimizing Delta, this implies a deterministic lower bound of Omega(root log n/beta log log n) for all beta <= c 3 root log n/log log n. Any randomized algorithm requires Omega(min{log Delta/beta log log Delta, log(Delta) n}) rounds, for all beta <= c . min{root log Delta/log log Delta, log(Delta) n}. By optimizing Delta, this implies a randomized lower bound of Omega(root log log n/beta log log log n) for all beta <= c 3 root log log n/log log log n. For beta > 1, this improves on the previously best lower bound of Omega(log* n) rounds that follows from the 30-year-old bounds of Linial [FOCS'87] and Naor [***.'91] (resp. Omega(1) rounds if beta is an element of omega(log* n)). For beta = 1, i.e., for the problem of computing a maximal independent set (which is nothing else than a (2, 1)-ruling set), our results improve on the previously best lower bound of Omega(log* n) on trees, as our bounds already hold on trees.
We overview a recent line of work [Rozho.n and Ghaffari at STOC 2020;Ghaffari, Harris, and Kuhn at FOCS 2018;and Ghaffari, Kuhn, and Maus at STOC 2017], which proved that any (locallycheckable) graph problem that admi...
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ISBN:
(纸本)9783030549206;9783030549213
We overview a recent line of work [Rozho.n and Ghaffari at STOC 2020;Ghaffari, Harris, and Kuhn at FOCS 2018;and Ghaffari, Kuhn, and Maus at STOC 2017], which proved that any (locallycheckable) graph problem that admits an efficient randomized distributed algorithm also admits an efficient deterministic distributed algorithm, thereby resolving several central and decades-old open problems in distributed graph algorithms. We present a short and self-contained version of the proofs, and conclude by discussing several related questions that remain open. This article accompanies a keynote talk of the author at the International Colloquium on Structural Information and Communication Complexity (SIROCCO) 2020. The writing is based on [24,28,45] and primarily targets non-experts.
We study the maximum cardinality matching problem in a standard distributed setting, where the nodes V of a given n-node network graph G = (V, E) communicate over the edges E in synchronous rounds. More specifically, ...
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Naor, Parter, and Yogev (SODA 2020) have recently demonstrated the existence of a distributed interactive proof for planarity (i.e., for certifying that a network is planar), using a sophisticated generic technique fo...
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ISBN:
(纸本)9781450375825
Naor, Parter, and Yogev (SODA 2020) have recently demonstrated the existence of a distributed interactive proof for planarity (i.e., for certifying that a network is planar), using a sophisticated generic technique for constructing distributed IP protocols based on sequential IP protocols. The interactive proof for planarity is based on a distributed certification of the correct execution of any given sequential linear-time algorithm for planarity testing. It involves three interactions between the prover and the randomized distributed verifier (i.e., it is a dMAM protocol), and uses small certificates, on O(logn) bits in n-node networks. We show that a single interaction from the prover suffices, and randomization is unecessary, by providing an explicit description of a proof-labeling scheme for planarity, still using certificates on just O(logn) bits. We also show that there are no proof-labeling schemes - in fact, even no locally checkable proofs - for planarity using certificates on o(logn) bits.
The weighted vertex cover problem revolves around selecting a subset of vertices that covers a target edge set while minimizing the total cost of the selected vertices. We consider a variant of this classic optimizati...
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The weighted vertex cover problem revolves around selecting a subset of vertices that covers a target edge set while minimizing the total cost of the selected vertices. We consider a variant of this classic optimization problem where the target edge set is not fully known;rather, it is characterized by a probability distribution. Adhering to the model of two-stage stochastic optimization, the execution is divided into two stages. In the first stage, the decision maker selects a vertex subset based on the probabilistic forecast of the target edge set. In the second stage, the target edge set is revealed, and the decision maker can augment the initial vertex subset with additional vertices to ensure coverage;however, this augmentation is more expensive due to increased vertex costs. This paper initiates the study of the two-stage stochastic vertex cover problem in the realm of distributed graph algorithms, where the decision-making process is distributed among the graph's vertices. We consider two known stochastic optimization variants: the independent sampling model, where the edges in the target set are drawn independently from some probability distribution;and the finite scenario model, where the probability distribution over the target edge set is provided explicitly. For both variants, we devise efficient distributedalgorithms based on a novel adaptation of the distributed primal-dual technique to linear programs resulting from the stochastic optimization problems' relaxation.
This keynote talk will provide an overview of a recent line of work [Rozhoň and Ghaffari at STOC 2020; Ghaffari, Harris, and Kuhn at FOCS 2018; and Ghaffari, Kuhn, and Maus at STOC 2017], which presented the first ef...
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ISBN:
(纸本)9781450389334
This keynote talk will provide an overview of a recent line of work [Rozhoň and Ghaffari at STOC 2020; Ghaffari, Harris, and Kuhn at FOCS 2018; and Ghaffari, Kuhn, and Maus at STOC 2017], which presented the first efficient deterministic network decomposition algorithm as well as a general derandomization result for distributed graph algorithms. Informally, the derandomization result shows that any (locally-checkable) graph problem that admits an efficient randomized distributed algorithm also admits an efficient deterministic distributed algorithm. These results resolve several central and decades-old open problems in distributed graph algorithms.
We focus on two classes of problems in graph mining: (1) finding trees and (2) anomaly detection in complex networks using scan statistics. These are fundamental problems in a broad class of applications. Most of the ...
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We focus on two classes of problems in graph mining: (1) finding trees and (2) anomaly detection in complex networks using scan statistics. These are fundamental problems in a broad class of applications. Most of the parallel algorithms for such problems are either based on heuristics, which do not scale very well, or use techniques like color coding, which have a high memory overhead. In this paper, we develop a novel approach for parallelizing both these classes of problems, using an algebraic representation of subgraphs as monomials-this methodology involves detecting multilinear terms in multivariate polynomials. Our algorithms show good scaling over a large regime, and they run on networks with close to half one billion edges. The resulting parallel algorithm for trees is able to scale to subgraphs of size 18, which has not been done before, and it significantly outperforms the best prior color coding based method (FASCIA) by more than two orders of magnitude. Our algorithm for network scan statistics is the first such parallelization, and it is able to handle a broad class of scan statistics functions with the same approach. Published by Elsevier Inc.
The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it in O(n) time, where n i...
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The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it in O(n) time, where n is the number of vertices in the input graph G. Peleg and Rubinovich [49] showed a lower bound of (Omega) over tilde (D + root n) for this problem, where D is the hop-diameter of G. Whether or not this problem can be solved in o(n) time when D is relatively small is a major open question. Despite intensive research [10, 17, 33, 41, 45] that yielded near-optimal algorithms for the approximate variant of this problem, no progress was reported for the original problem. In this article, we answer this question in the affirmative. We devise an algorithm that requires O((n log n)(5/6)) time, for D = O(root n logn), and O(D-1/3 . (n log n)(2/3)) time, for larger D. This running time is sublinear in n in almost the entire range of parameters, specifically, for D = o(n/log(2) n). We also generalize our result in two directions. One is when edges have bandwidth b >= 1, and the other is the s-sources shortest paths problem. For both problems, our algorithm provides bounds that improve upon the previous state-of-the-art in almost the entire range of parameters. In particular, we provide an all-pairs shortest paths algorithm that requires O(n(5/3) . log(2/3) n) time, even for b <= 1, for all values of D. We also devise the first algorithm with non-trivial complexity guarantees for computing exact shortest paths in the multipass semi-streaming model of computation. From the technical viewpoint, our distributed algorithm computes a hopset G '' of a skeleton graph G' of G without first computing G' itself. We then conduct a Bellman-Ford exploration in G'. G '', while computing the required edges of G' on the fly. As a result, our distributed algorithm computes exactly those edges of G' that it really needs, rather than computing approximately the entire G'.
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