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NearOptimal Distributed Approximation of MinimumWeight Connected Dominating Set
"... This paper presents a nearoptimal distributed approximation algorithm for the minimumweight connected dominating set (MCDS) problem. We use the standard distributed message passing model called the CONGEST model in which in each round each node can send O(log n) bits to each neighbor. The presente ..."
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This paper presents a nearoptimal distributed approximation algorithm for the minimumweight connected dominating set (MCDS) problem. We use the standard distributed message passing model called the CONGEST model in which in each round each node can send O(log n) bits to each neighbor. The presented algorithm finds an O(log n) approximation in Õ(D+√n) rounds, where D is the network diameter and n is the number of nodes. MCDS is a classical NPhard problem and the achieved approximation factor O(log n) is known to be optimal up to a constant factor, unless P = NP. Furthermore, the Õ(D +√n) round complexity is known to be optimal modulo logarithmic factors (for any approximation), following [Das Sarma et al.—STOC’11]. 1 Introduction and Related Work Connected dominating set (CDS) is one of the classical structures studied in graph optimization problems which also has deep roots in networked computation. For instance, CDSs have been used rather extensively in distributed algorithms for wireless networks (see e.g. [2, 3, 5–10, 30, 38, 39]), typically as a globalconnectivity backbone.
Distributed Computation of Largescale Graph Problems
, 2015
"... Abstract Motivated by the increasing need for fast distributed processing of largescale graphs such as the Web graph and various social networks, we study a number of fundamental graph problems in the messagepassing model, where we have k machines that jointly perform computation on an arbitrary ..."
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Abstract Motivated by the increasing need for fast distributed processing of largescale graphs such as the Web graph and various social networks, we study a number of fundamental graph problems in the messagepassing model, where we have k machines that jointly perform computation on an arbitrary nnode (typically, n ≫ k) input graph. The graph is assumed to be randomly partitioned among the k ≥ 2 machines (a common implementation in many real world systems). The communication is pointtopoint, and the goal is to minimize the time complexity, i.e., the number of communication rounds, of solving various fundamental graph problems. We present lower bounds that quantify the fundamental time limitations of distributively solving graph problems. We first show a lower bound of Ω(n/k) rounds for computing a spanning tree (ST) of the input graph. This result also implies the same bound for other fundamental problems such as computing a minimum spanning tree (MST), breadthfirst tree (BFS), and shortest paths tree (SPT). We also show an Ω(n/k 2 ) lower bound for connectivity, ST verification and other * Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371 & Centre for Quantum Technologies, Singapore 117543. Email: hklauck@gmail.com. This work is funded by the Singapore Ministry of Education (partly through the Academic Research Fund Tier 3 MOE2012T31009) and by the Singapore National Research Foundation. † KTH Royal Institute of Technology, Sweden, and University of Vienna, Austria Email: danupon@gmail.com. Work done while at ICERM, Brown University, USA, and Nanyang Technological University, Singapore. To complement our lower bounds, we also give algorithms for various fundamental graph problems, e.g., PageRank, MST, connectivity, ST verification, shortest paths, cuts, spanners, covering problems, densest subgraph, subgraph isomorphism, finding triangles, etc. We show that problems such as PageRank, MST, connectivity, and graph covering can be solved inÕ(n/k) time (the notationÕ hides polylog(n) factors and an additive polylog(n) term); this shows that one can achieve almost linear (in k) speedup, whereas for shortest paths, we present algorithms that run inÕ(n/ √ k) time (for (1 + ǫ)factor approximation) and inÕ(n/k) time (for O(log n)factor approximation) respectively. Our results are a step towards understanding the complexity of distributively solving largescale graph problems.
AlmostTight Distributed Minimum Cut Algorithms
, 2014
"... We study the problem of computing the minimum cut in a weighted distributed messagepassing networks (the CONGEST model). Let λ be the minimum cut, n be the number of nodes (processors) in the network, and D be the network diameter. Our algorithm can compute λ exactly in O(( n log ∗ n+D)λ4 log2 n) t ..."
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We study the problem of computing the minimum cut in a weighted distributed messagepassing networks (the CONGEST model). Let λ be the minimum cut, n be the number of nodes (processors) in the network, and D be the network diameter. Our algorithm can compute λ exactly in O(( n log ∗ n+D)λ4 log2 n) time. To the best of our knowledge, this is the first paper that explicitly studies computing the exact minimum cut in the distributed setting. Previously, nontrivial sublinear time algorithms for this problem are known only for unweighted graphs when λ ≤ 3 due to Pritchard and Thurimella’s O(D)time and O(D + n1/2 log ∗ n)time algorithms for computing 2edgeconnected and 3edgeconnected components [ACM Transactions on Algorithms 2011]. By using the edge sampling technique of Karger [STOC 1994], we can convert this algorithm into a (1 + )approximation O(( n log ∗ n + D)−5 log3 n)time algorithm for any > 0. This improves over the previous (2 + )approximation O((
Decremental SingleSource Shortest Paths on Undirected Graphs in NearLinear Total Update Time
"... AbstractThe decremental singlesource shortest paths (SSSP) problem concerns maintaining the distances between a given source node s to every node in an nnode medge graph G undergoing edge deletions. While its static counterpart can be easily solved in nearlinear time, this decremental problem ..."
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AbstractThe decremental singlesource shortest paths (SSSP) problem concerns maintaining the distances between a given source node s to every node in an nnode medge graph G undergoing edge deletions. While its static counterpart can be easily solved in nearlinear time, this decremental problem is much more challenging even in the undirected unweighted case. In this case, the classic O(mn) total update time of Even and Shiloach (JACM 1981) In contrast to the previous results which rely on maintaining a sparse emulator, our algorithm relies on maintaining a socalled sparse (d, )hop set introduced by Cohen (JACM 2000) in the PRAM literature. A (d, )hop set of a graph G = (V, E) is a set E of weighted edges such that the distance between any pair of nodes in G can be (1 + )approximated by their dhop distance (given by a path containing at most d edges) on G = (V, E ∪E ). Our algorithm can maintain an (n o(1) , )hop set of nearlinear size in nearlinear time under edge deletions. It is the first of its kind to the best of our knowledge. To maintain the distances on this hop set, we develop a monotone boundedhop EvenShiloach tree. It results from extending and combining the monotone EvenShiloach tree of Henzinger, Krinninger, and Nanongkai (FOCS 2013) with the boundedhop SSSP technique of Bernstein (STOC 2013). These two new tools might be of independent interest.
Ami Paz Technion
"... In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an O(n1−2/ω) round matrix multiplication algorithm, where ω & ..."
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In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an O(n1−2/ω) round matrix multiplication algorithm, where ω < 2.3728639 is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: – triangle and 4cycle counting in O(n0.158) rounds, improving upon the O(n1/3) triangle counting algorithm of Dolev et al. [DISC 2012], – a (1 + o(1))approximation of allpairs shortest paths in O(n0.158) rounds, improving upon the Õ(n1/2)round (2+o(1))approximation algorithm of Nanongkai [STOC 2014], and – computing the girth in O(n0.158) rounds, which is the first nontrivial solution in this model. In addition, we present a novel constantround combinatorial algorithm for detecting 4cycles.
Fast Partial Distance Estimation and Applications
"... We study approximate distributed solutions to the weighted allpairsshortestpaths (APSP) problem in the congest model. We obtain the following results. • A deterministic (1 + ε)approximation to APSP with running time O(ε−2n logn) rounds. The best previously known algorithm was randomized and slow ..."
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We study approximate distributed solutions to the weighted allpairsshortestpaths (APSP) problem in the congest model. We obtain the following results. • A deterministic (1 + ε)approximation to APSP with running time O(ε−2n logn) rounds. The best previously known algorithm was randomized and slower by a Θ(logn) factor. In many cases, routing schemes involve relabeling, i.e., assigning new names to nodes and that are used in distance and routing queries. It is known that relabeling is necessary to achieve running times of o(n / logn). In the relabeling model, we obtain the following results. • A randomized O(k)approximation to APSP, for any integer k> 1, running in Õ(n1/2+1/k +D) rounds, where D is the hop diameter of the network. This algorithm simplifies the best previously known result and reduces its approximation ratio from O(k log k) to O(k). Also, the new algorithm uses O(logn)bit labels, which is asymptotically optimal. • A randomized O(k)approximation to APSP, for any integer k> 1, running in time Õ((nD)1/2 · n1/k + D) and producing compact routing tables of size Õ(n1/k). The node labels consist of O(k logn) bits. This improves on the approximation ratio of Θ(k2) for tables of that size achieved by the best previously known algorithm, which terminates faster, in Õ(n1/2+1/k +D) rounds. In addition, we improve on the time complexity of the best known deterministic algorithm for distributed approximate Steiner forest.
NearOptimal Distributed Tree Embedding
"... Tree embeddings are a powerful tool in the area of graph approximation algorithms. Roughly speaking, they transform problems on general graphs into much easier ones on trees. Fakcharoenphol, Rao, and Talwar (FRT) [STOC’04] present a probabilistic tree embedding that transforms nnode metrics into (p ..."
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Tree embeddings are a powerful tool in the area of graph approximation algorithms. Roughly speaking, they transform problems on general graphs into much easier ones on trees. Fakcharoenphol, Rao, and Talwar (FRT) [STOC’04] present a probabilistic tree embedding that transforms nnode metrics into (probability distributions over) trees, while stretching each pairwise distance by at most anO(log n) factor in expectation. This O(log n) stretch is optimal. Khan et al. [PODC’08] present a distributed algorithm that implements FRT in O(SPD log n) rounds, where SPD is the shortestpathdiameter of the weighted graph, and they explain how to use this embedding for various distributed approximation problems. Note that SPD can be as large as Θ(n), even in graphs where the hopdiameter D is a constant. Khan et al. noted that it would be interesting to improve this complexity. We show that this is indeed possible. More precisely, we present a distributed algorithm that constructs a tree embedding that is essentially as good as FRT in Õ(min{n0.5+ε,SPD} + D) rounds, for any constant ε> 0. A lower bound of Ω̃(min{n0.5,SPD}+D) rounds follows from Das Sarma et al. [STOC’11], rendering our round complexity nearoptimal.
Distributed Approximation of Minimum Routing Cost Trees *
"... Abstract We study the NPhard problem of approximating a Minimum Routing Cost Spanning Tree in the message passing model with limited bandwidth (CONGEST model). In this problem one tries to nd a spanning tree of a graph G over n nodes that minimizes the sum of distances between all pairs of nodes. ..."
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Abstract We study the NPhard problem of approximating a Minimum Routing Cost Spanning Tree in the message passing model with limited bandwidth (CONGEST model). In this problem one tries to nd a spanning tree of a graph G over n nodes that minimizes the sum of distances between all pairs of nodes. In the considered model every node can transmit a dierent (but short) message to each of its neighbors in each synchronous round. We provide a randomized (2 + ε)approximation with runtime O(D + log n ε ) for unweighted graphs. Here, D is the diameter of G. This improves over both, the (expected) approximation factor O(log n) and the runtime O(D log 2 n) stated in
Can Quantum Communication Speed Up Distributed Computation?
"... The focus of this paper is on quantum distributed computation, where we investigate whether quantum communication can help in speeding up distributed network algorithms. Our main result is that for certain fundamental network problems such as minimum spanning tree, minimum cut, and shortest paths, ..."
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The focus of this paper is on quantum distributed computation, where we investigate whether quantum communication can help in speeding up distributed network algorithms. Our main result is that for certain fundamental network problems such as minimum spanning tree, minimum cut, and shortest paths, quantum communication does not help in substantially speeding up distributed algorithms for these problems compared to the classical setting. In order to obtain this result, we extend the technique of Das Sarma et al. [SICOMP 2012] to obtain a uniform approach to prove nontrivial lower bounds for quantum distributed algorithms for several graph optimization (both exact and approximate versions) as well as verification problems, some of which are new even in the classical setting, e.g. tight randomized lower bounds for Hamiltonian cycle and spanning tree verification, answering an open problem of Das Sarma et al., and a lower bound in terms of the weight aspect ratio, matching the upper bounds of Elkin [STOC 2004]. Our approach introduces the Server model and Quantum Simulation Theorem which together provide a connection between distributed algorithms and communication complexity. The Server model is the standard twoparty communication complexity model