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Distributed approximation algorithms for weighted shortest paths
- In Proc. of the Symp. on Theory of Comp. (STOC
"... A distributed network is modeled by a graph having n nodes (processors) and diameter D. We study the time complexity of approximating weighted (undirected) shortest paths on distributed networks with a O(log n) bandwidth restriction on edges (the standard synchronous CONGEST model). In this problem ..."
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A distributed network is modeled by a graph having n nodes (processors) and diameter D. We study the time complexity of approximating weighted (undirected) shortest paths on distributed networks with a O(log n) bandwidth restriction on edges (the standard synchronous CONGEST model). In this problem every node wants to know its distance to some others nodes. The question whether approximation algorithms help speed up the shortest paths computation (more precisely distance computation) was raised since at least 2004 by Elkin (SIGACT News 2004). The unweighted case of this problem is well-understood while its weighted counterpart is fundamental problem in the area of distributed approximation algorithms and remains widely open. We present new algorithms for computing both single-source shortest paths (SSSP) and all-pairs shortest paths (APSP) in the weighted case. Our main result is an algorithm for SSSP. Previous results are the classic O(n)-time Bellman-Ford algorithm and an Õ(n1/2+1/2k + D)-time (8kdlog(k + 1)e − 1)-approximation algorithm, for any integer k ≥ 1, which follows from the result of Lenzen and Patt-Shamir (STOC 2013). (Note that Lenzen and Patt-Shamir in fact solve a harder problem, and we use Õ(·) to hide the
Near-Optimal Distributed Approximation of Minimum-Weight Connected Dominating Set
"... This paper presents a near-optimal distributed approximation algorithm for the minimum-weight 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 near-optimal distributed approximation algorithm for the minimum-weight 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 NP-hard 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 global-connectivity backbone.
Improved Distributed Steiner Forest Construction Extended Abstract
"... We present new distributed algorithms for constructing a Steiner Forest in the congest model. Our deterministic algorithm finds, for any given constant ε> 0, a (2 + ε)-approximation in Õ(sk +√min {st, n}) rounds, where s is the shortest path diameter, t is the number of terminals, k is the numbe ..."
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We present new distributed algorithms for constructing a Steiner Forest in the congest model. Our deterministic algorithm finds, for any given constant ε> 0, a (2 + ε)-approximation in Õ(sk +√min {st, n}) rounds, where s is the shortest path diameter, t is the number of terminals, k is the number of terminal components in the input, and n is the number of nodes. Our randomized algorithm finds, with high probability, an O(logn)-approximation in time Õ(k + min {s,√n} + D), where D is the unweighted di-ameter of the network. We also prove a matching lower bound of Ω̃(k+min {s,√n}+D) on the running time of any distributed approximation algorithm for the Steiner Forest problem. Previous algorithms were randomized, and ob-tained either an O(logn)-approximation in Õ(sk) time, or an O(1/ε)-approximation in Õ((√n+ t)1+ε +D) time. 1.
Efficient Distributed Source Detection with Limited Bandwidth
"... Given a simple graph G = (V, E) and a set of sources S ⊆ V, denote for each node v ∈ V by L (∞) v the lexicographically ordered list of distance/source pairs (d(s, v), s), where s ∈ S. For integers d, k ∈ N∪{∞}, we consider the source detection, or (S, d, k)-detection task, requiring each node v to ..."
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Given a simple graph G = (V, E) and a set of sources S ⊆ V, denote for each node v ∈ V by L (∞) v the lexicographically ordered list of distance/source pairs (d(s, v), s), where s ∈ S. For integers d, k ∈ N∪{∞}, we consider the source detection, or (S, d, k)-detection task, requiring each node v to learn the (if for all of them d(s, v) ≤ d) or all entries (d(s, v), s) ∈ L (∞) v satisfying that d(s, v) ≤ d (otherwise). Solutions to this problem provide natural generalizations of concurrent breadth-first search (BFS) tree constructions. For example, the special case of k = ∞ requires each source s ∈ S to build a complete BFS tree rooted at s, whereas the special case of d = ∞ and S = V requires constructing a partial BFS tree comprising at least k nodes from every node in V. In this work, we give a simple, near-optimal solution for the source detection task in the CONGEST model, where messages contain at most O(log n) bits, running in d + k rounds. We demonstrate its utility for various routing problems, exact and approximate diameter computation, and spanner construction. For those problems, we obtain algorithms in the CONGEST model that are faster and in some cases much simpler than previous solutions. first k entries of L (∞)
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 multipli-cation 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 multipli-cation 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 algorith-mics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: – triangle and 4-cycle counting in O(n0.158) rounds, im-proving upon the O(n1/3) triangle counting algorithm of Dolev et al. [DISC 2012], – a (1 + o(1))-approximation of all-pairs 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 non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.
Fast Partial Distance Estimation and Applications
"... We study approximate distributed solutions to the weighted all-pairs-shortest-paths (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 all-pairs-shortest-paths (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 in-teger 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 approxima-tion 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 ap-proximation 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.
Near-Optimal 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 n-node 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 n-node 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 shortest-path-diameter 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 hop-diameter 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 near-optimal.
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 re-sult 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 re-sult 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 dis-tributed 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 non-trivial lower bounds for quantum distributed algo-rithms for several graph optimization (both exact and approximate versions) as well as verifica-tion 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 communica-tion complexity. The Server model is the standard two-party communication complexity model
Fast Routing Table Construction . . .
, 2013
"... We describe a distributed randomized algorithm to construct routing tables. Given 0 < ε ≤ 1/2, the algorithm runs in time Õ(n1/2+ε + HD), where n is the number of nodes and HD denotes the diameter of the network in hops (i.e., as if the network is unweighted). The weighted length of the produced ..."
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We describe a distributed randomized algorithm to construct routing tables. Given 0 < ε ≤ 1/2, the algorithm runs in time Õ(n1/2+ε + HD), where n is the number of nodes and HD denotes the diameter of the network in hops (i.e., as if the network is unweighted). The weighted length of the produced routes is at most O(ε −1 log ε −1) times the optimal weighted length. This is the first algorithm to break the Ω(n) complexity barrier for computing weighted shortest paths even for a single source. Moreover, the algorithm nearly meets the ˜ Ω(n 1/2 + HD) lower bound for distributed computation of routing tables and approximate distances (with optimality, up to polylog factors, for ε = 1 / log n). The presented techniques have many applications, including improved distributed approximation algorithms for Generalized Steiner Forest, all-pairs distance estimation, and estimation of the weighted diameter.
