Distributed Approximation of Minimum k-edge-connected Spanning Subgraphs

Dory, Michal

doi:10.1145/3212734.3212760

Cited by 15 publications

(49 citation statements)

References 39 publications

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“…Here, h M ST denotes the height of the minimum spanning tree; note that this can be as large as Θ(n), even in networks with small diameter D. Thus, this round complexity can be much higher than the lower bound of Ω(D + √ n) [4,7]. Dory [8] provided a randomized algorithm that runs in the near-optimal run time of O(D + √ n) and provides an O(log n)-approximation. Two comments are in order.…”

Section: First Contribution -Better Approximationmentioning

confidence: 99%

“…Two comments are in order. First, for general k, one can obtain an O(log k)-approximation but in round complexity of O(knD) [17,32], and an O(k log n)-approximation in O(k(D log 3 n + n)) rounds [8]. Both of these complexities are at least linear in the network size and well-above our target round complexity.…”

Section: First Contribution -Better Approximationmentioning

confidence: 99%

“…Our First Result: Above we mentioned a 3-approximation with suboptimal time [4] and an optimal time with suboptimal approximation of O(log n) [8]. Our first main result is to provide an algorithm that gets the best of these two worlds (using a different technique), namely near-optimal time and constant approximation.…”

Section: First Contribution -Better Approximationmentioning

confidence: 99%

“…We think that achieving a constant approximation in the near-optimal time is the main strength of this result. Another advantage of our approach is that is gives a deterministic algorithm, while the O(log n)-approximation algorithm of [8] was randomized. Our algorithm also gives an O((D + √ n) log 2 n)-round (4+ )-approximation for the closely related weighted tree augmentation problem (TAP), a similar result was known before only for the unweighed variant of the problem [4].…”

Section: First Contribution -Better Approximationmentioning

confidence: 99%

“…Previous Approaches and the Challenge: The connection between TAP and set cover is used in [8], where it is shown that using a certain decomposition of the graph it is possible to solve TAP by simulating a distributed greedy algorithm for set cover. This results in a randomized O((D + √ n) log 2 n)-round O(log n)-approximation for weighted TAP and weighted 2-ECSS.…”

Section: Technical Overviewmentioning

confidence: 99%

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Improved Distributed Approximations for Minimum-Weight Two-Edge-Connected Spanning Subgraph

Dory

Ghaffari

2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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The minimum-weight 2-edge-connected spanning subgraph (2-ECSS) problem is a natural generalization of the well-studied minimum-weight spanning tree (MST) problem, and it has received considerable attention in the area of network design. The latter problem asks for a minimum-weight subgraph with an edge connectivity of 1 between each pair of vertices while the former strengthens this edge-connectivity requirement to 2. Despite this resemblance, the 2-ECSS problem is considerably more complex than MST. While MST admits a linear-time centralized exact algorithm, 2-ECSS is NP-hard and the best known centralized approximation algorithm for it (that runs in polynomial time) gives a 2-approximation.In this paper, we give a deterministic distributed algorithm with round complexity of O(D + √ n) that computes a (5+ε)-approximation of 2-ECSS, for any constant ε > 0. Up to logarithmic factors, this complexity matches the Ω(D + √ n) lower bound that can be derived from the technique of Das Sarma et al. [STOC'11], as shown by Censor-Hillel and Dory [OPODIS'17]. Our result is the first distributed constant approximation for 2-ECSS in the nearly optimal time and it improves on a recent randomized algorithm of Dory [PODC'18], which achieved an O(log n)-approximation in O(D + √ n) rounds. Our algorithm also gives a deterministic O(D + √ n)-round (4 + ε)-approximation for the closely related weighted tree augmentation problem (TAP), a similar result was known before only for the unweighted variant of the problem.We also present an alternative algorithm for O(log n)-approximation, whose round complexity is linear in the low-congestion shortcut parameter of the network-following a framework introduced by Ghaffari and Haeupler [SODA'16]. This algorithm has round complexity O(D+ √ n) in worst-case networks but it provably runs much faster in many well-behaved graph families of interest. For instance, it runs in O(D) time in planar networks and those with bounded genus, bounded path-width or bounded tree-width.

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Section: First Contribution -Better Approximationmentioning

confidence: 99%

Section: First Contribution -Better Approximationmentioning

confidence: 99%

Section: First Contribution -Better Approximationmentioning

confidence: 99%

Section: First Contribution -Better Approximationmentioning

confidence: 99%

Section: Technical Overviewmentioning

confidence: 99%

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Improved Distributed Approximations for Minimum-Weight Two-Edge-Connected Spanning Subgraph

Dory

Ghaffari

2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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Fast distributed approximation for TAP and 2-edge-connectivity

Censor-Hillel

Dory

2019

Distrib. Comput.

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The tree augmentation problem (TAP) is a fundamental network design problem, in which the input is a graph G and a spanning tree T for it, and the goal is to augment T with a minimum set of edges Aug from G, such that T ∪ Aug is 2-edge-connected.TAP has been widely studied in the sequential setting. The best known approximation ratio of 2 for the weighted case dates back to the work of Frederickson and JáJá, SICOMP 1981. Recently, a 3/2-approximation was given for unweighted TAP by Kortsarz and Nutov, TALG 2016. Recent breakthroughs give an approximation of 1.458 for unweighted TAP [Grandoni et al., STOC 2018], and approximations better than 2 for bounded weights [Adjiashvili, SODA 2017; Fiorini et al., SODA 2018].In this paper, we provide the first fast distributed approximations for TAP. We present a distributed 2-approximation for weighted TAP which completes in O(h) rounds, where h is the height of T . When h is large, we show a much faster 4-approximation algorithm for the unweighted case, completing in O(D + √ n log * n) rounds, where n is the number of vertices and D is the diameter of G.Immediate consequences of our results are an O(D)-round 2-approximation algorithm for the minimum size 2-edge-connected spanning subgraph, which significantly improves upon the running time of previous approximation algorithms, and an O(h M ST + √ n log * n)-round 3approximation algorithm for the weighted case, where h M ST is the height of the MST of the graph. Additional applications are algorithms for verifying 2-edge-connectivity and for augmenting the connectivity of any connected spanning subgraph to 2.Finally, we complement our study with proving lower bounds for distributed approximations of TAP. * A preliminary version of this paper appeared in OPODIS 2017.

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A distributed algorithm for directed minimum-weight spanning tree

Fischer

Oshman

2021

Distrib. Comput.

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In the directed minimum spanning tree problem (DMST, also called minimum weight arborescence), the network is given a root node r, and needs to construct a minimum-weight directed spanning tree, rooted at r and oriented outwards. In this paper we present the first sub-quadratic DMST algorithms in the distributed CONGEST network model, where the messages exchanged between the network nodes are bounded in size. We consider three versions: a model where the communication links are bidirectional but can have different weights in the two directions; a model where communication is unidirectional; and the Congested Clique model, where all nodes can communicate directly with each other.Our algorithm is based on a variant of Lovász' DMST algorithm for the PRAM model, and uses a distributed single-source shortest-path (SSSP) algorithm for directed graphs as a black box. In the bidirectional CONGEST model, our algorithm has roughly the same running time as the SSSP algorithm; using the state-of-the-art SSSP algorithm, we obtain a running time of O(min( √ nD, √ nD 1/4 + n 3/5 + D)) rounds for the bidirectional communication case. For the unidirectional communication model we give an O(n) algorithm, and show that it is nearly optimal. And finally, for the Congested Clique, our algorithm again matches the best known SSSP algorithm: it runs in O(n 1/3 ) rounds.On the negative side, we adapt an observation of Chechik in the sequential setting to show that in all three models, the DMST problem is at least as hard as the (s, t)-shortest path problem. Thus, in terms of round complexity, distributed DMST lies between single-source shortest path and (s, t)-shortest path.

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Distributed Approximation of Minimum k-edge-connected Spanning Subgraphs

Cited by 15 publications

References 39 publications

Improved Distributed Approximations for Minimum-Weight Two-Edge-Connected Spanning Subgraph

Improved Distributed Approximations for Minimum-Weight Two-Edge-Connected Spanning Subgraph

Fast distributed approximation for TAP and 2-edge-connectivity

A distributed algorithm for directed minimum-weight spanning tree

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