Hopsets with Constant Hopbound, and Applications to Approximate Shortest Paths

Elkin, Michael; Neiman, Ofer

doi:10.1109/focs.2016.22

Cited by 51 publications

(198 citation statements)

References 31 publications

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“…7 More precisely, following Elkin and Neiman [EN16], one constructs an overlay network of size √ sn instead of √ n as done in this paper. and for a node and set of nodes A ⊆ V by d(u, A, R,G) = min ∈A d(u, , R,G).…”

Section: Notationmentioning

confidence: 99%

A deterministic almost-tight distributed algorithm for approximating single-source shortest paths

Henzinger

Krinninger²,

Nanongkai

2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

137

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We present a deterministic (1 + o(1))-approximation O(n 1/2+o(1) + D 1+o(1) )-time algorithm for solving the single-source shortest paths problem on distributed weighted networks (the CONGEST model); here n is the number of nodes in the network and D is its (hop) diameter. This is the first non-trivial deterministic algorithm for this problem. It also improves (i) the running time of the randomized (1+o (1) In achieving this result, we develop two techniques which might be of independent interest and useful in other settings: (i) a deterministic process that replaces the "hitting set argument" commonly used for shortest paths computation in various settings, and (ii) a simple, deterministic, construction of an (n o(1) , o(1))-hop set of size O(n 1+o(1) ). We combine these techniques with many distributed algorithmic techniques, some of which from problems that are not directly related to shortest paths, e.g. ruling sets [Goldberg et al. STOC 1987], source detection [Lenzen, Peleg PODC 2013], and partial distance estimation [Lenzen, Patt-Shamir PODC 2015]. Our hop set construction also leads to single-source shortest paths algorithms in two other settings:

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Section: Notationmentioning

confidence: 99%

A deterministic almost-tight distributed algorithm for approximating single-source shortest paths

Henzinger

Krinninger²,

Nanongkai

2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

137

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“…In addition, we show that the superclustering and interconnection steps themselves can be implemented efficiently. This part of the algorithm is based on our recent work on hopsets [34], where we showed that [35] approach is extremely useful in that context as well, and that it can be made efficient.…”

Section: Technical Overview Linial and Saksmentioning

confidence: 99%

Efficient Algorithms for Constructing Very Sparse Spanners and Emulators

Elkin

Neiman

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

Self Cite

155

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Miller et al. [43] devised a distributed 1 algorithm in the CONGEST model, that given a parameter k = 1, 2, . . ., constructs an O(k)-spanner of an input unweighted nvertex graph with O(n 1+1/k ) expected edges in O(k) rounds of communication. In this paper we improve the result of [43], by showing a k-round distributed algorithm in the same model, that constructs a (2k − 1)-spanner with O(n 1+1/k / ) edges, with probability 1 − , for any > 0. Moreover, when k = ω(log n), our algorithm produces (still in k rounds) ultra-sparse spanners, i.e., spanners of size n(1 + o(1)), with probability 1 − o(1). To our knowledge, this is the first distributed algorithm in the CONGEST or in the PRAM models that constructs spanners or skeletons (i.e., connected spanning subgraphs) that sparse. Our algorithm can also be implemented in linear time in the standard centralized model, and for large k, it provides spanners that are sparser than any other spanner given by a known (near-)linear time algorithm.We also devise improved bounds (and algorithms realizing these bounds) for (1 + , β)-spanners and emulators. In particular, we show that for any unweighted n-vertex graph and any > 0, there exists a (1 + , ( log log n ) log log n )-emulator with O(n) edges. All previous constructions of (1 + , β)-spanners and emulators employ a superlinear number of edges, for all choices of parameters.Finally, we provide some applications of our results to approximate shortest paths' computation in unweighted graphs. * Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel. Email: elkinm@cs.bgu.ac.il. 1 They actually showed a PRAM algorithm. The distributed algorithm with these properties is implicit in [43].

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“…In some cases, different edges of the graph may have different attributes, which can be represented using edge weights. The existence of edge weights has been extensively studied in various tasks, such as finding or approximating lightest paths [20,37,27,21,34,31,4,25], finding a minimum spanning tree (MST) in the graph [5,23,13], finding a maximum matching [36,13], and more. However, as far as we are aware, no study addresses the problem of constructing multiple weighted BFS (WBFS) trees, where the goal is not to find the lightest paths from the sources to the nodes, but rather the lightest shortest paths.…”

Section: Introductionmentioning

confidence: 99%

The sparsest additive spanner via multiple weighted BFS trees

Censor-Hillel

Paz

Ravid

2020

Theoretical Computer Science

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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 tradeoff 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 ofÕ(n 4/3) edges. We then show a distributed implementation of our algorithm, answering an open problem in [10]. 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 |S| + D − 1 rounds, where S is the set of sources and D is the diameter of the network graph.

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Hopsets with Constant Hopbound, and Applications to Approximate Shortest Paths

Cited by 51 publications

References 31 publications

A deterministic almost-tight distributed algorithm for approximating single-source shortest paths

A deterministic almost-tight distributed algorithm for approximating single-source shortest paths

Efficient Algorithms for Constructing Very Sparse Spanners and Emulators

The sparsest additive spanner via multiple weighted BFS trees

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