Efficient algorithms for constructing very sparse spanners and emulators

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Abstract

Miller et al. [43] devised a distributed1 algorithm in the CONGEST model, that given a parameter k = 1; 2; : : :, constructs an O(k)-spanner of an input unweighted n- vertex graph with 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 edges, with probability, for any . Moreover, when, 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 spanners and emulators. In particular, we show that for any unweighted n-vertex graph and any there exists a emulator with O(n) edges. All previous constructions of 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.

Original languageEnglish
Title of host publication28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017
EditorsPhilip N. Klein
PublisherAssociation for Computing Machinery
Pages652-669
Number of pages18
ISBN (Electronic)9781611974782
DOIs
StatePublished - 1 Jan 2017
Event28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 - Barcelona, Spain
Duration: 16 Jan 201719 Jan 2017

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume0

Conference

Conference28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017
Country/TerritorySpain
CityBarcelona
Period16/01/1719/01/17

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