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Distributed algorithms for ultrasparse spanners and linear size skeletons

Published: 18 August 2008 Publication History

Abstract

We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some non-trivial lower bounds on the trade-off between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages may be communicated in each time step. Our first result is an O(log n)1+o(1)-time algorithm for finding a (2O(log* n)log n)-spanner with size O(n). Besides being nearly optimal in time and distortion, this algorithm appears to be the first that constructs a O(n)-size skeleton without requiring unbounded length messages or time proportional to the diameter of the network. Our second result is a new class of efficiently constructible (α,β)-spanners called Fibonacci spanners whose distortion improves with the distance being approximated. At their sparsest Fibonacci spanners can have nearly linear size O(n(log log n)φ) where φ = 1+☂5/2 is the golden ratio. As the distance increases the Fibonacci spanner's multiplicative distortion passes through four discrete stages, moving from logarithmic to doubly logarithmic, then into a period where it is constant, tending to 3, followed by another period tending to 1. On the lower bound side we prove that many recent sequential spanner constructions have no efficient counterparts in distributed networks, even if the desired distortion only needs to be achieved on the average or for a tiny fraction of the vertices. In particular, any distance preservers, purely additive spanners, or spanners with sublinear additive distortion must either be very dense, slow to construct, or have very weak guarantees on distortion.

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cover image ACM Conferences
PODC '08: Proceedings of the twenty-seventh ACM symposium on Principles of distributed computing
August 2008
474 pages
ISBN:9781595939890
DOI:10.1145/1400751
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 18 August 2008

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  • (2021)Ultra-Sparse Near-Additive EmulatorsProceedings of the 2021 ACM Symposium on Principles of Distributed Computing10.1145/3465084.3467926(235-246)Online publication date: 21-Jul-2021
  • (2019)Improved Distributed Approximations for Minimum-Weight Two-Edge-Connected Spanning SubgraphProceedings of the 2019 ACM Symposium on Principles of Distributed Computing10.1145/3293611.3331617(521-530)Online publication date: 16-Jul-2019
  • (2015)Improved Parallel Algorithms for Spanners and HopsetsProceedings of the 27th ACM symposium on Parallelism in Algorithms and Architectures10.1145/2755573.2755574(192-201)Online publication date: 13-Jun-2015
  • (2014)Distributed approximation algorithms for weighted shortest pathsProceedings of the forty-sixth annual ACM symposium on Theory of computing10.1145/2591796.2591850(565-573)Online publication date: 31-May-2014
  • (2014)Near-Optimal Distributed Approximation of Minimum-Weight Connected Dominating SetAutomata, Languages, and Programming10.1007/978-3-662-43951-7_41(483-494)Online publication date: 2014
  • (2010)Sublinear Fully Distributed Partition with ApplicationsTheory of Computing Systems10.1007/s00224-009-9190-x47:2(368-404)Online publication date: 1-Aug-2010
  • (2010)Multipath spannersProceedings of the 17th international conference on Structural Information and Communication Complexity10.1007/978-3-642-13284-1_17(211-223)Online publication date: 7-Jun-2010

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