NDLI: Distributed algorithms for ultrasparse spanners and linear size skeletons

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Distributed algorithms for ultrasparse spanners and linear size skeletons

Content Provider	ACM Digital Library
Author	Pettie, Seth
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.
Starting Page	253
Ending Page	262
Page Count	10
File Format	PDF
ISBN	9781595939890
DOI	10.1145/1400751.1400786
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2008-08-18
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Spanner
Content Type	Text
Resource Type	Article

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