129 research outputs found
Distributed Construction of Lightweight Spanners for Unit Ball Graphs
Resolving an open question from 2006 [Damian et al., 2006], we prove the existence of light-weight bounded-degree spanners for unit ball graphs in the metrics of bounded doubling dimension, and we design a simple ?(log^*n)-round distributed algorithm in the LOCAL model of computation, that given a unit ball graph G with n vertices and a positive constant ? < 1 finds a (1+?)-spanner with constant bounds on its maximum degree and its lightness using only 2-hop neighborhood information. This immediately improves the best prior lightness bound, the algorithm of Damian, Pandit, and Pemmaraju [Damian et al., 2006], which runs in ?(log^*n) rounds in the LOCAL model, but has a ?(log ?) bound on its lightness, where ? is the ratio of the length of the longest edge to the length of the shortest edge in the unit ball graph. Next, we adjust our algorithm to work in the CONGEST model, without changing its round complexity, hence proposing the first spanner construction for unit ball graphs in the CONGEST model of computation. We further study the problem in the two dimensional Euclidean plane and we provide a construction with similar properties that has a constant average number of edge intersections per node. Lastly, we provide experimental results that confirm our theoretical bounds, and show an efficient performance from our distributed algorithm compared to the best known centralized construction
Optimal Spanners for Unit Ball Graphs in Doubling Metrics
Resolving an open question from 2006, we prove the existence of light-weight
bounded-degree spanners for unit ball graphs in the metrics of bounded doubling
dimension, and we design a simple -round distributed
algorithm in the LOCAL model of computation, that given a unit ball graph
with vertices and a positive constant finds a
-spanner with constant bounds on its maximum degree and its
lightness using only 2-hop neighborhood information. This immediately improves
the best prior lightness bound, the algorithm of Damian, Pandit, and Pemmaraju,
which runs in rounds in the LOCAL model, but has a
bound on its lightness, where is the ratio
of the length of the longest edge to the length of the shortest edge in the
unit ball graph. Next, we adjust our algorithm to work in the CONGEST model,
without changing its round complexity, hence proposing the first spanner
construction for unit ball graphs in the CONGEST model of computation. We
further study the problem in the two dimensional Euclidean plane and we provide
a construction with similar properties that has a constant average number of
edge intersections per node. Lastly, we provide experimental results that
confirm our theoretical bounds, and show an efficient performance from our
distributed algorithm compared to the best known centralized construction
Light Spanners
A -spanner of a weighted undirected graph , is a subgraph
such that for all . The sparseness of
the spanner can be measured by its size (the number of edges) and weight (the
sum of all edge weights), both being important measures of the spanner's
quality -- in this work we focus on the latter.
Specifically, it is shown that for any parameters and ,
any weighted graph on vertices admits a
-stretch spanner of weight at most , where is the weight of a minimum
spanning tree of . Our result is obtained via a novel analysis of the
classic greedy algorithm, and improves previous work by a factor of .Comment: 10 pages, 1 figure, to appear in ICALP 201
Parallel Graph Decompositions Using Random Shifts
We show an improved parallel algorithm for decomposing an undirected
unweighted graph into small diameter pieces with a small fraction of the edges
in between. These decompositions form critical subroutines in a number of graph
algorithms. Our algorithm builds upon the shifted shortest path approach
introduced in [Blelloch, Gupta, Koutis, Miller, Peng, Tangwongsan, SPAA 2011].
By combining various stages of the previous algorithm, we obtain a
significantly simpler algorithm with the same asymptotic guarantees as the best
sequential algorithm
07151 Abstracts Collection -- Geometry in Sensor Networks
From 9.4.2007 to 13.4.07, the Dagstuhl Seminar 07151 ``Geometry in Sensor
Networks\u27\u27 was held in the International Conference and Research Center
(IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first
section describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
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