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LIPIcs - Leibniz International Proceedings in Informatics. 25th International Symposium on Theoretical Aspects of Computer Science
Doi
Abstract
We consider the problem of constructing bounded-degree planar
geometric spanners of Euclidean and unit-disk graphs. It is well
known that the Delaunay subgraph is a planar geometric spanner with
stretch factor C_{delapprox 2.42; however, its degree may not be
bounded. Our first result is a very simple linear time algorithm
for constructing a subgraph of the Delaunay graph with stretch
factor
ho =1+2pi(kcos{frac{pi{k)^{-1 and degree bounded by
k, for any integer parameter kgeq14. This result immediately
implies an algorithm for constructing a planar geometric spanner of
a Euclidean graph with stretch factor
ho cdot C_{del and
degree bounded by k, for any integer parameter kgeq14.
Moreover, the resulting spanner contains a Euclidean Minimum
Spanning Tree (EMST) as a subgraph. Our second contribution lies
in developing the structural results necessary to transfer our
analysis and algorithm from Euclidean graphs to unit disk graphs,
the usual model for wireless ad-hoc networks. We obtain a very
simple distributed, {em strictly-localized algorithm that, given a
unit disk graph embedded in the plane, constructs a geometric
spanner with the above stretch factor and degree bound, and also
containing an EMST as a subgraph. The obtained results
dramatically improve the previous results in all aspects, as shown
in the paper
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