96 research outputs found
Sparse Hop Spanners for Unit Disk Graphs
A unit disk graph G on a given set of points P in the plane is a geometric graph where an edge exists between two points p,q ? P if and only if |pq| ? 1. A subgraph G\u27 of G is a k-hop spanner if and only if for every edge pq ? G, the topological shortest path between p,q in G\u27 has at most k edges. We obtain the following results for unit disk graphs.
1) Every n-vertex unit disk graph has a 5-hop spanner with at most 5.5n edges. We analyze the family of spanners constructed by Biniaz (2020) and improve the upper bound on the number of edges from 9n to 5.5n.
2) Using a new construction, we show that every n-vertex unit disk graph has a 3-hop spanner with at most 11n edges.
3) Every n-vertex unit disk graph has a 2-hop spanner with O(nlog n) edges. This is the first nontrivial construction of 2-hop spanners.
4) For every sufficiently large n, there exists a set P of n points on a circle, such that every plane hop spanner on P has hop stretch factor at least 4. Previously, no lower bound greater than 2 was known.
5) For every point set on a circle, there exists a plane 4-hop spanner. As such, this provides a tight bound for points on a circle.
6) The maximum degree of k-hop spanners cannot be bounded from above by a function of k
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
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
On Geometric Spanners of Euclidean and Unit Disk Graphs
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_{del\approx
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 \rho =1+2\pi(k\cos{\frac{\pi{k)^{-1 and degree bounded by
, for any integer parameter . This result immediately implies an
algorithm for constructing a planar geometric spanner of a Euclidean graph with
stretch factor \rho \cdot C_{del and degree bounded by , for any integer
parameter . 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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