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 $k$, for any integer parameter $k\geq 14$. 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 $k$, for any integer parameter $k\geq 14$. 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

Spanners for Geometric Intersection Graphs

Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time. For unit ball graphs, the spanners have a small separator decomposition which can be used to obtain efficient algorithms for approximating proximity problems like diameter and distance queries. The results on compressed quadtrees, geometric graph separators, and diameter approximation might be of independent interest.Comment: 16 pages, 5 figures, Late

On a family of strong geometric spanners that admit local routing strategies

We introduce a family of directed geometric graphs, denoted \paz, that depend on two parameters $\lambda$ and $\theta$. For $0\leq \theta<\frac{\pi}{2}$ and ${1/2} < \lambda < 1$, the \paz graph is a strong $t$-spanner, with $t=\frac{1}{(1-\lambda)\cos\theta}$. The out-degree of a node in the \paz graph is at most $\lfloor2\pi/\min(\theta, \arccos\frac{1}{2\lambda})\rfloor$. Moreover, we show that routing can be achieved locally on \paz. Next, we show that all strong $t$-spanners are also $t$-spanners of the unit disk graph. Simulations for various values of the parameters $\lambda$ and $\theta$ indicate that for random point sets, the spanning ratio of \paz is better than the proven theoretical bounds

Lower bounds on the dilation of plane spanners

(I) We exhibit a set of 23 points in the plane that has dilation at least $1.4308$, improving the previously best lower bound of $1.4161$ for the worst-case dilation of plane spanners. (II) For every integer $n\geq13$, there exists an $n$-element point set $S$ such that the degree 3 dilation of $S$ denoted by $\delta_0(S,3) \text{ equals } 1+\sqrt{3}=2.7321\ldots$ in the domain of plane geometric spanners. In the same domain, we show that for every integer $n\geq6$, there exists a an $n$-element point set $S$ such that the degree 4 dilation of $S$ denoted by $\delta_0(S,4) \text{ equals } 1 + \sqrt{(5-\sqrt{5})/2}=2.1755\ldots$ The previous best lower bound of $1.4161$ holds for any degree. (III) For every integer $n\geq6$, there exists an $n$-element point set $S$ such that the stretch factor of the greedy triangulation of $S$ is at least $2.0268$.Comment: Revised definitions in the introduction; 23 pages, 15 figures; 2 table

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 $\mathcal{O}(\log^*n)$-round distributed algorithm in the LOCAL model of computation, that given a unit ball graph $G$ with $n$ vertices and a positive constant $\epsilon < 1$ finds a $(1+\epsilon)$-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 $\mathcal{O}(\log^*n)$ rounds in the LOCAL model, but has a $\mathcal{O}(\log \Delta)$ bound on its lightness, where $\Delta$ 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