Theta-3 is connected
In this paper, we show that the $θ$-graph with three cones is connected. We also provide an alternative proof of the connectivity of the Yao graph with three cones.
cs.CG↗
arXiv subjects
Sander Verdonschot
Publications and source records attributed to Sander Verdonschot.
Towards Tight Bounds on Theta-Graphs
We present improved upper and lower bounds on the spanning ratio of $θ$-graphs with at least six cones. Given a set of points in the plane, a $θ$-graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $θ=2π/m$, and adds an edge to the `closest' vertex in each cone. We show that for any integer $k \geq 1$, $θ$-graphs with $4k+2$ cones have a spanning ratio of $1+2\sin(θ/2)$ and we provide a matching lower bound, showing that this spanning ratio tight. Next, we show that for any integer $k \geq 1$, $θ$-graphs with $4k+4$ cones have spanning ratio at most $1+2\sin(θ/2)/(\cos(θ/2)-\sin(θ/2))$. We also show that $θ$-graphs with $4k+3$ and $4k+5$ cones have spanning ratio at most $\cos(θ/4)/(\cos(θ/2)-\sin(3θ/4))$. This is a significant improvement on all families of $θ$-graphs for which exact bounds are not known. For example, the spanning ratio of the $θ$-graph with 7 cones is decreased from at most 7.5625 to at most 3.5132. These spanning proofs also imply improved upper bounds on the competitiveness of the $θ$-routing algorithm. In particular, we show that the $θ$-routing algorithm is $(1+2\sin(θ/2)/(\cos(θ/2)-\sin(θ/2)))$-competitive on $θ$-graphs with $4k+4$ cones and that this ratio is tight. Finally, we present improved lower bounds on the spanning ratio of these graphs. Using these bounds, we provide a partial order on these families of $θ$-graphs. In particular, we show that $θ$-graphs with $4k+4$ cones have spanning ratio at least $1+2\tan(θ/2)+2\tan^2(θ/2)$. This is somewhat surprising since, for equal values of $k$, the spanning ratio of $θ$-graphs with $4k+4$ cones is greater than that of $θ$-graphs with $4k+2$ cones, showing that increasing the number of cones can make the spanning ratio worse.
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A History of Flips in Combinatorial Triangulations
Given two combinatorial triangulations, how many edge flips are necessary and sufficient to convert one into the other? This question has occupied researchers for over 75 years. We provide a comprehensive survey, including full proofs, of the various attempts to answer it.
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On the Average Number of Edges in Theta Graphs
Theta graphs are important geometric graphs that have many applications, including wireless networking, motion planning, real-time animation, and minimum-spanning tree construction. We give closed form expressions for the average degree of theta graphs of a homogeneous Poisson point process over the plane. We then show that essentially the same bounds---with vanishing error terms---hold for theta graphs of finite sets of points that are uniformly distributed in a square. Finally, we show that the number of edges in a theta graph of points uniformly distributed in a square is concentrated around its expected value.
cs.CG↗
On the stretch factor of the Theta-4 graph
In this paper we show that the θ-graph with 4 cones has constant stretch factor, i.e., there is a path between any pair of vertices in this graph whose length is at most a constant times the Euclidean distance between that pair of vertices. This is the last θ-graph for which it was not known whether its stretch factor was bounded.
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Making triangulations 4-connected using flips
We show that any combinatorial triangulation on n vertices can be transformed into a 4-connected one using at most floor((3n - 9)/5) edge flips. We also give an example of an infinite family of triangulations that requires this many flips to be made 4-connected, showing that our bound is tight. In addition, for n >= 19, we improve the upper bound on the number of flips required to transform any 4-connected triangulation into the canonical triangulation (the triangulation with two dominant vertices), matching the known lower bound of 2n - 15. Our results imply a new upper bound on the diameter of the flip graph of 5.2n - 33.6, improving on the previous best known bound of 6n - 30.
cs.CG↗