André van Renssen

On plane constrained bounded-degree spanners

Let P be a set of points in the plane and S a set of non-crossing line segments with endpoints in P. The visibility graph of P with respect to S, denoted Vis(P,S), has vertex set P and an edge for each pair of vertices u,v in P for which no line segment of S properly intersects uv. We show that the constrained half-θ6-graph (which is identical to the constrained Delaunay graph whose empty visible region is an equilateral triangle) is a plane 2-spanner of Vis(P,S). We then show how to construct a plane 6-spanner of Vis(P,S) with maximum degree 6 + c, where c is the maximum number of segments adjacent to a vertex.

On the spanning ratio of theta-graphs

We present improved upper bounds on the spanning ratio of a large family of θ-graphs. A θ-graph partitions the plane around each vertex into m disjoint cones, each having aperture θ=2 π/m. We show that for any integer k≥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. We also improve the upper bounds on the competitiveness of the θ-routing algorithm for these graphs to 1+2 sin(θ/2) / (cos(θ/2)−sin(θ/2)) on θ-graphs with 4k+4 cones and to 1+2 sin(θ/2) ·cos(θ/4) / (cos(θ/2)−sin(3θ/4)) on θ-graphs with 4k+3 and 4k+5 cones. For example, the routing ratio of the θ-graph with 7 cones is decreased from at most 7.5625 to at most 4.0490.

New and Improved Spanning Ratios for Yao Graphs

For a set of points in the plane and a fixed integer k > 0, the Yao graph Yk partitions the space around each point into k equiangular cones of angle &thetas; = 2π/k, and connects each point to a nearest neighbor in each cone. It is known for all Yao graphs, with the sole exception of Y5, whether or not they are geometric spanners. In this paper we close this gap by showing that for odd k ≥ 5, the spanning ratio of Yk is at most 1/(1−2sin(3&thetas;/8)), which gives the first constant upper bound for Y5, and is an improvement over the previous bound of 1/(1−2sin(&thetas;/2)) for odd k ≥ 7. We further reduce the upper bound on the spanning ratio for Y5 from 10.9 to 2 + √3 ≈ 3.74, which falls slightly below the lower bound of 3.79 established for the spanning ratio of ⊝5 (⊝-graphs differ from Yao graphs only in the way they select the closest neighbor in each cone). This is the first such separation between a Yao and ⊝-graph with the same number of cones. We also give a lower bound of 2.87 on the spanning ratio of Y5. Finally, we revisit the Y6 graph, which plays a particularly important role as the transition between the graphs (k > 6) for which simple inductive proofs are known, and the graphs (k ≤ 6) whose best spanning ratios have been established by complex arguments. Here we reduce the known spanning ratio of Y6 from 17.6 to 5.8, getting closer to the spanning ratio of 2 established for ⊝6.

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.

Area-preserving subdivision schematization

We describe an area-preserving subdivision schematization algorithm: the area of each region in the input equals the area of the corresponding region in the output. Our schematization is axis-aligned, the final output is a rectilinear subdivision. We first describe how to convert a given subdivision into an area-equivalent rectilinear subdivision. Then we define two area-preserving contraction operations and prove that at least one of these operations can always be applied to any given simple rectilinear polygon. We extend this approach to subdivisions and showcase experimental results. Finally, we give examples for standard distance metrics (symmetric difference, Hausdorff- and Frechet-distance) that show that better schematizations might result in worse shapes.

Optimal Local Routing on Delaunay Triangulations Defined by Empty Equilateral Triangles

We present a deterministic local routing algorithm that is guaranteed to find a path between any pair of vertices in a half-

The θ 5 -graph is a spanner

\theta_6

Towards tight bounds on theta-graphs

-graph (the half-

Area-Preserving Simplification and Schematization of Polygonal Subdivisions

\theta_6

The θ 5 -Graph is a Spanner

-graph is equivalent to the Delaunay triangulation where the empty region is an equilateral triangle). The length of the path is at most