We prove that the intersection hypergraph of a family of n pseudo-disks with respect to another family of pseudo-disks admits a proper coloring with 4 colors and a conflict-free coloring with O(log n) colors. Along the way we prove that the respective Delaunay-graph is planar. We also prove that the intersection hypergraph of a family of n regions with linear union complexity with respect to a ...

We consider two classes of higher order proximity graphs defined on a set of points in the plane, namely, the k-Delaunay graph and the k-Gabriel graph. We give bounds on the following combinatorial and geometric properties of these graphs: spanning ratio, diameter, connectivity, chromatic number, and minimum number of layers necessary to partition the edges of the graphs so that no two edges of...

This paper shows how Voronoi diagrams and their dual Delaunay complexes, defined with geodesic distances over 2D Reimannian manifolds, can be used to solve two important problems encountered in computer vision and graphics. The first problem studied is perceptual grouping which is a curve reconstruction problem where one should complete in a meaningful way a sparse set of noisy curves. From thi...

a b c d Visualization of the moving least squares surface (magenta) over a 2D point cloud (black dots) based on different distance functions: (a,c) Euclidean, (b,d) ours based on proximity graphs. We present a new definition of an implicit surface over a noisy point cloud. It can be evaluated very fast, but, unlike other definitions based on the moving least squares approach, it does not suffer...

We consider the problem of constructing a bounded-degree planar geometric spanner for a unit disk graph modeling a wireless network. The related problem of constructing a planar geometric spanner of a Euclidean graph has been extensively studied in the literature. It is well known that the Delaunay subgraph is a planar geometric spanner with stretch factor ; however, its degree may not be bound...

We consider how continuous-time quantum walks can be used to give a measure of graph similarity. Our approach is to simulate the quantum walk on the two graphs in parallel by using an auxiliary graph that incorporates both graphs. The auxiliary graph allows quantum interference to take place between the two walks. Modeling the resultant interference amplitudes, which result from the differences...

Let S be a finite set of points in the Euclidean plane, and let E be the complete graph whose point-set is S. Chew, in 1986, proved a lower bound of π/2 on the stretch factor of the Delaunay triangulation of S (with respect to E), and conjectured that this bound is tight. Dobkin, Friedman, and Supowit, in 1987, showed that the stretch factor of the Delaunay triangulation of S is at most π( √ 5 ...

Several localized routing protocols [1] guarantee the delivery of the packets when the underlying network topology is the Delaunay triangulation of all wireless nodes. However, it is expensive to construct the Delaunay triangulation in a distributed manner. Given a set of wireless nodes, we often model the network as a unit-disk graphUDG . In this paper, we present a novel localized networking ...

We give a unified view to greedy geometric routing algorithms in ad hoc networks. For this, we first present a general form of greedy routing algorithm using a class of objective functions which are invariant under congruent transformations of a point set. We show that several known greedy routing algorithms such as Greedy Routing, Compass Routing, and Midpoint Routing can be regarded as specia...

Let G be a graph, s ∈ G be a source node and t ∈ G be a target node. The sequence of adjacent nodes (graph walk) visited by a routing algorithm is a c-competitive route if its length in G is at most c times the length of the shortest path from s to t in G. We present 21.766-, 17.982and 15.479-competitive online routing algorithms on the Delaunay triangulation of an arbitrary given set of points...