Regions, distances and graphs

The region-counting distances, introduced by Demaine, Iacono and Langerman, associate to any pair of points (p,q) the number of items of a dataset S contained in a region R(p,q) surrounding (p,q). We define region-counting disks and circles, and study the complexity of these objects. Algorithms to compute epsilon-approximations of region-counting distances and approximations of region-counting circles are presented.

We propose a definition of the locality for properties of geometric graphs. We measure the local density of graphs using the region-counting distances between pairs of vertices, and we use this density to define local properties of classes of graphs.

We illustrate the locality by introducing the local diameter of geometric graphs: we define it as the upper bound on the size of the shortest path between any pair of vertices, expressed as a function of the density of the graph around those vertices. We determine the local diameter of several well-studied graphs such as the Theta-graph, the Ordered Theta-graph and the Skip List Spanner. We also show that various operations, such as path and point queries using geometric graphs as data structures, have complexities which can be expressed as local properties.

A family of proximity graphs, called Empty Region Graphs (ERG) is presented. The vertices of an ERG are points in the plane, and two points are connected if their neighborhood, defined by a region, does not contain any other point. The region defining the neighborhood of two points is a parameter of the graph. This family of graphs includes several known proximity graphs such as Nearest Neighbor Graphs, Beta-Skeletons or Theta-Graphs. We concentrate on ERGs that are invariant under translations, rotations and uniform scaling of the vertices. We give conditions on the region defining an ERG to ensure a number of properties that might be desirable in applications, such as planarity, connectivity, triangle-freeness, cycle-freeness, bipartiteness and bounded degree. These conditions take the form of what we call tight regions: maximal or minimal regions that a region must contain or be contained in to make the graph satisfy a given property. We show that every monotone property has at least one corresponding tight region; we discuss possibilities and limitations of this general model for constructing a graph from a point set.

We introduce and analyze sigma-local graphs, based on a definition of locality by Erickson, to illustrate efficient construction algorithm on a subclass of ERGs.