Bipartite Grid Partitioning of a Random Geometric Graph

We investigate the problem of efficient computation of a partition of a Random Geometric Graph (RGG) into a limited number of densely packed bipartite grid subgraphs. The study focuses on the collection of subgraphs each individually having similar size and structure and the union employing most (e.g. over 85%) of the vertices. The residual vertices we seek to minimize are attributed to the inherent variations in densities of the randomly placed vertices and to any shortcomings of our greedy algorithms. RGG's have been extensively employed in recent times to model the deployment of numerous instances of Wireless Sensor Networks (WSN's). The properties investigated in our selected bipartite grid backbones are those deemed most relevant for applications to the foundations of this widely growing field. Distributed algorithms are primarily used to determine backbones. Our results review what backbone grid partitions exist in the data. This provides a metric to measure the effectiveness of any distributed algorithm against an existing optimal result. The visual display of selected backbone grids suggests local algorithm design strategies. Furthermore, these partitions must be efficiently computable for highly scalable computation, e.g. WSN's with 100's of thousands of vertices and millions of edges in the resulting RGG. We consider distributions over a segment of the plane and over the surface of the sphere to model sensor distributions both in limited planar regions, all around the globe or on distant planets.