A novel geometric diagram and its applications in wireless networks

Wireless networks have a lot of geometric properties since the signal strength corresponds directly to the distance. Earlier research findings in geometry, such as Voronoi diagram, have found a lot of applications in wireless networks. In this paper, a new geometric diagram, referred to as the umbrella diagram, is proposed. The umbrella diagram is comparable to the Voronoi diagram. The Voronoi diagram deals with a set of points, whereas the umbrella diagram deals with a set of different networks. The umbrella diagram of a set of hexagonal networks is to divide the plane into regions such that the points in a region have a larger distance to one network than to the other networks. Unlike the Voronoi diagram, the regions in the umbrella diagram are not necessarily convex. The paper gives an efficient solution to compute the umbrella diagram of n networks with the same cell size and orientation. Like the Voronoi diagram, the umbrella diagram has potential to be used in many areas. To illustrate its usefulness, the paper further gives two applications in wireless networks: the optimal network deployment and the maximum base station reuse. The optimal network deployment intends to find the best position to fix the network such that the network is closest to all existing base stations. The maximum base station reuse problem is to maximize the number of base stations that can be reused under a given bound.