Connected Coverage in Three-Dimensional Wireless Sensor Networks Using Convex Polyhedral Space-Fillers

The coverage problem of a three-dimensional (3D) space has similarity with the tiling problem in the same space, which can be formulated as follows: How can a 3D space be tiled by replicas of tiles? This is an instance of the second part of Hilbert's eighteenth problem [14], which is stated as follows: "What convex polyhedra exist for which a complete filling of all space is possible by juxtaposition of congruent copies?" In this paper, we propose a polyhedral framework to investigate the connected coverage problem in 3D homogeneous wireless sensor networks. First, we restrict the sensors' sensing sphere to a variety of convex polyhedral space-fillers. Our study aims to find the largest enclosed convex polyhedron space-filler in the sensors' sensing sphere, with a goal to maximize their utilized sensing volume. Second, based on this analysis, we select a minimum number of sensors to cover a 3D space for deterministic and random sensor deployment strategies. Third, we compute the ratio of the communication range to the sensing range of the sensors to ensure network connectivity. Fourth, we corroborate our analysis with various simulation results.