Topology Control and Network Lifetime in Three-Dimensional Wireless   Sensor Networks
  Coverage and connectivity issues of three-dimensional (3D) networks are addressed in [2], but that work assumes that a node can be placed at any arbitrary location. In this work, we drop that assumption and rather assume that nodes are uniformly and densely deployed in a 3D space. We want to devise a mechanism that keeps some nodes active and puts other nodes into sleep so that the number of active nodes at a time is minimized (and thus network life time is maximized), while maintaining full coverage and connectivity. One simple way to do that is to partition the 3D space into cells, and only one node in each cell remains active at a time. Our results show that the number of active nodes can be minimized if the shape of each cell is a truncated octahedron. It requires the sensing range to be at least 0.542326 times the transmission radius. This value is 0.5, 0.53452 and 0.5 for cube, hexagonal prism, and rhombic dodecahedron, respectively. However, at a time the number of active nodes for cube, hexagonal prism and rhombic dodecahedron model is respectively 2.372239, 1.82615 and 1.49468 times of that of truncated octahedron model. So clearly truncated octahedron model has the highest network lifetime. We also provide a distributed topology control algorithm that can be used by each sensor node to determine its cell id using a constant number of local arithmetic operations provided that the sensor node knows its location. We also validate our results by simulation. 
S. M. Nazrul Alam and Zygmunt J. Haas