Space Filling Curves for 3D Sensor Networks with Complex Topology

Several aspects of managing a sensor network (e.g., motion planning for data mules, serial data fusion and inference) benefit once the network is linearized to a path. The linearization is often achieved by constructing a space filling curve inside the domain. But existing methods cannot handle networks distributed on surfaces of complex topology. This paper presents a novel method for generating space filling curves to 3D sensor networks that are distributed densely on some two-dimensional geometric surface. Our algorithm is completely distributed and constructs a path which gets uniformly, progressively denser as the path becomes longer. We analyze the algorithm mathematically and prove that the curve we obtain is dense. Our method is based on Hodge Decomposition theorem and uses holomorphic differentials on Riemann surfaces. The underlying high genus surface is conformally mapped to a union of flat tori and then a proportionally-dense space filling curve on this union is constructed. The pullback of this curve to the original network gives us the desired curve. We show via simulations that our method handles complicated topologies and performs much better than the alternative methods on both density and coverage rate.