Spherical Representation and Polyhedron Routing for Load Balancing in Wireless Sensor Networks
In this paper, the authors address the problem of scalable and load balanced routing for wireless sensor networks. Motivated by the analog of the continuous setting that geodesic routing on a sphere gives perfect load balancing, they embed sensor nodes on a convex polyhedron in 3D and use greedy routing to deliver messages between any pair of nodes with guaranteed success. This embedding is known to exist by the Koebe-Andreev-Thurston Theorem for any 3-connected planar graphs. In their paper they use Ricci flow to develop a distributed algorithm to compute this embedding. Further, such an embedding is not unique and differ by one another with a Mobius transformation.