Spherical Representation & Polyhedron Routing for Load Balancing in Wireless Sensor Networks Xiaokang Yu Xiaomeng Ban Wei Zeng Rik Sarkar Xianfeng David Gu Jie Gao Load Balanced Routing in Sensor Networks • Goal: Min Max # messages any node delivers. – Prolong network lifetime • A difficult problem – NP-hard, unsplittable flow problem. – Existing approximation algorithms are centralized. – Practical solutions use heuristic methods. • Curveball Routing [Popa et. al. 2007] • Routing in Outer Space [Mei et. al. 2008] •… A Simple Case • A disk shape network. • greedy routing (send to neighbor closer to dest) ≈ Shortest path routing • Uniform traffic: All pairs of node have 1 message. • Center load is high! Curveball Routing • Use stereographic projection and perform greedy routing on the sphere • The center load is alleviated. • But greedy routing may fail on sparse networks Routing in Outer Spaces i.e., Torus Routing • A rectangular network • Wrapped up as a torus. Flip • Route on the torus. • With equal prob to each of the 4 images. Flip • Again, delivery is not guaranteed! Our Approach • Embed the network as a convex polytope (Thurston’s theorem) – Greedy routing guarantees delivery • Embedding is subject to a Möbius transformation f – Optimize f for load balancing. • Explore different network density, battery level, traffic pattern, etc. Thurston’s Theorem • Koebe-Andreev-Thurston Theorem: Any 3-connected graph can be embedded as a convex polyhedron – Circle packing with circles on vertices. – all edges are tangent to a unit sphere. • Compared to stereographic mapping, vertices are lifted up from the sphere. Polyhedron Routing • [Papadimitriou & Ratajczak] Greedy routing with d(u, v)= – c(u) · c(v) guarantees delivery. 3D coordinates of v • Route along the surface of a convex polytope. Compute Thurston’s Embedding 1. Extract a planar graph G of a sensor network – Many prior algorithms exist. 2. Compute a pair of circle packings, for G and its dual graph Ĝ using curvature flow. – Variation definition of the Thurston’s embedding – Vertex circle is orthogonal to the adjacent face circle. – Use Curvature flow on the reduced graph = G + Ĝ. Prepare the Reduced Graph • Input graph Prepare the Reduced Graph Vertex node • Overlay G and the Edge node dual graph Ĝ, add Face node intersection vertices Edge node as edge nodes. • Each “face” becomes a quadrilateral • Triangulate each quadrilateral by adding a virtual edge. Compute Circle Packing Using Curvature Flow • Goal: find radius of vertex circle and the radius of the face circle that are orthogonal & embedding is flat on the plane. Idea: start from some initial values that guarantee orthogonality & run Ricci flow to flatten it. Circle Packing Results • Use stereographic projection to map circles to the sphere. • Compute the supporting planes of the face circles • Their intersection is the convex polytope Different Möbius transformation • Möbius transformation preserves the circle packings. • Optimize for “uniform vertex distribution” ≈ uniform vertex circle size. Simulations • Compare with Curveball Routing and Torus Routing Delivery Rate and Load Balancing • Delivery Rate: – Dense network: all methods can deliver. • Load balancing, tested on dense network – Torus routing: most uniform load; but avg load is 80% higher than simple greedy methods. – Ours v.s Curveball: slightly higher avg load, but solves the center-dense problem better. Adjust Node Density wrt Battery Level • Find the Möbius transformation st circle size ~ battery level. With optimization Battery level: High to Low No optimization Routes prefer high battery nodes Network with Non-Uniform Density • Dense region spans wider area. Birdeye view Uniform density Conclusion & Future Work • Bend a network for better load balancing. • Open Question: How to deform a surface such that the geodesic paths have uniform density? – Saddles attract geodesic paths, peaks/valleys repel. – Uniformizing curvature always leads to better load balancing? Questions and Comments?
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