1. Spherical Representation & Polyhedron Routing for Load Balancing in Wireless Sensor Networks Xiaokang Yu Xiaomeng Ban Wei Zeng Rik Sarkar Xianfeng David Gu Jie Gao

2. 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] …

3. 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!

4. 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

5. Routing in Outer Spaces i.e., Torus Routing A rectangular network Wrapped up as a torus. Route on the torus. With equal prob to each of the 4 images. Again, delivery is not guaranteed! Flip

6. 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.

7. 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.

8. Polyhedron Routing [Papadimitriou & Ratajczak] Greedy routing with d(u, v)= – c(u) · c(v) guarantees delivery. Route along the surface of a convex polytope. 3D coordinates of v

9. 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.

10. Prepare the Reduced Graph Input graph

11. Prepare the Reduced Graph Overlay G and the dual graph Ĝ, add intersection vertices as edge nodes. Each “face” becomes a quadrilateral Triangulate each quadrilateral by adding a virtual edge. Vertex node Edge node Face node

12. 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.

13. 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

14. Different Möbius transformation Möbius transformation preserves the circle packings. Optimize for “uniform vertex distribution” ≈ uniform vertex circle size.

15. Simulations Compare with Curveball Routing and Torus Routing

16. 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.

17. Adjust Node Density wrt Battery Level Find the Möbius transformation st circle size ~ battery level. Battery level: High to LowNo optimization With optimization Routes prefer high battery nodes

18. Network with Non-Uniform Density Dense region spans wider area. Birdeye viewUniform density

19. 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?

20. Questions and Comments?