Spherical Representation & Polyhedron Routing for Load Balancing in Wireless Sensor Networks

Spherical Representation & Polyhedron Routing forLoad 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. • Route on the torus. • With equal prob to each of the 4 images. • Again, delivery is not guaranteed! Flip Flip

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. • Route along the surface of a convex polytope. 3D coordinates of v

Compute Thurston’s Embedding • Extract a planar graph G of a sensor network • Many prior algorithms exist. • 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 • 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 Edge node

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.sCurveball: 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 Routes prefer high battery nodes Battery level: High to Low No optimization

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?

Spherical Representation & Polyhedron Routing for Load Balancing in Wireless Sensor Networks