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

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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] …

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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!

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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

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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

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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.

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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.

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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

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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.

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Prepare the Reduced Graph Input graph

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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

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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.

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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

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

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Simulations Compare with Curveball Routing and Torus Routing

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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.

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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

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

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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?

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Questions and Comments?

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