Spherical Representation & Polyhedron Routing for

Load Balancing in Wireless Sensor Networks

Xiaokang Yu Xiaomeng Ban

Wei ZengRik Sarkar

Xianfeng David GuJie 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

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

• 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 nodeEdge node

Edge node

Face 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.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.

Battery level: High to Low No optimizationWith optimizationRoutes 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?