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Spherical Representation & Polyhedron Routing for
Load Balancing in Wireless Sensor Networks
Xiaokang Yu Xiaomeng Ban
Wei ZengRik Sarkar
Xianfeng David GuJie 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
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 nodeEdge 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 Low No optimizationWith optimizationRoutes prefer high battery nodes
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Network with Non-Uniform Density
• Dense region spans wider area.
Birdeye view Uniform 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?