spherical representation & polyhedron routing for load balancing in wireless sensor...
DESCRIPTION
Spherical Representation & Polyhedron Routing for Load 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 M ax # messages any node delivers. Prolong network lifetime - PowerPoint PPT PresentationTRANSCRIPT
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?