# Spherical Representation _ Polyhedron Routing for Load Balancing by hcj

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```									 Spherical Representation &
Polyhedron Routing for
in Wireless Sensor Networks
Xiaokang Yu
Xiaomeng Ban
Wei Zeng
Rik Sarkar
Xianfeng David Gu
Jie Gao
Networks
• Goal: Min Max # messages any node delivers.
• 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.
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.              Flip

•   Route on the torus.
•   With equal prob to each of
the 4 images.                Flip

• Again, delivery is not
guaranteed!
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
Greedy routing with
d(u, v)= – c(u) · c(v)
guarantees delivery.

3D coordinates of v

• Route along the surface of a
convex polytope.
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
Vertex node
• Overlay G and the                 Edge node
Face node
intersection vertices
Edge node
as edge nodes.
• Each “face” becomes
• Triangulate each
Compute Circle Packing Using
Curvature Flow
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:
– Dense network: all methods can deliver.

• Load balancing, tested on dense network
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.

With optimization
Battery level: High to Low   No optimization   Routes 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.