# Spherical Representation &

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21 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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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
M
ax # 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

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

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

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
intersection vertices
as edge nodes.

Each “face” becomes

Triangulate each

Vertex node

Edge node

Edge node

Face node

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

Torus routing: most uniform load; but
avg

80% higher than simple greedy methods.

Ours
v.s

Curveball: slightly higher
avg

solves the center
-
dense problem better.

wrt

Battery Level

Find the
Möbius

transformation
st

circle size ~
battery level.

Battery level: High to Low

No optimization

With 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?