Spherical Representation &

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21 Νοε 2013 (πριν από 3 χρόνια και 10 μήνες)

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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


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 node

Edge 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 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?


Saddles attract geodesic paths, peaks/valleys
repel.


Uniformizing

curvature always leads to better load
balancing?

Questions and Comments?