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Oblivious Routing in Wireless networks

Costas BuschRensselaer Polytechnic Institute

Joint work with: Malik Magdon-Ismail and Jing Xi

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Length of chosen pathLength of shortest path

uv

Stretch=

5.1812

stretchshortest path

chosen path

3

source destination

4

Pick random node

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Pick random node

6

Pick random node

7

Pick random node

8

Pick random node

9

Pick random node

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Pick random node

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Adjacent nodes may follow long paths

Big stretchProblem:

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An Impossibility Result

Stretch and congestion cannot be minimized simultaneously in arbitrary graphs

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)( nEach path has length

n paths

Length 1

Source of packetsn

Destinationof all packets

Example graph:

nodesn

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n packets in one path

Stretch =

Edge congestion =

1

n

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1 packet per path

n

1

Stretch =

Edge congestion =

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Contribution

Oblivious algorithm for special graphsembedded in the 2-dimensional plane

Constant stretch

Small congestion

)log( * nCOC nodenode

)log( * nCOC edgeedge

degree

Busch, Magdon-Ismail, Xi [SPAA 2005]:

)1(Ostretch

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Basic Idea

source destination

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Pick a random intermediate node

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Construct path through intermediate node

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Outline of Presentation

Introduction

Network Model

Oblivious Algorithm

Analysis

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Network G Surrounding area

A

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spacepoint space

point

Perpendicular bisector

geodesic

xy

yx ,

yx ,

A

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spacepoint space

point

yx ,

s

xy

yxs

yx,

),(

A

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Area wideness: ),(min,

yxAyx

A

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x

Rspace pointgraph node

Coverage Radius :Rmaximum distance from a space point to the closest node

A

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Auv

vuvudistG

,),(

there exist :,

6.158

,),(

vuvudistG

For all pair of nodes

vu,

),( vudistGShortest path length:

Euclidian distance:

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Consequences of

u v

(max transmission radius in wireless networks)

edge

1

, vuMax Euclidian distancebetween adjacent nodes

vuvudistG

,),(

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Consequences of vuvudistG

,),(

1

, vu

u vr

2)( rO nodes

Min Euclidian Distancebetween any pair of

nodes:

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Outline of Presentation

Introduction

Network Model

Oblivious Algorithm

Analysis

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Auv

z w

Every pair of nodes is assigned a default path

default path

default path

Examples: •Shortest paths

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As

t

The algorithm

sourcedestination

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As

tgeodesic

Perpendicular bisector

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As

t

y

Pick random space pointy

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As

t

R

Find closest node to pointy

wy

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As

t

wdefault path

default path

Connect intermediate node to source and destination

w

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Outline of Presentation

Introduction

Network Model

Oblivious Algorithm

Analysis

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Consider an arbitrary set of packets:

N ,,1

NppP ,,1

Suppose the oblivious algorithm gives paths:

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We will show:

1Ostretch

nCOC nodenode log*

optimal congestion

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Theorem: 1Ostretch

Proof: Consider an arbitrary pathand show that:

Pp

1)( Opstretch

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sA

tdefaultpath default

pathw

y1q

2qp

),()()(

),()(

)( 21

tsdistqlengthqlength

tsdistplength

pstretchGG

shortest path

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),()()(

)( 21

tsdistqlengthqlength

pstretchG

),(),(),(

)(tsdist

twdistwsdistpstretch

G

GG

we show this is constant

when default paths are shortest paths

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RtsRyswswsdistG ,,,),(

sA

t

w

yDefault path (shortest)

wswsdistG

,),(R

RtstwdistG ,),( Similarly:

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tstsdistG ,),(

sA

tshortest path

tstsdistG

,),(

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ts

Rtstsdist

twdistwsdistpstretch

G

GG

,,2

),(),(),(

)(

For constants:R,,

1)( Opstretch

End of Proof

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Theorem:

nCOC log*

nodeC *nodeCdenotes