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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
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Pick random node
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Pick random node
11
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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