1

Oblivious Routing in Wireless networks

Costas BuschRensselaer Polytechnic Institute

Joint work with: Malik Magdon-Ismail and Jing Xi

2

Outline of Presentation

Introduction

Network Model

Oblivious Algorithm

Discussion

Analysis

3

1u

1v

2u2v

3u

3v

Routing: choose paths from

sources to destinations

4

Edge congestion

edgeC

maximum number of

paths that use any edge

Node congestion

nodeC

maximum number of

paths that use any node

5

Length of chosen path

Length of shortest path

uv

Stretch=

5.18

12stretch

shortest path

chosen path

6

Oblivious Routing

Each packet path choice is independent

of other packet path choices

7

1q

2q

3q

Path choices:

4q

4q

5q

kqq ,,1

Probability of choosing a path: ]Pr[ iq

1]Pr[1

k

iiq

8

Benefits of oblivious routing:

•Appropriate for dynamic packet arrivals

•Distributed

•Needs no global coordination

9

Related Work

Valiant [SICOMP’82]:

First oblivious routing algorithms

for permutations on butterfly and hypercube

butterfly butterfly (reversed)

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d-dimensional Grid: nCdOC edgeedge log*

d

nCC

edge

edge

log*Lower bound

for oblivious routing:

Maggs, Meyer auf der Heide,

Voecking, Westermann [FOCS’97]:

11

Azar et al. [STOC03]

Harrelson et al. [SPAA03]

Bienkowski et al. [SPAA03]

Arbitrary Graphs: nCOC edgeedge 3* log

constructive

Racke [FOCS’02]:

existential result

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Hierarchical clusteringApproach:

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14

At the lowest level every node is a cluster

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source destination

16

Pick random node

17

Pick random node

18

Pick random node

19

Pick random node

20

Pick random node

21

Pick random node

22

Pick random node

23

24

Adjacent nodes may follow long paths

Big stretchProblem:

25

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

Destination

of all packets

Example graph:

nodesn

27

n packets in one path

Stretch =

Edge congestion =

1

n

28

1 packet per path

n

1

Stretch =

Edge congestion =

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Contribution

Oblivious algorithm for special graphs

embedded 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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Embeddings in wide, closed-curved areas

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Our algorithm is appropriate

for various wireless network topologies

Transmission radius

32

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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nCdOC edgeedge log*

Stretch = )( 2dO

Previous results for Grids:

Busch, Magdon-Ismail, Xi [IPDPS’05]

For d=2, a similar result given by C. Scheideler

36

Outline of Presentation

Introduction

Network Model

Oblivious Algorithm

Discussion

Analysis

37

Network G Surrounding area

A

38

space

point space

point

Perpendicular bisector

xy

yx ,

yx ,

A

39

space

point space

point

yx ,

s

xy

yx

syx

,),(

A

40

Area wideness: ),(min,

yxAyx

A

41

x

Rspace pointgraph node

Coverage Radius :Rmaximum distance from a space point

to the closest node

A

42

Au v

vu

vudistG,

),(

there exist :,

6.15

8

,

),(

vu

vudistG

For all pair of nodes

vu ,

),( vudistGShortest path length:

Euclidian distance:

43

Consequences of

u v

(max transmission radius in wireless networks)

edge

1, vu

Max Euclidian distance

between adjacent nodes

vu

vudistG,

),(

44

Consequences of vu

vudistG,

),(

1, vu

u vr

2)( rO nodesMin Euclidian Distancebetween any pair of

nodes:

45

Small and large R,,

Good Network embeddings:

Suppose they are constants

46

Outline of Presentation

Introduction

Network Model

Oblivious Algorithm

Discussion

Analysis

47

Au v

z w

Every pair of nodes is assigned a default path

default path

default path

Examples: •Shortest paths

•Geographic routing paths (GPSR)

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As

t

The algorithm

sourcedestination

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As

t

Perpendicular bisector

50

As

t

y

Pick random space point y

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As

t

R

Find closest node to point y

wy

52

As

t

wdefault

pathdefault

path

Connect intermediate node

to source and destination

w

53

Outline of Presentation

Introduction

Network Model

Oblivious Algorithm

Discussion

Analysis

54

Consider an arbitrary set of packets:

N ,,1

NppP ,,1

Suppose the oblivious algorithm gives paths:

55

We will show:

1Ostretch

nCOC nodenode log*

optimal congestion

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Theorem: 1Ostretch

Proof: Consider an arbitrary path

and show that:

Pp

1)( Opstretch

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sA

tdefault

path default

pathw

y1q

2qp

),(

)()(

),(

)()( 21

tsdist

qlengthqlength

tsdist

plengthpstretch

GG

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),(

)()()( 21

tsdist

qlengthqlengthpstretch

G

),(

),(),()(

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) ws

wsdistG,

),(R

RtstwdistG ,),( Similarly:

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tstsdistG ,),(

sA

t

ts

tsdistG,

),(

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ts

Rts

tsdist

twdistwsdistpstretch

G

GG

,

,2

),(

),(),()(

For constants:R,,

1)( Opstretch

End of Proof

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Theorem:

nCOC log*

nodeC

Proof: Consider some arbitrary node

and estimate congestion on

Expected case:

vv

*nodeCdenotes

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Au v

z w

Deviation of default paths:

)( 1qdeviation

)(max iq

qdeviationdeviationi

maximum distance from geodesic

)( 2qdeviation

geodesic

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Consider some path from to

s

t

s t

v

65

s

tv

vthe use of depends on the choice of space point

one choice

R y

y

w

66

s

t

w another choice

v

Ry

67

s

tv

wv

deviation

If you choose node in the cone

the respective path may use vw

68

s

tv

wv

deviation

If you choose node outside the cone

the respective path does not usevw

69

s

tv

y

R

R

wRdeviation

v

Segment of space points affecting v

1

70

Probability of using node :v

2

1]Pr[

v

A

s

tv

yw

R)(Qdeviation

v

1

2

R

R

71

It can be shown that:

vs

deviation

ts

Rkv

,,]Pr[ 1

2

1

constant

72

)(,, QdeviationRtsvs

s

tv

ts ,

vs ,

R

R

ts ,

deviation

deviation

tsvs ,,

for simplicity

assume:

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tsvs ,,

vs

deviation

ts

Rkv

,,]Pr[ 1

vs

deviationRkv

,]Pr[ 1

deviationR ,, : constants

vs

kv

,]Pr[ 2

74

0rv

1r

2r

3r

i

ir2

Divide area into concentric circlesA

A

0A1A

2A

3A

75

A

Max Euclidian distance

between any two nodes =

n

1, 1 ii uu

Longest path has at most nodes

1u2u

3u

nu

1nu

n

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v0r

1r

2r

i

ir2

0A1A

2A

nAlog

nr n log

Maximum ring radius

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v

iN = number of packets that can affectv

iC = number of paths that use v

iriARing

We will bound

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v

1ir

ir

1iA

s t

w

1

22

,]Pr[

ir

k

vs

kv

vs ,

iA

79

v

1

2]Pr[][

i

iii r

NkvNCEExpected congestion:

1ir

ir

1iA

iA

80

1

][i

ii r

NOCE

1

*

i

i

r

NC

)(][ *COCE i

We have proven

we prove next

11 4 ii rr

81

v

tsvsri ,,1

1, irts

1, irvs

st

we showed

earlier

1ir

ir

1iA

iA

82

v

Similarly, each packet that affects

traverses distance at least

1ir

1ir

1ir

1ir

1ir

1ir

1ir

1ir

1ir

v

1ir

1ir

ir

1iA

iA

83

1, irts

v

1ir1iA

ir

1ir

1ir

1ir

1ir 1ir

1ir

1ir

1ir

1ir

1ir1iA

1ir

ts

tsdistG,

),(

1),( iG rtsdist

iA

XArea

84

1 ii rN Total number of nodes used

v

1ir

1ir

1ir

1ir 1ir

1ir

1ir

1ir

1ir

1ir

XArea

1ir1iA

ir

1ir1iA

iA

85

v

1ir

1ir

1ir

1ir 1ir

1ir

1ir

1ir

1ir

1ir

X area in nodes#1 ii

rN Average node utilization

XArea

1ir1iA

ir

1ir1iA

iA

86

21)( irO

v

#nodes in area =X

XArea

1iA1ir

87

12

1

1

)( i

i

i

ii

r

N

rO

rN

Average node utilization

average node utilization*C

1

*

i

i

r

NC

88

1

][i

ii r

NOCE

1

*

i

i

r

NC

)(][ *COCE i

We have proven:

11 4 ii rr

89

Considering all the rings:

)log(

log

][)(

*

*

log

0

nCO

nCO

CECE

n

ii

End of Proof

90

Recap

Constant stretch Small congestion

)log( * nCOC nodenode

)log( * nCOC edgeedge

We presented a simple oblivious

algorithm which has:

1Ostretch

when the parameters of the

Euclidian embedding are constants

91

Outline of Presentation

Introduction

Network Model

Oblivious Algorithm

Discussion

Analysis

92

Holes

93

Arbitrary closed shapes

there is no