1
Oblivious Routing in Wireless networks
Costas BuschRensselaer Polytechnic Institute
Joint work with: Malik Magdon-Ismail and Jing Xi
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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
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Oblivious Routing
Each packet path choice is independent
of other packet path choices
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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
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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]:
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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
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Pick random node
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Pick random node
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Pick random node
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Pick random node
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Pick random node
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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
Destination
of 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 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
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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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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
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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
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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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Au v
vu
vudistG,
),(
there exist :,
6.15
8
,
),(
vu
vudistG
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, vu
Max Euclidian distance
between adjacent nodes
vu
vudistG,
),(
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Consequences of vu
vudistG,
),(
1, vu
u vr
2)( rO nodesMin Euclidian Distancebetween any pair of
nodes:
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Small and large R,,
Good Network embeddings:
Suppose they are constants
46
Outline of Presentation
Introduction
Network Model
Oblivious Algorithm
Discussion
Analysis
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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
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As
t
y
Pick random space point y
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As
t
R
Find closest node to point y
wy
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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:
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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
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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
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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
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s
tv
y
R
R
wRdeviation
v
Segment of space points affecting v
1
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Probability of using node :v
2
1]Pr[
v
A
s
tv
yw
R)(Qdeviation
v
1
2
R
R
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It can be shown that:
vs
deviation
ts
Rkv
,,]Pr[ 1
2
1
constant
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)(,, 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
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0rv
1r
2r
3r
i
ir2
Divide area into concentric circlesA
A
0A1A
2A
3A
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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
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1, irts
v
1ir1iA
ir
1ir
1ir
1ir
1ir 1ir
1ir
1ir
1ir
1ir
1ir1iA
1ir
ts
tsdistG,
),(
1),( iG rtsdist
iA
XArea
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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
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Considering all the rings:
)log(
log
][)(
*
*
log
0
nCO
nCO
CECE
n
ii
End of Proof
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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
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Outline of Presentation
Introduction
Network Model
Oblivious Algorithm
Discussion
Analysis
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Holes
93
Arbitrary closed shapes
there is no