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Optimal Oblivious Routing in Hole-Free Networks
Costas BuschLouisiana State University
Malik Magdon-IsmailRensselaer Polytechnic Institute
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1u
1v
2u2v
3u
3v
Routing: choose paths from sources to destinations
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Edge congestion
edgeC
maximum number of paths that use any edge
Node congestion
nodeC
maximum number of paths that use any node
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Length of chosen pathLength of shortest path
uv
Stretch=
5.18
12stretch
shortest path
chosen path
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Oblivious RoutingEach 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
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Benefits of oblivious routing:
•Appropriate for dynamic packet arrivals
•Distributed
•Needs no global coordination
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Hole-free network
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Our contribution in this work:Oblivious routing in hole-free networks
Constant stretch
Small congestion
)log( * nCOC nodenode )1(stretch O
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Holes
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Related Work
Valiant [SICOMP’82]:First oblivious routing algorithmsfor permutations on butterfly and hypercube
butterfly butterfly (reversed)
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d-dimensional Grid:
nCdOC edgeedge log*
d
nCC edgeedge
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 (existential result): nCOC edgeedge
3* log
Constructive Results:
Racke [FOCS’02]:
nCOC edgeedge log* Racke [STOC’08]:
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Hierarchical clusteringGeneral Approach:
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Hierarchical clusteringGeneral Approach:
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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
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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nCdOC edgeedge log*
)(stretch 2dO
Result for Grids:
Busch, Magdon-Ismail, Xi [TC’08]
For d=2, a similar result given by C. Scheideler
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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(stretch O
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Embeddings in wide, closed-curved areas
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Graph models 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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However, algorithm does not extend to arbitrary closed shapes
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Our contribution in this work:Oblivious routing in hole-free networks
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Approach: route within square areas
)1(stretch O )log( * nCOC nodenode
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nn grid
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simple area in grid (hole-free area)
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Hole-free network
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Canonical square decomposition
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Canonical square decomposition
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Canonical square decomposition
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Canonical square decomposition
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u
v
Shortest path
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u
v
Canonical square sequence
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u
v
A random path in canonical squares
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u
v
Path has constant stretch
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Random 2-bend pathsor 1-bend paths in square sequence