Optimal Oblivious Routing in Hole-Free Networks

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Optimal Oblivious Routing in Hole-Free Networks

• Costas Busch
• Louisiana State University
• Malik Magdon-Ismail
• Rensselaer Polytechnic Institute

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Routing choose paths from sources
to destinations
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Edge congestion
Node congestion
maximum number of paths that use any edge
maximum number of paths that use any node
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Length of chosen path Length of shortest path
Stretch
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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Path choices
Probability of choosing a path
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Benefits of oblivious routing

• Distributed

• Needs no global coordination

• Appropriate for dynamic packet arrivals

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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
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Holes
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Related Work
Valiant SICOMP82
First oblivious routing algorithms for
permutations on butterfly and hypercube
butterfly
butterfly (reversed)
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Maggs, Meyer auf der Heide, Voecking, Westermann
FOCS97
d-dimensional Grid
Lower bound for oblivious routing
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Arbitrary Graphs (existential result)
Racke FOCS02
Racke STOC08
Constructive Results
Azar et al. STOC03 Harrelson et al.
SPAA03 Bienkowski et al. SPAA03
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General Approach
Hierarchical clustering
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General Approach
Hierarchical clustering
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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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Problem
Big stretch
Adjacent nodes may follow long paths
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An Impossibility Result
Stretch and congestion cannot be minimized
simultaneously in arbitrary graphs
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Example graph
Each path has length
paths
nodes
Length 1
Destination of all packets
Source of packets
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Stretch
Edge congestion
packets in one path
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Stretch
Edge congestion
1 packet per path
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Result for Grids
Busch, Magdon-Ismail, Xi TC08
For d2, a similar result given by C. Scheideler
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Special graphs embedded in the 2-dimensional
plane
Busch, Magdon-Ismail, Xi SPAA 2005
Constant stretch
Small congestion
degree
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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
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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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Shortest path
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Canonical square sequence
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A random path in canonical squares
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Path has constant stretch
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Random 2-bend paths or 1-bend paths in square
sequence