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Optimal Oblivious Routing in Hole-Free Networks Costas Busch - - PowerPoint PPT Presentation
Optimal Oblivious Routing in Hole-Free Networks Costas Busch - - PowerPoint PPT Presentation
Optimal Oblivious Routing in Hole-Free Networks Costas Busch Louisiana State University Malik Magdon-Ismail Rensselaer Polytechnic Institute 1 Routing: choose paths from sources to destinations v 3 u 1 u v 2 2 u 3 v 1 2 Edge
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Edge congestion
edge
C
maximum number of paths that use any edge
Node congestion
node
C
maximum number of paths that use any node
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Length of chosen path Length of shortest path
u v
Stretch=
5 . 1 8 12 stretch
shortest path
chosen path
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Oblivious Routing
Each packet path choice is independent
- f other packet path choices
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1
q
2
q
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q
Path choices:
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q
k
q q , ,
1
Probability of choosing a path:
] Pr[ i q
1 ] Pr[
1
k i i
q
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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 (
*
n C O C
node node
) 1 ( stretch O
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Holes
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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:
n C d O C
edge edge
log
*
d n C C
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 (existential result):
n C O C
edge edge 3 *
log
Constructive Results: Racke [FOCS’02]:
n C O C
edge edge
log
*
Racke [STOC’08]:
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Hierarchical clustering General Approach:
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Hierarchical clustering General 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 stretch
Problem:
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An Impossibility Result Stretch and congestion cannot be minimized simultaneously in arbitrary graphs
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) ( n
Each path has length n paths
Length 1
Source of packets
n
Destination
- f all packets
Example graph: nodes
n
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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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n C d O C
edge edge
log
*
) ( stretch
2
d O
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 (
*
n C O C
node node
) log (
*
n C O C
edge edge
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 (
*
n C O C
node node
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n n
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 paths
- r 1-bend paths