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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