WALCOM 2013 1

Broadcasting in Conflict Aware Multi-Channel Networks

WALCOM 2013 – Feb 14, 2013

Shahin Kamali1

Joint work with Francisco Claude1, Reza Dorrigiv2, Alejandro Lopez-Ortiz1, Pawel Pralat1, Jazmin Romero1, Alejandro Salinger1, and Diego Seco3

1 David R. Cheriton School of Computer Science, University of Waterloo, Canada. 2 Faculty of Computer Science, Dalhousie University, Canada 3 Department of Mathematics, Ryerson University, Toronto, Canada4 Database Laboratory, University of A Coruna, Spain.

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Outline

• Introduction– Broadcasting problem– Multi-channel networks• Conflict-aware model

– Problem statement• Graph families– Trees, Grids, Complete graphs

• Channel assignment

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Broadcasting Problem

• A network is modelled by an undirected, unweighted graph

• Broadcasting problem– A single message is sent from a ‘source’ of a network to all

other vertices– Communication occurs in discrete rounds– In each round informed vertices inform ‘some’ uninformed

vertices– The goal is to find a scheme which completes in minimum

number of rounds

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Classical Model (Telephone Model)

– In each round, each informed node can send the message to at most one neighbor

A

B

E

C

D F

A

B

D

C

E

F

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Classical Model (Telephone Model)

• Under the telephone model– The problem is NP-hard• Remains NP-hard for planar graphs, etc. [Jakobi, et al]

• Polynomial solvable for decomposable graphs, etc. [Jakobi, et al]

– The best approximation algorithm has ratio lg n/lg lg n [Elkin, Kortsarz]

• A constant approximation?

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t=1t=1

Multi-channel Networks

• At each round a message can be sent on a channel (multiple edges)– Frequencies in Wireless Networks

A

B

E

C

D F

A

B C

E

1 1,3

2,3

11

2

2

22,3

D F

t=21 1

22

2

221

1

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Multi-channel Networks with Conflicts

• A conflict occurs when– Two or more neighbors of u send data to u

through the same channel in the same round– u does not receive message from that channel

t=1t=1

A

B

E

C

D F

A

B C

E

1 1,3

2,3

11

2

2

22,3

D F

t=21 1

22

2

221

1

✓✓

✗

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Previous Work

• Geometric graphs [Mahojiran, et al, Zheng, et al]

– No theoretical analysis• An extension of telephone model– Hardness, etc.

AB

E

C

D F

1 74

35

6 810

9

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Summary of Results

• Trees– A polynomial optimal algorithm

• Complete graph– Hardness proof

AB

E

C

D F

1 74

35

6 810

9

Single channel on each edge(simplified model)

AB

E

C

D F

1 1,32,3

11

2 22,3

2

Multiple channels on each edge(generalized model)

• Trees– Hardness proof

• Grids– A polynomial optimal algorithm

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Simplified Model (Single Channel on Edges)

• Optimal polynomial algorithm for trees– Extension from telephone model

A

B C D E

F G H I J

K L

= 4

2 1 2 0

0 1 0 0 1

0 0

max{1+2, 2+2, 0+3}

1 2 2 3

12 3 1 3

1 1

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Generalized Model (Multiple Channels on Edges)

• The problem is NP-hard for trees• Reduction from the set cover problem– Example:

• U = {1, 2, 3, 4, 5}• Subsets: {W = {1,2,3}, X ={2, 4}, Y = {3, 4}, Z = {4, 5}}

– There is a set cover of size k if and only if the broadcast completes in k rounds

A

1 3 52 4

W ZW,XW,Y

X,Y,Z

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Generalized Model(multiple channels each edges)

• Polynomial algorithm for grids• Find splitters

1,2 11,3

3

2,3

23

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Complete Graphs

• It is NP-Hard to find the optimum broadcast scheme– Even if there is only one channel on each edge– Reduction series:• Exact cover Exact cover with neighborhood

Broadcasting in complete bipartite graph Broadcasting in complete graphs

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Hardness for Complete Graphs

• Exact Cover– Given a bipartite graph, is there a subset on left

which exactly covers all vertices on right

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Hardness for Complete Graphs (ctd)

• Exact cover with neighborhood– Given a bipartite graph, is there a vertex u on the left an also a subset

X of vertices on the left such that all neighbors of u are exactly covered by X

– Ex: u = {a4}, X={a1,a3} is a solution

• Exact cover with neighborhood is NP-hard– Reduction from Exact cover

a1

a2

a3

a4

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Hardness for Complete Graphs (ctd)

• The broadcasting problem is NP-hard for complete bipartite graphs • Even in the special case when

– There are a total of 2 channels– source is connected to all its neighbors with the same channel.

• Reduction from Exact Cover with Neighborhood– Broadcasting completes in two rounds iff the answer to exact coverwith neighborhood is yes

a1

a2

a3

a4

v

a2

a1

a3

a4

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Hardness for Complete Graphs (ctd)

• The broadcasting problem is NP-hard for complete graphs under the restricted model– Reduction from broadcasting in special instances of

complete bipartite graph instances– Assuming there are at least 8 channels in the network

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Summary of Results

• Trees– A polynomial optimal algorithm

• Complete graph– Hardness proof

AB

E

C

D F

1 74

35

6 810

9

Single channel on each edge(simplified model)

AB

E

C

D F

1 1,32,3

11

2 22,3

2

Multiple channels on each edge(generalized model)

• Trees– Hardness proof

• Grids– A polynomial optimal algorithm

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Channel Assignment

• Assign channels to the given network– Fast communication (minimize broadcast time)– Given k channels

• Complete Graphs– Assign a single channel on all edges

• Good for broadcasting • Bad when there are more than one source

– Minimum broadcast time is 2

1

1

1

11

11

1

1

1

1

1

1

1

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A Desired Channel Assignment

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A Desired Channel Assignment

• k (=3) classes of vertices• Vertices in the same class

connected with the same channel

• All edges between two classes are assigned the same channel

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A desired Channel Assignment

• Broadcasting from any node completes in 2 rounds– Having k channels and at least

k2- 2k + 1 nodes

• Gossiping completes in 3 rounds

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Concluding Remarks

• The problem is hard, even for simple class families– Approximation algorithm

• Channel assignment– Other graph families (trees?)

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Thanks !

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