1. Broadcasting in Conflict Aware Multi-Channel Networks WALCOM 2013 – Feb 14, 2013 Shahin Kamali 1 Joint work with Francisco Claude 1, Reza Dorrigiv 2, Alejandro Lopez-Ortiz 1, Pawel Pralat 1, Jazmin Romero 1, Alejandro Salinger 1, and Diego Seco 3 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, Canada 4 Database Laboratory, University of A Coruna, Spain. 14/02/2013WALCOM 20131

2. Outline Introduction – Broadcasting problem – Multi-channel networks Conflict-aware model – Problem statement Graph families – Trees, Grids, Complete graphs Channel assignment 14/02/2013WALCOM 20132

3. 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 14/02/2013WALCOM 20133

4. Classical Model (Telephone Model) – In each round, each informed node can send the message to at most one neighbor A B E C DF A B D C E F 14/02/2013WALCOM 20134

5. 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? 14/02/2013WALCOM 20135

6. t=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 DF A BC E 1 1,3 2,3 1 1 2 2 2 DF t=2 1 1 2 2 2 2 2 1 1 14/02/2013WALCOM 20136

7. 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=1 A B E C DF A BC E 1 1,3 2,3 1 1 2 2 2 DF t=2 1 1 2 2 2 2 2 1 1 ✓ ✓ ✗ 14/02/2013WALCOM 20137

8. Previous Work Geometric graphs [Mahojiran, et al, Zheng, et al] – No theoretical analysis An extension of telephone model – Hardness, etc. A B E C DF 1 7 4 3 5 6 8 10 9 14/02/2013WALCOM 20138

9. Summary of Results Trees – A polynomial optimal algorithm Complete graph – Hardness proof A B E C DF 1 7 4 3 5 6 8 10 9 Single channel on each edge (simplified model) A B E C DF 1 1,3 2,3 1 1 2 2 2 Multiple channels on each edge (generalized model) Trees – Hardness proof Grids – A polynomial optimal algorithm 14/02/2013WALCOM 20139

10. Simplified Model (Single Channel on Edges) Optimal polynomial algorithm for trees – Extension from telephone model A BCDE FGHIJ KL = 4 2120 01001 0 0 max{1+2, 2+2, 0+3} 1 2 2 3 12313 1 1 14/02/2013WALCOM 201310

11. 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 35 2 4 W ZW,X W,Y X,Y,Z 14/02/2013WALCOM 201311

12. Generalized Model (multiple channels each edges) Polynomial algorithm for grids Find splitters 1,2 1 1,3 3 2,3 2 3 14/02/2013WALCOM 201312

13. 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 14/02/2013WALCOM 201313

14. Hardness for Complete Graphs Exact Cover – Given a bipartite graph, is there a subset on left which exactly covers all vertices on right 14/02/2013WALCOM 201314

15. 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 = {a 4 }, X={a 1,a 3 } is a solution Exact cover with neighborhood is NP-hard – Reduction from Exact cover a1a1 a2a2 a3a3 a4a4 14/02/2013WALCOM 201315

16. 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 cover with neighborhood is yes a1a1 a2a2 a3a3 a4a4 v a2a2 a1a1 a3a3 a4a4 14/02/2013WALCOM 201316

17. 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 14/02/2013WALCOM 201317

18. Summary of Results Trees – A polynomial optimal algorithm Complete graph – Hardness proof A B E C DF 1 7 4 3 5 6 8 10 9 Single channel on each edge (simplified model) A B E C DF 1 1,3 2,3 1 1 2 2 2 Multiple channels on each edge (generalized model) Trees – Hardness proof Grids – A polynomial optimal algorithm 14/02/2013WALCOM 201318

19. 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 1 1 1 1 1 1 1 1 1 1 1 14/02/2013WALCOM 201319

20. A Desired Channel Assignment 14/02/2013WALCOM 201320

21. 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 14/02/2013WALCOM 201321

22. A desired Channel Assignment Broadcasting from any node completes in 2 rounds – Having k channels and at least k 2 - 2k + 1 nodes Gossiping completes in 3 rounds 14/02/2013WALCOM 201322

23. Concluding Remarks The problem is hard, even for simple class families – Approximation algorithm Channel assignment – Other graph families (trees?) 14/02/2013WALCOM 201323

24. Thanks ! 14/02/2013WALCOM 201324