Download presentation

Presentation is loading. Please wait.

Published byHaley Weir Modified over 2 years ago

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

Similar presentations

OK

CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch.

CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on retail marketing mix Ppt on eia report environmental impacts Free download ppt on the road not taken Ppt on the art of war audio Ppt on project tiger free Ppt on the rise of nationalism in europe class 10 Ppt on indian army weapons Free ppt on forest society and colonialism in india Free ppt on water cycle Ppt on our country india class 6