Constructing a m-connected k-Dominating Set in Unit Disc Graphs

Slides:



Advertisements
Similar presentations
6.896: Topics in Algorithmic Game Theory Lecture 21 Yang Cai.
Advertisements

Chapter 4 Partition I. Covering and Dominating.
CS 336 March 19, 2012 Tandy Warnow.
Connectivity - Menger’s Theorem Graphs & Algorithms Lecture 3.
Chapter 8 Topics in Graph Theory
Walks, Paths and Circuits Walks, Paths and Circuits Sanjay Jain, Lecturer, School of Computing.
Bayesian Networks, Winter Yoav Haimovitch & Ariel Raviv 1.
Introduction to Graphs
IKI 10100: Data Structures & Algorithms Ruli Manurung (acknowledgments to Denny & Ade Azurat) 1 Fasilkom UI Ruli Manurung (Fasilkom UI)IKI10100: Lecture10.
1 Maximal Independent Set. 2 Independent Set (IS): In a graph G=(V,E), |V|=n, |E|=m, any set of nodes that are not adjacent.
Approximating the Domatic Number Feige, Halldorsson, Kortsarz, Srinivasan ACM Symp. on Theory of Computing, pages , 2000.
CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch.
1 Maximal Independent Set. 2 Independent Set (IS): In a graph, any set of nodes that are not adjacent.
1 On Constructing k- Connected k-Dominating Set in Wireless Networks Department of Computer Science and Information Engineering National Cheng Kung University,
CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch.
1 Discrete Structures & Algorithms Graphs and Trees: II EECE 320.
CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch.
A general approximation technique for constrained forest problems Michael X. Goemans & David P. Williamson Presented by: Yonatan Elhanani & Yuval Cohen.
Is the following graph Hamiltonian- connected from vertex v? a). Yes b). No c). I have absolutely no idea v.
What is the next line of the proof? a). Assume the theorem holds for all graphs with k edges. b). Let G be a graph with k edges. c). Assume the theorem.
K-Coloring k-coloring: A k-coloring of a graph G is a labeling f: V(G)  S, where |S|=k. The labels are colors; the vertices of one color form a color.
Connected Dominating Sets in Wireless Networks My T. Thai Dept of Comp & Info Sci & Engineering University of Florida June 20, 2006.
Minimum Spanning Trees. Subgraph A graph G is a subgraph of graph H if –The vertices of G are a subset of the vertices of H, and –The edges of G are a.
Special Topics on Algorithmic Aspects of Wireless Networking Donghyun (David) Kim Department of Mathematics and Computer Science North Carolina Central.
Introduction to Graph Theory
V. V. Vazirani. Approximation Algorithms Chapters 3 & 22
Message-Optimal Connected Dominating Sets in Mobile Ad Hoc Networks Paper By: Khaled M. Alzoubi, Peng-Jun Wan, Ophir Frieder Presenter: Ke Gao Instructor:
Approximating the Minimum Degree Spanning Tree to within One from the Optimal Degree R 陳建霖 R 宋彥朋 B 楊鈞羽 R 郭慶徵 R
Minimum Average Routing Path Clustering Problem in Multi-hop 2-D Underwater Sensor Networks Presented By Donghyun Kim Data Communication and Data Management.
1 Maximal Independent Set. 2 Independent Set (IS): In a graph G=(V,E), |V|=n, |E|=m, any set of nodes that are not adjacent.
On Non-Disjoint Dominating Sets for the Lifetime of Wireless Sensor Networks Akshaye Dhawan.
Connected Dominating Sets. Motivation for Constructing CDS.
5.2 Trees  A tree is a connected graph without any cycles.
CS774. Markov Random Field : Theory and Application Lecture 02
Fall 2015 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University 1.
Connectivity and Paths 報告人:林清池. Connectivity A separating set of a graph G is a set such that G-S has more than one component. The connectivity of G,
Topics Paths and Circuits (11.2) A B C D E F G.
Two Connected Dominating Set Algorithms for Wireless Sensor Networks Overview Najla Al-Nabhan* ♦ Bowu Zhang** ♦ Mznah Al-Rodhaan* ♦ Abdullah Al-Dhelaan*
Maximal Independent Set and Connected Dominating Set Xiaofeng Gao Research Group on Mobile Computing and Wireless Networking Univ. of Texas at Dallas.
Introduction to Graph Theory
Constructing K-Connected M-Dominating Sets in Wireless Sensor Networks Yiwei Wu, Feng Wang, My T. Thai and Yingshu Li Georgia State University Arizona.
NOTE: To change the image on this slide, select the picture and delete it. Then click the Pictures icon in the placeholder to insert your own image. Fast.
Chromatic Coloring with a Maximum Color Class Bor-Liang Chen Kuo-Ching Huang Chih-Hung Yen* 30 July, 2009.
Introduction Wireless Ad-Hoc Network  Set of transceivers communicating by radio.
Theory of Computational Complexity Probability and Computing Chapter Hikaru Inada Iwama and Ito lab M1.
Khaled M. Alzoubi, Peng-Jun Wan, Ophir Frieder
Computing Connected Components on Parallel Computers
Minimum Spanning Tree 8/7/2018 4:26 AM
Maximal Independent Set
Great Theoretical Ideas in Computer Science
Greedy Algorithms / Minimum Spanning Tree Yin Tat Lee
Chapter 5. Optimal Matchings
Planarity Testing.
Discrete Mathematics for Computer Science
CSE 421: Introduction to Algorithms
Maximal Independent Set
Connected Dominating Sets
CS 583 Analysis of Algorithms
V11 Metabolic networks - Graph connectivity
Introduction Wireless Ad-Hoc Network
Algorithms (2IL15) – Lecture 7
V11 Metabolic networks - Graph connectivity
at University of Texas at Dallas
A Better Approximation for Minimum Total Routing Path Clustering Problem in 2-D Underwater Sensor Networks Wei Wang, Donghyun Kim, and Weili Wu, A Better.
V11 Metabolic networks - Graph connectivity
On Constructing k-Connected k-Dominating Set in Wireless Networks
GRAPHS.
at University of Texas at Dallas
Locality In Distributed Graph Algorithms
Minimum Spanning Trees
Presentation transcript:

Constructing a m-connected k-Dominating Set in Unit Disc Graphs W. Shang, F. Yao, P. Wan, and X. Hu, On minimum m-connected k-dominating set problem in unit disc graphs, Journal of Combinatorial Optimization, Dec. 2007. Presented By Donghyun Kim June 11, 2008 Mobile Computing and Wireless Networking Research Group at University of Texas at Dallas

Virtual Backbone In simulation, DSR/AODV over virtual backbones performs better than plain DSR/AODV. Size does matter! Communication overhead can be reduced. Increase the convergence speed (in routing). Simplify the connectivity management. Support broadcasting or multicasting. Reduce the overall energy consumption. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Notations and Definitions is a connected graph with vertex-set and edge-set . For is called the (open) neighborhood of . is called the closed neighborhood of . A cut-vertex of a connected graph is a vertex such that the graph is disconnected. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Notations and Definitions – cont’ A block is a maximal connected subgraph having no cut- vertex. The block-cut-vertex graph of is a graph where consists of all cut-vertices of and all blocks of , with a cut-vertex adjacent to a block if is a vertex of . A leaf block of a connected graph is a block with only one cut-vertex. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Notations and Definitions – cont’ For a graph , a Dominating Set (DS) of is a subset of such that each node in is adjacent to at least one node in . Computing an Maximal Independent Set (MIS) is the most popular way to get DS. A Connected Dominating Set (CDS) of is a dominating set of which induces a connected subgraph of . Nodes in are called as dominaters. Others are called as dominatees. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Notations and Definitions – cont’ Coloring Technique for MIS computation Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Notations and Definitions – cont’ The problem of computing a virtual backbone for a wireless network is modeled as one of calculating a CDS of a graph which represents the network. A -dominating set of is a set of vertices such that each vertex is either in or has at least neighbors in . Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Notations and Definitions – cont’ -Vertex Connectivity A network is -vertex connected if it is connected and removing any 1, 2, …, nodes from will not cause partition in . -Connected -Dominating Set Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Fault Tolerant Virtual Backbone Fault-Tolerance for dominaters With -connectivity, communication may not be disrupted even when up to paths fail. Fault-Tolerance for dominatees Each dominatee node has at least neighboring dominators in a CDS. It also provides routing flexibility. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Overview How to compute ( , )-CDS? Algorithm for (1, 1)-CDS Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Relationship between an MIS and k-Dominating Set Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Relationship between an MIS and k-Dominating Set – cont’ Lemma 1 Let be a UDG and a natural number such that minimum degree . Let be a minimum -dominating set of and an MIS of . Then . Proof of Lemma 1 Let Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Relationship between an MIS and k-Dominating Set – cont’ For each , let . Then, has to be true by the property of -dominating set, and . For each , let Then, since is a UDG, for each , there are at most 5 independent vertices in its neighborhood and , and . Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (1,k)-CDS Basic Idea Compute (1,1)-CDS Sequentially produce an MIS times. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (1,k)-CDS – cont’ Theorem 1 Algorithm A is an approximation algorithm for the minimum connected -dominating set problem with performance ratios for and 7 for . Correctness of Algorithm A is (1, 1)-CDS A node has at least neighbors in . Therefore, is (1, )-CDS. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (1,k)-CDS – cont’ Proof of Theorem 1 Let for is an independent set and . By lemma1, Then, Then, and hence for | and for . Since By combining the two estimation above, the theorem holds true. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (2,k)-CDS Step 1: Apply Algorithm A to construct a 1- connected -dominating set . Step 2: Compute all the blocks in by computing the 2-connected components through the depth first search. Step 3: Produce the shortest path in the original graph such that it can connect a leaf block in with other part of but does not contain any vertices in except the two endpoints. Then add all intermediate vertices in this path to . Step 4: Repeat Step 2 and Step 3 until is 2- connected. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (2,k)-CDS – cont’ Basic Idea Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (2,k)-CDS – cont’ Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (2,k)-CDS – cont’ Lemma 2 For , at most two new vertices are added into at each augmenting step. Lemma 3 The number of cut-vertices in the connected -dominating set by Algorithm A is no bigger than the number of vertices in generated at the first step of Algorithm A. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (2,k)-CDS – cont’ Theorem 2 Algorithm B is an approximation algorithm for the minimum 2-connected -dominating set problem with performance ratio Proof of Theorem 2 At most nodes are added by Lemma 2 for 2- connectivity by Lemma 2 and 3. From Theorem1, we have and for and for . The theorem hold true by combining the results above. Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas

Algorithm for (m,k)-CDS – cont’ This algorithm assumes the existence of -CDS computation algorithm and . Theorem 3. If there exists an -approximation algorithm A for the case of , then there exists an ( +6)-approximation algorithm for the case of . Presented by Donghyun Kim on June 11, 2008 Mobile Computing and Wireless Networking Research Group at The University of Texas at Dallas