maximal independent set and connected dominating set xiaofeng gao research group on mobile computing...

Maximal Independent Set Maximal Independent Set and Connected Dominating and Connected Dominating SetSet

Xiaofeng GaoResearch Group on Mobile Computing and Wireless NetworkingUniv. of Texas at Dallas

CatalogCatalogIntroductionPreliminaryVoronoi DivisionEuler’s FormulaDiscussion for Holes

Xiaofeng Gao, Univ. of Texas at Dallas 26/6/2008

INTRODUCTIONINTRODUCTIONSection 1

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Connected Dominating Connected Dominating SetSetA virtual backbone for wireless

ad hoc networkGarey and Johnson (1978) proved

that finding a Minimum Connected Dominating Set in a general graph is NP-hard.

Researchers always use approximation algorithms to select a feasible solution (as close as possible to OPT)

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Communication ModelCommunication ModelUnit Disk Graph (UDG)

◦ G=(V, E)◦ Each v in V denotes a center of a disk with

radius 1◦ Each (u, v) in E denotes the distance

between u and v is less than or equal to 1.

Disk Graph (DG)◦ G=(V, A)◦ Each v in V denotes a center of a disk with

radius rv ◦ Each (u, v) in E means v lies within u’s

range. Say, dist(u,v) <= ru

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Common TechniquesCommon TechniquesFind a Dominating Set D

◦Partition◦Final Maximal Independent Set (MIS)

Connect D as a CDS◦Steiner Tree◦Spanning Tree

Greedy Approach◦Spider (Guha & Khuller)

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Approximation RatioApproximation RatioGiven graph Gmcds(G): the size of optimal CDS

solution for Gmis(G): the size of selected MIS

set for Gconnect(G): the size of disks

selected to connect the MIS into CDS.

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Approximation Ratio (2)Approximation Ratio (2)Formula

Compare the relationships between mis(G) and mcds(G) is crucial to reduce the approximation ratio

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MilestonesMilestonesWan et.al. (2002)

◦mis(G) <= 4 ∙ mcds(G) + 1Wu et.al. (2006)

◦mis(G) <= 3.8 ∙ mcds(G) +1.2Funke et.al. (2006)

◦mis(G) <= 3.748 ∙ mcds(G) + 9◦mis(G) <= 3.453 ∙ mcds(G) + 8.291

Yao et.al. (2008)◦mis(G) <= 11/3 ∙ mcds(G) + Constant

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Our AccomplishmentOur AccomplishmentResult

◦mis(G) <= 3.453 ∙ mcds(G) + 4.839◦mis(G) <= 3.339 ∙ mcds(G) + 4.974

if there’s no hole in the special graph◦mis(G) <= 3.478 ∙ mcds(G) + 4.874

if there exists holes in the induced graphStrategy

◦Voronoi Division◦Euler’s Formula

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PRELIMINARIESPRELIMINARIESSection 2

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Induced GraphInduced GraphOPT: the optimal CDS in G.

◦OPT can dominate the whole graph.◦For any disks u, v in OPT, dist(u, v) <= 1

S: a selected independent set◦For any disk d in S, there exist a disk t in

OPT such that dist(d, t) <= 1◦For any two disks d1, d2 in S, dist(d1,d2)>1

Make Induced Graph G’◦Change radius of disks in OPT into 1.5◦Change radius of disks in V\OPT into 0.5

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Properties for Induced Properties for Induced GraphGraphThe areas covered by OPT will

contain all the other disks.For any selected independent set

S, any two disks will not intersect each other.

If a disk belongs to both OPT and S, we can consider it as two distinct disks, one belongs to OPT and another belongs to S.

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Example – Original GraphExample – Original Graph

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Example – Induced GraphExample – Induced Graph

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Rough BoundRough Bound

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CalculationCalculation

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VORONOI DIVISIONVORONOI DIVISIONSection 3

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DefinitionDefinitionLet S be a set of n nodes in

Eucliean space. For each node pi in S, the Voronoi Cell V(pi) of pi is the set of points that are closer to pi than to any other nodes of S.

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ExampleExample|OPT| = 2; |MIS| = 7

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AnalysisAnalysisEach disk in MIS will occupy a

Voronoi Cell with more area than a circle with radius 0.5

The minimum Voronoi Cell inside the graph should be a heptagon.

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Minimum Area for TriangleMinimum Area for TriangleNon-Boundary

◦ area(P3)=1.299Boundary

◦ area(E3)=1.178

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Minimum Area for Minimum Area for QuadrangleQuadrangle

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area(P4) = 1

area(A4) > 1.136

area(E4) = 0.9726/6/2008

Minimum Area for Minimum Area for PentagonPentagon

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area(P5) = 0.908

area(A5) > 0.950

area(P5) = 0.9086/6/2008

Minimum Area for Minimum Area for HexagonHexagon

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area(P6) = 0.866

area(E6) = 0.855

area(A6) = 0.886

6/6/2008

Heptagon and OthersHeptagon and Othersarea(Pk)> area(P6) (k>=7)

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Minimum AreaMinimum AreaArea(A7) = 0.853

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Modified Upper BoundModified Upper Bound

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EULER’S FORMULAEULER’S FORMULASection 4

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SymbolsSymbolssi: the minimum area of the non-

boundary i-polygon cell;si': the minimum area of the boundary

i-polygon cell. We have

◦s3 >= s4 >= s5 >= s6 <= s7 <= s8 <= …◦s3' >= s4' >= s5' >= s6' >= s7' <= s8'

<= ...Set si = s6 (i >=7) and si' = s7' (i >= 8)

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3-Regularization3-RegularizationMake the degree of every vertex

in Voronoi Division be exactly 3.

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Euler’s FormulaEuler’s Formula : the outer boundary of MCDSfi: no. of non-boundary Voronoi

Cells with i edges;fi': no. of boundary VC with i

edges.m: no. of edgesn: no. of verticesBy Euler’s Formula:

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EquationsEquationsG' is a cubic graph

◦2m = 3n◦ (1)

be no. of edges in outer face,every edge is in two faces exactly, (2)

For boundary cell, it must have one edge belonging to the outer face (3)

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CalculationCalculationCombining (1), (2), and (3), we

have

Together with Euler’s Formula, remove n, we have

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Calculation – cont.Calculation – cont.Since all Voronoi cells are

contained in the area constructed by MCDS,

Combining two formula together,

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Calculation – cont.Calculation – cont.Since we have

Finally, we get that

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DISCUSSION WITH DISCUSSION WITH HOLESHOLES

Section 5

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ModificationModificationk: no. of holes in G‘ : no. of edges in all holes.Then we have

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Modification – cont.Modification – cont.

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We have

Since

GuessGuess

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ExampleExample

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ReferenceReferenceXianyue Li, Xiaofeng Gao, Weili Wu, A Better 

Theoretical Bound to Approximate Connected Dominating Set in Unit Disk Graph, Submitted to The 3rd International Conference on Wireless Algorithms, Systems and Applications (WASA 2008), Oct. 26-28, 2008.

S.Funke, A.Kesselman and U.Meyer, A Simple Improved Distributed Algorithm for Minimum CDS in Unit Disk Graphs, ACM Transactions on Sensor Networks, 2(3), 444-453, (2006).

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Questions?Questions?

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