Minimum Dominating Set Approximation in Graphs of Bounded Arboricity
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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
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CatalogCatalogIntroductionPreliminaryVoronoi DivisionEuler’s FormulaDiscussion for Holes
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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
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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
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Minimum Area for Minimum Area for HexagonHexagon
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area(P6) = 0.866
area(E6) = 0.855
area(A6) = 0.886
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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
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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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