1. Maximal Independent Set and Connected Dominating Set Xiaofeng Gao Research Group on Mobile Computing and Wireless Networking Univ. of Texas at Dallas

2. Catalog Introduction Preliminary Voronoi Division Euler’s Formula Discussion for Holes Xiaofeng Gao, Univ. of Texas at Dallas26/6/2008

3. INTRODUCTION Section 1 Xiaofeng Gao, Univ. of Texas at Dallas36/6/2008

4. Connected Dominating Set A virtual backbone for wireless ad hoc network Garey 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) Xiaofeng Gao, Univ. of Texas at Dallas46/6/2008

5. Communication Model Unit 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 r v ◦ Each (u, v) in E means v lies within u’s range. Say, dist(u,v) <= r u Xiaofeng Gao, Univ. of Texas at Dallas56/6/2008

6. Common Techniques Find a Dominating Set D ◦ Partition ◦ Final Maximal Independent Set (MIS) Connect D as a CDS ◦ Steiner Tree ◦ Spanning Tree Greedy Approach ◦ Spider (Guha & Khuller) Xiaofeng Gao, Univ. of Texas at Dallas66/6/2008

7. Approximation Ratio Given graph G mcds(G): the size of optimal CDS solution for G mis(G): the size of selected MIS set for G connect(G): the size of disks selected to connect the MIS into CDS. Xiaofeng Gao, Univ. of Texas at Dallas76/6/2008

8. Approximation Ratio (2) Formula Compare the relationships between mis(G) and mcds(G) is crucial to reduce the approximation ratio Xiaofeng Gao, Univ. of Texas at Dallas86/6/2008

9. Milestones Wan et.al. (2002) ◦ mis(G) <= 4 ∙ mcds(G) + 1 Wu et.al. (2006) ◦ mis(G) <= 3.8 ∙ mcds(G) +1.2 Funke 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 Xiaofeng Gao, Univ. of Texas at Dallas96/6/2008

10. Our Accomplishment Result ◦ 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 graph Strategy ◦ Voronoi Division ◦ Euler’s Formula Xiaofeng Gao, Univ. of Texas at Dallas106/6/2008

11. PRELIMINARIES Section 2 Xiaofeng Gao, Univ. of Texas at Dallas116/6/2008

12. Induced Graph OPT: 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 d 1, d 2 in S, dist(d 1,d 2 )>1 Make Induced Graph G’ ◦ Change radius of disks in OPT into 1.5 ◦ Change radius of disks in V\OPT into 0.5 Xiaofeng Gao, Univ. of Texas at Dallas126/6/2008

13. Properties for Induced Graph The 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. Xiaofeng Gao, Univ. of Texas at Dallas136/6/2008

14. Example – Original Graph Xiaofeng Gao, Univ. of Texas at Dallas146/6/2008

15. Example – Induced Graph Xiaofeng Gao, Univ. of Texas at Dallas156/6/2008

16. Rough Bound Xiaofeng Gao, Univ. of Texas at Dallas166/6/2008

17. Calculation Xiaofeng Gao, Univ. of Texas at Dallas176/6/2008

18. VORONOI DIVISION Section 3 Xiaofeng Gao, Univ. of Texas at Dallas186/6/2008

19. Definition Let S be a set of n nodes in Eucliean space. For each node pi in S, the Voronoi Cell V(p i ) of p i is the set of points that are closer to p i than to any other nodes of S. Xiaofeng Gao, Univ. of Texas at Dallas196/6/2008

20. Example |OPT| = 2; |MIS| = 7 Xiaofeng Gao, Univ. of Texas at Dallas206/6/2008

21. Analysis Each 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. Xiaofeng Gao, Univ. of Texas at Dallas216/6/2008

22. Minimum Area for Triangle Non-Boundary ◦ area(P 3 )=1.299 Boundary ◦ area(E 3 )=1.178 Xiaofeng Gao, Univ. of Texas at Dallas226/6/2008

23. Minimum Area for Quadrangle Xiaofeng Gao, Univ. of Texas at Dallas23 area(P 4 ) = 1 area(A 4 ) > 1.136 area(E 4 ) = 0.972 6/6/2008

24. Minimum Area for Pentagon Xiaofeng Gao, Univ. of Texas at Dallas24 area(P 5 ) = 0.908 area(A 5 ) > 0.950 area(P 5 ) = 0.908 6/6/2008

25. Minimum Area for Hexagon Xiaofeng Gao, Univ. of Texas at Dallas25 area(P 6 ) = 0.866area(E 6 ) = 0.855 area(A 6 ) = 0.886 6/6/2008

26. Heptagon and Others area(P k )> area(P 6 ) (k>=7) Xiaofeng Gao, Univ. of Texas at Dallas266/6/2008

27. Minimum Area Area(A 7 ) = 0.853 Xiaofeng Gao, Univ. of Texas at Dallas276/6/2008

28. Modified Upper Bound Xiaofeng Gao, Univ. of Texas at Dallas286/6/2008

29. EULER’S FORMULA Section 4 Xiaofeng Gao, Univ. of Texas at Dallas296/6/2008

30. Symbols s i : the minimum area of the non-boundary i-polygon cell; s i ': the minimum area of the boundary i- polygon cell. We have ◦ s 3 >= s 4 >= s 5 >= s 6 <= s 7 <= s 8 <= … ◦ s 3 ' >= s 4 ' >= s 5 ' >= s 6 ' >= s 7 ' <= s 8 ' <=... Set s i = s 6 (i >=7) and s i ' = s 7 ' (i >= 8) Xiaofeng Gao, Univ. of Texas at Dallas306/6/2008

31. 3-Regularization Make the degree of every vertex in Voronoi Division be exactly 3. Xiaofeng Gao, Univ. of Texas at Dallas316/6/2008

32. Euler’s Formula : the outer boundary of MCDS f i : no. of non-boundary Voronoi Cells with i edges; f i ': no. of boundary VC with i edges. m: no. of edges n: no. of vertices By Euler’s Formula: Xiaofeng Gao, Univ. of Texas at Dallas326/6/2008

33. Equations G' 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) 6/6/2008Xiaofeng Gao, Univ. of Texas at Dallas33

34. Calculation Combining (1), (2), and (3), we have Together with Euler’s Formula, remove n, we have 6/6/2008Xiaofeng Gao, Univ. of Texas at Dallas34

35. Calculation – cont. Since all Voronoi cells are contained in the area constructed by MCDS, Combining two formula together, 6/6/2008Xiaofeng Gao, Univ. of Texas at Dallas35

36. Calculation – cont. Since we have Finally, we get that 6/6/2008Xiaofeng Gao, Univ. of Texas at Dallas36

37. DISCUSSION WITH HOLES Section 5 6/6/2008Xiaofeng Gao, Univ. of Texas at Dallas37

38. Modification k: no. of holes in G‘ : no. of edges in all holes. Then we have 6/6/2008Xiaofeng Gao, Univ. of Texas at Dallas38

39. Modification – cont. 6/6/2008Xiaofeng Gao, Univ. of Texas at Dallas39 We have Since

40. Guess 6/6/2008Xiaofeng Gao, Univ. of Texas at Dallas40

41. Example 6/6/2008Xiaofeng Gao, Univ. of Texas at Dallas41

42. Reference Xianyue Li, Xiaofeng Gao, Weili Wu, A Better Theoretical Bound to Approximate Connected Dominating Set in Unit Disk Graph, Submitted to The 3rd International Conf erence on Wireless Algorithms, Systems and Applications (W ASA 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). 6/6/2008Xiaofeng Gao, Univ. of Texas at Dallas42

43. Questions? 6/6/2008Xiaofeng Gao, Univ. of Texas at Dallas43