Chapter 4 Partition (3) Double Partition

1. Chapter 4 Partition(3) Double Partition Ding-Zhu Du

2. It is a natural idea. Partition a big thing (hard to deal) into small ones (easy to deal).

3. Partition is also an important technique in design of approximation algorithms. Example: To find a dominating set, we may find a dominating set in each small area.

5. Weighted Dominating Set in unit disk graphs Given a unit disk graph G=(D,E) with node weight c:D→R, find a dominating set with minimum total weight. + <1

6. Backgroud • 72-approximation (Ambuhl, et al. 2006).

7. 72-approximation (Ambuhl, et al. 2006). • (6+ε)-approximation (Gao, et al. 2008). Double Partition

8. General Case Partition into big cells

9. No node lies on a cut-line. Partition

10. 1 Dominating area Problem A(i,j)

11. Lemma Problem A(i,j) has 2-approximation. Theorem

12. 16 ? 14 !

14. 2-approximation for A(I,j) Case 1 Minimum weight of node in Dij

15. 2-approximation for A(I,j) Case 2. nodes in N(Dij) dominate nodes in Dij ? AL AM AR CR CL BR BM BL

16. A problem on strip: outside disks cover inside points L p2 p1 pi Ti(D,D’) : minimum weight set with D, D’, dominating p1, …, pisuch that (1) D (lowest intersection point on L) among disks above the strip (2) D’(highest intersection point on L) among disks below the strip

17. Dynamic Programming

18. Dynamic Programming

19. D1 (lowest intersection point on L’) among disks above the strip, in Ti(D,D’) L’ p2 p1 pi-1 D2 (highest intersection point on L’) among disks below the strip, in Ti(D,D’)

20. D1 pi-1 pi pj D

21. 2-approximation for A(I,j) Case 2. nodes in N(Dij) dominate nodes in Dij ?

22. LemmaIf p is dominated by u in LM area, then every point in is dominated by u. LM

23. p p v v u u

24. LemmaIf p and p’ can be dominated by nodes in BM but not nodes in CL and CR, then every node in can be dominated in nodes in A and B. A CL CR p p’ B

25. Consider OPT is the leftmost one for p dominated by a node in BM, but not any node in CL and CR is the rightmost one for p’ dominated by a node in LM, but not any node in CL and CR p p’ contains all nodes dominated by nodes in BM but not nodes in CL and CR.

26. Consider OPT is the leftmost one for p dominated by a node in UM, but not any node in CL and CR is the rightmost one for p’ dominated by a node in UM, but not any node in CL and CR q q’ contains all nodes dominated by nodes in UM but not nodes in CL and CR.

27. Consider OPT U R L R

28. How do we find p, p’, q, q’? Try all possibilities. How many possibilities?

29. Idea: Combine cells into a strip Each strip contains m cells.

30. 6-approximation for a special case: For every subset C of cells, • every cell e in C is in case 1; • every cell e not in C is in case 2.

31. 6! 1 2 3 4 5 6

32. General Case Partition into big cells

33. (6+ε)-approximation in general case Shafting to minimize # of disks on boundaries

34. (9.875+ε)-approximation for minimum weight connected dominating set in unit disk graph. Connecting a dominating set into a cds needs to add at most 3.875 optnodes. (Zou et al, 2008) (improved 17opt)

35. Thanks, end.