1. Chapter 4 Partition (1) Shifting Ding-Zhu Du

2. Disk Covering Given a set of n points in the Euclidean plane, find the minimum number of unit disks to cover the n given points.

3. (x,x) Partition P(x) a

4. Construct Minimum Unit Disk Cover in Each Cell 1/√2 Each square with edge length 1/√2 can be covered by a unit disk. Hence, each cell can be covered By at most disks. Suppose a cell contains n i points. Then there are n i (n i -1) possible positions for each disk. Minimum cover can be computed In time n i O(a ) 2

5. Solution S(x) associated with P(x) For each cell, construct minimum cover. S(x) is the union of those minimum covers. Suppose n points are distributed into k cells containing n 1, …, n k points, respectively. Then computing S(x) takes time n 1 + n 2 + ··· + n k < n O(a ) 2222

6. Approximation Algorithm For x=0, -2, …, -(a-2), compute S(x). Choose minimum one from S(0), S(-2), …, S(-a+2).

7. Analysis Consider a minimum cover. Modify it to satisfy the restriction, i.e., a union of disk covers each for a cell. To do such a modification, we need to add some disks and estimate how many added disks.

8. Added Disks Count twice Count four times 2

9. 2 Shifting

10. Estimate # of added disks Shifting

11. Estimate # of added disks Vertical strips Each disk appears once.

12. Estimate # of added disks Horizontal strips Each disk appears once.

13. Estimate # of added disks # of added disks for P(0) + # of added disks for P(-2) + ··· + # of added disks for P(-a+2) < 3 opt where opt is # of disk in a minimum cover. There is a x such that # of added disks for P(x) < (6/a) opt.

14. Performance Ratio P.R. < 1 + 6/a < 1 + ε when we choose a = 6 ⌠1/ε. Running time is n. O(1/ε ) 2

15. Unit disk graph < 1

16. Dominating set in unit disk graph Given a unit disk graph, find a dominating set with the minimum cardinality. Theorem This problem has PTAS.

17. Connected Dominating Set in Unit Disk Graph Given a unit disk graph G, find a minimum connected dominating set in G. Theorem There is a PTAS for connected dominating set in unit disk graph.

19. Boundary area central area h h+1

20. Why overlapping? cds for G cds for each connected component 1

21. 1. In each cell, construct MCDS for each connected component in the inner area. Construct PTAS 2. Connect those minimum connected dominating sets with a part of 8-approximation lying in boundary area. For each partition P(a,a), construct C(a) as follows: Choose smallest C(a) for a = 0, h+1, 2(h+1), ….

22. Existence of 8-approximation 1.There exists (1+ε)-approximation for minimum dominating set in unit disk graph. 2. We can reduce one connected component with two nodes. Therefore, there exists 3(1+ε)-approximation for mcds.

23. 8-approximation 1.A maximal independent set has size at most 4 mcds +1. 2. There exists a maximal independent set, connecting it into cds need at most 4mcds nodes.

24. MCDS (Time) 1.In a square of edge length, any node can dominate every bode in the square. Therefore, minimum dominating set has size at most. a

25. MCDS (Time) 2. The total size of MCDSs for connected components in an inner square area is at most. a

27. MCDS (Size) Modify a mcds for G into MCDSs in each cell. mcds(G): mcds for G mcds cell (inner): MCDS in a cell for connected components in inner area

28. Connect & Charge charge

29. Multiple Charge charge How many possible charges for each node? How many components can each node be adjacent to?

30. 1. How many independent points can be packed by a disk with radius 1? 1 >1 5!

31. Each node can be charged at most 10 times!!!

32. Shifting 3 a/(2(h+1)) = integer Time=n O(a ) 2 h=2

33. Weighted Dominating Set Given a unit disk graph with vertex weight, find a dominating set with minimum total weight. Can the partition technique be used for the weighted dominating set problem?

34. Dominating Set in Intersection Disk Graph An intersection disk graph is given by a set of points (vertices) in the Euclidean plane, each associated with a disk and an edge exists between two points iff two disks associated with them intersects. Can the partition technique be used for dominating set in intersection disk graph?

35. Thanks, End