Chapter 4. Partition

1. Chapter 4. Partition Ding-Zhu Du (2) Multi-layer Partition

2. Intersection Disk Graph Consider n points in the Euclidean plane, each is associated with a disk. An edge exists between two points if and only if their associated disks have nonempty intersection.

3. Maximum Independent Set in Intersection Disk Graph Given a intersection disk graph D, find a maximum Independent set opt(D).

4. Multi-layer Suppose the largest disk has diameter 1-ε. Let dmin be The diameter of smallest disk. Fix an integer k > 0. Let Put all disks into m+1 layers. For 0 < j < m, layer j consists of all disks with diameter di,

5. Partition P(0,0) in layer j (0,0)

6. Partition P(0) in layer j and layer j+1

7. Partition P(a,b) in layer j

8. Layer j Layer j+1

9. A disk hits a cut line. At each layer, a disk can hit at most one among Parallel lines apart each other with distance .

10. D(a,b) In partition P(a,b), delete all disks each hitting a cut line in the same layer. The remaining disks form a collection D(a,b). Maximum Independent set opt(D(a,b)) can be computed in time Why use it? by dynamic programming.

11. Dynamic Programming j-cell is a cell in layer j. For any j-cell S and a set I of independent disks in layers < j, intersecting S, Table(S,I) = maximum independent set of disks layers > j, contained in S, and disjoint from I. opt(D(a, b)) = US Table(S, Ǿ) where S is over all cells in layer 0.

12. Recursive Relation For j-cell S and I,

13. # of Table(S,I) # of S = too large How do we overcome this difficulty? Relevant cell: A j-cell is relevant if it contains a disk in layer j.

14. Dynamic Programming j-cell is a cell in layer j. For any relevant j-cell S and a set I of independent disks in layers < j, intersecting S, Table(S,I) = maximum independent set of disks layers > j, contained in S, and disjoint from I. opt(D(a, b)) = USTable(S, Ǿ) where S is over all maximal relevant cells.

15. Children of a relevant cell S S’’ S’

16. Maximal relevant cell A relevant cell is maximal if it is not contained by Another relevant cell.

17. Recursive Relation For j-cell S,

18. # of Table(S,I) # of relevant S = n. # of I = # of Table(S,I) =

19. # of I S # of I’s =

20. Computation Time of Recursion # of S’ = # of J = Time = Running Time of dynamic programming

21. # of J S

22. (1+ε)-Approximation Choose k = ?. Compute opt(D(0,0)), opt(D(1,1)), …, opt(D(k-1,k-1). Choose maximum one among them.

23. Analysis • Consider an optimal solution D*. • For each partition P(a,b), let H(a,b) be the collection of all disks hitting cut line in the same layer. • Estimate |H(0,0)|+|H(1,1)|+···+|H(k-1,k-1)|.

24. |H(0,0)|+|H(1,1)|+···+|H(k-1,k-1)| Each disk appears in at most two terms in this sum. There exists i such that |H(2i,2i)| < 2|D*|/k.

25. Performance ratio Opt/approx =1/(1-2/k) = 1 + 2/(k-4) Choose We obtain a (1+ε)-approximation With time

26. Thanks, End