Chapter 5 Guillotine Cut (3) Quadtree Partition

1. Chapter 5 Guillotine Cut(3) Quadtree Partition Ding-Zhu Du

2. P(0,0)

3. P(a,b)

4. Quadtree Partition

5. p-portals

6. Restriction A Steiner tree T is restricted if there exists a Quadtree partition such that (a) every edge crosses a cut line at a portal, and (b) at every cut segment, there are at most m cross-points.

7. T(a,b) For any P(a,b), a minimum tree T(a,b) satisfying restriction provided by P(a,b) can be computed by dynamic programming in time # of possible set of (at most m) crosspoints: # of subproblems:

8. # of Subproblems # of nonempty cells: # of possible set of used portals on boundary: # of connected patterns

9. Approximation q q Compute T(0,0), T(1,1), …, T(2 -1, 2 -1). Choose the shortest one from above trees.

10. Analysis (idea) • Consider a MRST T. • Choose a quadtree partition P(a,a). • Modify it into a restricted RST by moving cross-points to portals and reduce # of cross-points to ≤ m. • Estimate the total cost of moving cross-points and reducing cross-points.

11. Lemma There is a RSMT T # of cross-points = length(T) Proof. Hannan Theorem Hannan grid

13. Computation of Cost for moving Cross-points to Portals

14. Moving of a cross-point 2 2 2 1 2 1 0 Once at level 0 Once at level 1 Twice at level 2 4 times at level 3

15. Computation of Cost for moving Cross-points to Portals

16. Patch

17. Patching Procedure

22. Thanks, End