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

P(0,0)

P(a,b)

Quadtree Partition

p-portals

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.

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 T(a,b) # of possible set of (at most m) crosspoints: # of subproblems:

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

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

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.

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

Computation of Cost for moving Cross-points to Portals

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

Computation of Cost for moving Cross-points to Portals

Patch

Patching Procedure

Thanks, End