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 at most m cross-points.
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:
# of Subproblems # of nonempty cells: # of possible set of used portals on boundary: # of connected patterns
Approximation q q Compute T(0, 0), T(1, 1), …, T(2 -1, 2 -1). 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 crosspoints and reducing cross-points.
Lemma There is a RSMT T # of cross-points = length(T) Proof. Hannan Theorem Hannan grid
Computation of Cost for moving Cross-points to Portals
Moving of a cross-point 2 1 2 0 2 1 2 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