Chapter 5 Guillotine Cut

1. Chapter 5 Guillotine Cut Ding-Zhu Du (2) Portals

2. Rectilinear Steiner Tree • Given a set of points in the rectilinear plane, find a minimum length tree interconnecting them. • Those given points are called terminals.

3. Initially Edge length < RSMT

4. Initially L Total moving Length: n = # of terminals 2 2 n x n grid If PTAS exists for grid points, then it exists for general case.

5. (1/3-2/3)-cut Longer edge 1/3 2/3 Shorter edge > 1/3 Longer edge

6. Cut line position L Cut line always passes through the center of a cell. 2 2 n x n grid 1 ( assume)

7. Depth of (1/3-2/3)-cut Note that every two parallel cut lines has distance at least one. Therefore, the smallest rectangle has area 1. After one cut, each resulting rectangle has area Within a factor of 2/3 from the original one. Hence, depth of cuts < (4 log n)/(log (3/2)) = O(log n) since depth 4 (2/3) n > 1

8. (1/3-2/3)-Partition O(log n)

9. Portals m portals divide a cut segment equally.

10. Restriction A rectilinear Steiner tree T is restricted if there exists a (1/3-2/3)-partition such that If a segment of T passes through a cut Line, it passes at a portal.

11. Minimum Restricted RST can be computed in time n2 by dynamic programming O(m) 26 2 Choices of each cut line = O(n ) O(m) 24 # of subproblems = n 2

12. # of subproblem Each subproblems can be described by three facts: 8 O(n ) 1. Position of for edges of a rectangle. 4 O(n ) 2. Position of portals at each edge. O(m) 3. Set of using portals. 2 4. Partition of using portals on the boundary. (In each part of the partition, all portals are connected and every terminal inside of the rectangle is connected to some tree containing a portal. ) O(m) 2

13. Position of portals 2 2 O(n ) O(n )

14. # of partitions

15. N(k) = # of partitions N(0)=1 N(k) = N(k-1) + N(k-2)N(1) + ··· + N(1)N(k-2) + N(k-1) = N(k-1)N(0) + N(k-2)N(1) + ··· + N(0)N(k-1) 2 k f(x) = N(0) + N(1)x + N(2)x + ··· + N(k)x + ··· 1 k 2 xf(x) = f(x) - 1

17. Analysis (idea) • Consider a MRST T. • Choose a (1/3-2/3)-partition. • Modify it into a restricted RST by moving cross-points to portals. • Estimate the total cost of moving cross-points.

18. Choice of (1/3-2/3)-partition 1/3 2/3 Each cut is chosen to minimize # of cross-points. (# of cross-points) x (1/3 longer edge length) < (length of T lying in rectangle).

19. Moving cross-points to portals Cost = (# of cross-points) x ( edge length/(m+1)) < (3/(m+1)) x (length of T lying in rectangle)

20. Moving cost at each level of (1/3-2/3)-Partition <(3/(m+1)) x (length of T ) O(log n) Total cost < O(log n)(3 / (m+1)) x (length of T) O(m) O(1/ε) Choose m = (1/ε) O(log n). Then 2 = n .

21. RSMT has (1+ε)-approximation with running Time n . O(1/ε)

22. Thanks, End