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

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 n x n grid 22 n = # of terminals L Total moving Length: 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 Longer edge > 1/3

6. Cut line position n x n grid 22 L Cut line always passes through the center of a cell. 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 (2/3) n > 1 depth4

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 n 2 by dynamic programming O(m) Choices of each cut line = O(n ) 2 # of subproblems = n 2 O(m) 24 26

12. # of subproblem Each subproblems can be described by three facts: 1. Position of for edges of a rectangle. 2. Position of portals at each edge. 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(n ) 8 4 2 O(m) 3. Set of using portals. 2 O(m)

13. Position of portals O(n ) 2 2

14. # of partitions

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

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 Each cut is chosen to minimize # of cross-points. (# of cross-points) x (1/3 longer edge length) < (length of T lying in rectangle). 1/32/3

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) Choose m = (1/ε) O(log n). Then 2 = n. O(m) O(1/ε)

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

22. Thanks, End