1. A simple linear algorithm for computing rectilinear 3-centers Michael Hoffmann Computational Geometry 31(2005)150-165 Reporter: Lincong Fang Nov 3,2005

2. Outline About the author Problem statement Previous works One-dimensional 3-centers Planar rectilinear 3-centers Experimental results Conclusions

3. About the author Institute for Theoretical Computer Science, ETH Zürich, Switzerland. Senior researcher. Scientific interests: algorithms and data structures, in particular from computational geometry, software design and combinatorial game theory.

4. Problem statement K-center problem: given demand points, locate k facilities, such that for any point the nearest facility is as close as possible. Rectilinear k-center problem: distance between points is measured according to the rectilinear ( l 1 or l ∞ ) metric.

5. Problem statement k: positive integer. : congruent closed axis parallel squares of side length 2. k=3 k-radius:

6. Previous works Z.Drezner, On the rectangular p-center problem, gave a linear time algorithm for k=2. M.Sharir, E.Welzl, Rectilinear and polygonal p-piercing and p-center problems, gave a linear time algorithm for k≤3. M.Blum, R.W.Floyd, V.Pratt, R.L.Rivest, R.E.Tarjan, Time bounds for selection.

7. One-dimensional 3-centers Left endpoint of I l is the smallest value of P. Right endpoint of I r is the largest value of P.

8. One-dimensional 3-centers The one-dimensional 3-center decision problem can be solved in linear time. Given

9. One-dimensional 3-centers

10. Knowing the smallest feasible radius, the optimal radius can be solved in linear time. :the smallest feasible radius from :the predecessor of

11. One-dimensional 3-centers Algorithm 1 Input:of n real numbers. Output: 3-covering for. While Compute and

12. One-dimensional 3-centers Test feasibility of If is feasible If is infeasible Solve the problem brute-force.

13. One-dimensional 3-centers The one-dimensional 3-center problem can be solved in linear time. Complexity:

14. Planar rectilinear 3-centers

15. Type 1:Type 2:

16. Planar rectilinear 3-centers

17. Type 2 can be computed in linear time: Then it can be compute just like one-dimension.

18. Planar rectilinear 3-centers Type 1

19. Planar rectilinear 3-centers

21. Algorithm 2

22. Planar rectilinear 3-centers

26. Type 1 can be computed in linear time.

27. Experimental results Points from the unit square Points from three clusters

28. Experimental results Points from the unit square Points from three clusters

29. Conclusions A new linear time algorithm for the rectilinear 3-center problem. Two heuristics to improve its performance in practice. The implementation appeared as part of the CGAL since Release2.1(January 2000).

30. Thank you!