1. UMass Lowell Computer Science 91.504 Advanced Algorithms Computational Geometry Prof. Karen Daniels Spring, 2005 2-Center Decision Problem

2. Aspect Hershberger Title A Faster Algorithm for the Two-Center Decision Problem Source Information Processing Letters Application Areas Operations Research Input Objects Planar point set, circle radii Aspect Hershberger Dimensional ity 2D Problem/ Task covering Theory? Implementat ion? theory ADTs & Data Structures arrangement of circles

3. Goal ä Can k disks of radius r cover the set of n 2D points? ä NP-complete if k is part of input. ä For fixed k, can be solved in polynomial time. Source: “A Faster Algorithm for the Two-Center Decision Problem” by Hershberger, Information Processing Letters 47(1993) p. 23-29. + 3-center example

4. Goal (continued) ä Generalization: Can k disks of radii r 1, r 2,…, r k cover the set of n 2D points? ä Still NP-complete for k part of input. ä For fixed k, can be solved in polynomial time. ä Improve O(n 2 logn) time algorithm for 2-center to O(n 2 ). Source: “A Faster Algorithm for the Two-Center Decision Problem” by Hershberger, Information Processing Letters 47(1993) p. 23-29. + Generalized 2- center example Answer = no

5. 2-Center Assumptions & Approach ä Develop lemma about intersections of fixed-radius circles ä W.l.o.g. assume ä Construct arrangement of radius-r circles centered at points Source: “A Faster Algorithm for the Two-Center Decision Problem” by Hershberger, Information Processing Letters 47(1993) p. 23-29. r

6. 2-Center Approach (continued) ä Each face of arrangement corresponds to points that can be covered by an r-disk whose center lies in the face. ä Why? A circle can cover a point iff its center is inside circle of radius r centered on that point. Source: “A Faster Algorithm for the Two-Center Decision Problem” by Hershberger, Information Processing Letters 47(1993) p. 23-29.

7. 2-Center Approach (continued) ä For each face of arrangement, see which points associated circle can cover. ä Test whether points not covered by associated circle can be covered by remaining circle. ä Roughly O(n 3 ) time complexity. ä Use fact that neighboring faces differ by only one covering disk to achieve O(n 2 ) time. Source: “A Faster Algorithm for the Two-Center Decision Problem” by Hershberger, Information Processing Letters 47(1993) p. 23-29.