UMass Lowell Computer Science 91.504 Advanced Algorithms Computational Geometry Prof. Karen Daniels Spring, 2005

1
UMass Lowell Computer Science 91.504 Advanced
AlgorithmsComputational Geometry Prof. Karen
Daniels Spring, 2005

• 2-Center Decision Problem

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2-Center Decision Problem
Hershberger
Hershberger
Aspect






Aspect






2D

Title

A Faster Algorithm for the
Dimensional
Two-Center Decision Problem
ity


Problem/
covering
Source

Information Processing Letters
Task



Theory?
theory

Application

Operations Research

Implementat
Areas

ion?

Input
Planar point set, circle radii

Objec
ts

ADTs
arrangement of circles

Data
Structures


3
Goal

• Can k disks of radius r cover the set of n 2D
  points?
• NP-complete if k is part of input.
• For fixed k, can be solved in polynomial time.

3-center example
Source A Faster Algorithm for the Two-Center
Decision Problem by Hershberger, Information
Processing Letters 47(1993) p. 23-29.
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Goal (continued)

• Generalization Can k disks of radii r1, r2,, rk
  cover the set of n 2D points?
• Still NP-complete for k part of input.
• For fixed k, can be solved in polynomial time.
• Improve O(n2logn) time algorithm for 2-center to
  O(n2).

Generalized 2-center example

Answer no
Source A Faster Algorithm for the Two-Center
Decision Problem by Hershberger, Information
Processing Letters 47(1993) p. 23-29.
5
2-Center Assumptions Approach

• Develop lemma about intersections of fixed-radius
  circles
• W.l.o.g. assume
• Construct arrangement of radius-r circles
  centered at points

r
Source A Faster Algorithm for the Two-Center
Decision Problem by Hershberger, Information
Processing Letters 47(1993) p. 23-29.
6
2-Center Approach (continued)

• Each face of arrangement corresponds to points
  that can be covered by an r-disk whose center
  lies in the face.
• 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)

• For each face of arrangement, see which points
  associated circle can cover.
• Test whether points not covered by associated
  circle can be covered by remaining circle.
• Roughly O(n3) time complexity.
• Use fact that neighboring faces differ by only
  one covering disk to achieve O(n2) time.

Source A Faster Algorithm for the Two-Center
Decision Problem by Hershberger, Information
Processing Letters 47(1993) p. 23-29.