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Algorithmica

, Volume 33, Issue 4, pp 411–435 | Cite as

Some Aperture-Angle Optimization Problems

  •  Bose
  •  Hurtado-Diaz
  •  Omaña-Pulido
  •  Snoeyink
  •  Toussaint

Abstract. Let P and Q be two disjoint convex polygons in the plane with m and n vertices, respectively. Given a point x in P , the aperture angle of x with respect to Q is defined as the angle of the cone that: (1) contains Q , (2) has apex at x and (3) has its two rays emanating from x tangent to Q . We present algorithms with complexities O(n log m) , O(n + n log (m/n)) and O(n + m) for computing the maximum aperture angle with respect to Q when x is allowed to vary in P . To compute the minimum aperture angle we modify the latter algorithm obtaining an O(n + m) algorithm. Finally, we establish an Ω(n + n log (m/n)) time lower bound for the maximization problem and an Ω(m + n) bound for the minimization problem thereby proving the optimality of our algorithms.

Key words. Aperture angle, Convexity, Unimodality, Discrete optimization, Algorithms, Complexity, Computational geometry, Robotics, Visibility. 

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Copyright information

© Springer-Verlag New York Inc. 2002

Authors and Affiliations

  •  Bose
    • 1
  •  Hurtado-Diaz
    • 2
  •  Omaña-Pulido
    • 3
  •  Snoeyink
    • 4
  •  Toussaint
    • 5
  1. 1.School of Computer Science, Herberg Room 5302, Carleton University, Ottawa, Ontario, Canada K1S 5B6.CA
  2. 2.Departamento de Matemáticas 2, Universitat Politècnica de Catalunya, Barcelona, Spain.ES
  3. 3.Departmento de Matemáticas, Universidad Autónoma de Mexico, Mexico City, Mexico.MX
  4. 4.Department of Computer Science, Campus Box 3175, Sitterson Hall, University of North Carolina—Chapel Hill, Chapel Hill, NC 27599-3175, USA.US
  5. 5.School of Computer Science, McGill University, 3480 University Street, Montreal, Quebec, Canada H3A 2A7. godfried@opus.cs.mcgill.ca.CA

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