BIT Numerical Mathematics

, Volume 25, Issue 1, pp 76–90 | Cite as

The power of geometric duality

  • Bernard Chazelle
  • Leo J. Guibas
  • D. T. Lee
Part I Computer Science Ordinary Papers

Abstract

This paper uses a new formulation of the notion of duality that allows the unified treatment of a number of geometric problems. In particular, we are able to apply our approach to solve two long-standing problems of computational geometry: one is to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane; the other is to produce an optimal algorithm for the half-plane range query problem. This problem is to preprocessn points in the plane, so that given a test half-plane, one can efficiently determine all points lying in the half-plane. We describe an optimalO(k + logn) time algorithm for answering such queries, wherek is the number of points to be reported. The algorithm requiresO(n) space andO(n logn) preprocessing time. Both of these results represent significant improvements over the best methods previously known. In addition, we give a number of new combinatorial results related to the computation of line arrangements.

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Copyright information

© BIT Foundations 1985

Authors and Affiliations

  • Bernard Chazelle
    • 1
  • Leo J. Guibas
    • 2
  • D. T. Lee
    • 3
  1. 1.Department of Computer ScienceBrown UniversityProvidenceUSA
  2. 2.Computer Science Laboratory, Xerox PARCPalo Alto Research CenterPalo AltoUSA
  3. 3.Northwestern UniversityEvanstonUSA

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