Discrete & Computational Geometry

, Volume 7, Issue 1, pp 45–58

Finding minimum areak-gons

  • David Eppstein
  • Mark Overmars
  • Günter Rote
  • Gerhard Woeginger
Article

DOI: 10.1007/BF02187823

Cite this article as:
Eppstein, D., Overmars, M., Rote, G. et al. Discrete Comput Geom (1992) 7: 45. doi:10.1007/BF02187823

Abstract

Given a setP ofn points in the plane and a numberk, we want to find a polygon
https://static-content.springer.com/image/art%3A10.1007%2FBF02187823/MediaObjects/454_2005_BF02187823_Fig1_HTML.gif
with vertices inP of minimum area that satisfies one of the following properties: (1)
https://static-content.springer.com/image/art%3A10.1007%2FBF02187823/MediaObjects/454_2005_BF02187823_Fig2_HTML.gif
is a convexk-gon, (2)
https://static-content.springer.com/image/art%3A10.1007%2FBF02187823/MediaObjects/454_2005_BF02187823_Fig3_HTML.gif
is an empty convexk-gon, or (3)
https://static-content.springer.com/image/art%3A10.1007%2FBF02187823/MediaObjects/454_2005_BF02187823_Fig4_HTML.gif
is the convex hull of exactlyk points ofP. We give algorithms for solving each of these three problems in timeO(kn3). The space complexity isO(n) fork=4 andO(kn2) fork≥5. The algorithms are based on a dynamic programming approach. We generalize this approach to polygons with minimum perimeter, polygons with maximum perimeter or area, polygons containing the maximum or minimum number of points, polygons with minimum weight (for some weights added to vertices), etc., in similar time bounds.

Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • David Eppstein
    • 1
  • Mark Overmars
    • 2
  • Günter Rote
    • 3
  • Gerhard Woeginger
    • 3
  1. 1.Department of Computer ScienceUniversity of CaliforniaIrvineUSA
  2. 2.Department of Computer ScienceUtrecht UniversityUtrechtThe Netherlands
  3. 3.Institut für MathematikTechnische Universität GrazGrazAustria