Minimum-perimeter enclosures

Authors:
Joseph S. B. Mitchell;Valentin Polishchuk
Affiliations:
Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794-3600, USA;Helsinki Institute for Information Technology, University of Helsinki and Helsinki University of Technology, Helsinki, P.O. Box 68, FI-00014, Finland
Venue:
Information Processing Letters
Year:
2008

Citing 3
Cited 1

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Note: Sweeping an oval to a vanishing point

Discrete Applied Mathematics

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Abstract

We give the first polynomial-time algorithm for the problem of finding a minimum-perimeter k-gon that encloses a given n-gon. Our algorithm is based on a simple structural result, that an optimal k-gon has at least one ''flush'' edge with the n-gon. This allows one to reduce the problem to computing a shortest k-link path in a simple polygon. As a by-product we observe that the minimum-perimeter ''envelope''-a convex polygon with a specified sequence of interior angles-can also be found in polynomial time. Finally, we introduce the problem of finding optimal convex polygons restricted to lie in the region between two nested convex polygons. We give polynomial-time algorithms for the problems of finding the minimum restricted envelopes.