We develop algorithms for the approximation of convex polygons with n vertices by convex polygons with fewer (k) vertices. The approximating polygons either contain or are contained in the approximated ones. The distance function between convex bodies which we use to measure the quality of the approximation is the Hausdorff metric. We consider two types of problems: min-#, where the goal is to minimize the number of vertices of the output polygon, for a given distance ε, and min-ε, where the goal is to minimize the error, for a given maximum number of vertices. For min-# problems, our algorithms are guaranteed to be within one vertex of the optimal, and run in O(nlogn) and O(n) time, for inner and outer approximations, respectively. For min-ε problems, the error achieved is within an arbitrary factor α>1 from the best possible one, and our inner and outer approximation algorithms run in O(f(α,P)ṡnlogn) and O(f(α,P)ṡn) time, respectively. Where the factor f(α,P) has reciprocal logarithmic growth as α decreases to 1, this factor depends on the shape of the approximated polygon P.
Bibliographical noteFunding Information:
* Corresponding author. E-mail addresses: email@example.com (M.A. Lopez), firstname.lastname@example.org (S. Reisner). 1 Has been supported in part by the NSF, under Grant DMS-0107628. 2 Has been supported in part by the NSF, under Grant DMS-0107628 and by NATO Collaborative Linkage Grant PST.CLG 979701.
- Approximation algorithms
- Convex polygons
- Hausdorff distance
ASJC Scopus subject areas
- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics