Fitting a two-joint orthogonal chain to a point set

Authors:
J. M. Díaz-Báñez;M. A. López;M. Mora;C. Seara;I. Ventura
Affiliations:
Departamento de Matemática Aplicada II, Universidad de Sevilla, Spain;Department of Mathematics, University of Denver, 2360 South Gaylord Street, Denver, CO 80208, USA;Departament de Matemítica Aplicada II, Universitat Politècnica de Catalunya, Spain;Departament de Matemítica Aplicada II, Universitat Politècnica de Catalunya, Spain;Departamento de Matemática Aplicada II, Universidad de Sevilla, Spain
Venue:
Computational Geometry: Theory and Applications
Year:
2011

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Abstract

We study the problem of fitting a two-joint orthogonal polygonal chain to a set S of n points in the plane, where the objective function is to minimize the maximum orthogonal distance from S to the chain. We show that this problem can be solved in @Q(n) time if the orientation of the chain is fixed, and in @Q(nlogn) time when the orientation is not a priori known. Moreover, our algorithm can be used to maintain the rectilinear convex hull of S while rotating the coordinate system in O(nlogn) time and O(n) space, improving on a recent result (Bae et al., 2009 [4]). We also consider some variations of the problem in three-dimensions where a polygonal chain is interpreted as a configuration of orthogonal planes. In this case we obtain O(n), O(nlogn), and O(n^2) time algorithms depending on which plane orientations are fixed.