Rectilinear Convex Hull with minimum area

TL;DR

This paper presents an improved algorithm for computing the orientation of a plane that minimizes the area of the rectilinear convex hull of a given set of points, reducing the time complexity from quadratic to near-linear.

Contribution

The authors develop a new algorithm that computes the minimum-area rectilinear convex hull orientation in Θ(n log n) time, improving upon previous quadratic-time solutions.

Findings

Algorithm achieves Θ(n log n) time complexity.

Successfully computes orientation for minimum-area rectilinear convex hull.

Improves efficiency over previous quadratic algorithms.

Abstract

Let $P$ be a planar set of $n$ points in general position. We consider the problem of computing an orientation of the plane for which the Rectilinear Convex Hull of $P$ has minimum area. Bae et al. (Computational Geometry: Theory and Applications, Vol. 42, 2009) solved the problem in quadratic time and linear space. We describe an algorithm that reduces this time complexity to $\Theta(n \log n)$.