Polygon Queries for Convex Hulls of Points

TL;DR

This paper introduces a data structure for efficiently computing convex hulls of point subsets within polygons of fixed orientations, optimizing range searching queries based on polygon shape and orientation.

Contribution

It presents a novel data structure that preprocesses points for fast convex hull queries within oriented polygons, with efficient space, construction, and query times.

Findings

Data structure supports efficient convex hull queries within oriented polygons.

Query time depends on the hull complexity and polygon size.

Perimeter and area of convex hulls can be computed quickly using the data structure.

Abstract

We study the following range searching problem: Preprocess a set $P$ of $n$ points in the plane with respect to a set $\mathcal{O}$ of $k$ orientations % , for a constant, in the plane so that given an $\mathcal{O}$-oriented convex polygon $Q$, the convex hull of $P\cap Q$ can be computed efficiently, where an $\mathcal{O}$-oriented polygon is a polygon whose edges have orientations in $\mathcal{O}$. We present a data structure with $O(nk^3\log^2n)$ space and $O(nk^3\log^2n)$ construction time, and an $O(h+s\log^2 n)$-time query algorithm for any query $\mathcal{O}$-oriented convex $s$-gon $Q$, where $h$ is the complexity of the convex hull. Also, we can compute the perimeter or area of the convex hull of $P\cap Q$ in $O(s\log^2n)$ time using the data structure.