An Optimal Algorithm for Half-plane Hitting Set

TL;DR

This paper introduces an optimal $O(n \,\log n)$ algorithm for the half-plane hitting set problem, improving previous solutions and matching the theoretical lower bound, while also emphasizing simplicity.

Contribution

The paper presents a simple, optimal $O(n \,\log n)$ algorithm for computing the smallest point set hitting all half-planes, matching the lower bound and improving upon prior methods.

Findings

Achieves optimal $O(n \,\log n)$ time complexity.

Matches the theoretical lower bound for the problem.

Provides a simpler algorithm compared to previous solutions.

Abstract

Given a set $ P $ of $n$ points and a set $ H $ of $n$ half-planes in the plane, we consider the problem of computing a smallest subset of points such that each half-plane contains at least one point of the subset. The previously best algorithm solves the problem in $O(n^3 \log n)$ time. It is also known that $\Omega(n \log n)$ is a lower bound for the problem under the algebraic decision tree model. In this paper, we present an $O(n \log n)$ time algorithm, which matches the lower bound and thus is optimal. Another virtue of the algorithm is that it is relatively simple.