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arXiv · 2501.02195

An Optimal Algorithm for Half-plane Hitting Set

Gang Liu·Haitao Wang
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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.

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Gang Liu, Haitao Wang. 2025-01-04. An Optimal Algorithm for Half-plane Hitting Set. https://arxiv.org/abs/2501.02195

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