An Optimal Algorithm for Half-plane Hitting Set

TL;DR

The paper addresses the half-plane hitting set problem for a set $P$ of $n$ points and a set $H$ of $n$ half-planes, seeking a smallest subset of points that hits every half-plane. It introduces a reduction to a circular-point coverage problem on a unit circle, where each half-plane defines a circle point and each $p\in P$ yields a small set of arcs on the circle; an optimal arc cover corresponds to a minimum hitting set. A key insight is that only a subset $\hat{\mathcal{A}}$ of at most $4n$ arcs suffices, and these arcs can be computed in $O(n \log n)$ time, allowing the circular-point coverage subproblem to be solved in $O(n \log n)$ time. Consequently, the overall half-plane hitting set problem is solved in $O(n \log n)$ time, matching the algebraic decision tree lower bound and offering a simpler, near-optimal approach.

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 $Ω(n \log n)$ 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.

Figures (4)

• Figure 1: Illustration of $h_{i-1,i}$, $h_{i,i+1}$, and $\rho$.
• Figure 2: Illustration of the relative positions of the four arcs of $\hat{A}(p)$. The red points are all on the upper half circle $C_u$ while the blue points are all on the lower half circle $C_l$.
• Figure 3: Illustration of the definition of $H^*_l[i_p,j_p]$ when $\ell_p$ contains an edge $e$ of $\mathcal{U}$. $i_p$ and $j_p$ in the figure represent the points $h^*_{i_p}$ and $h^*_{j_p}$, respectively.
• Figure 4: Illustration of the definition of $H^*_l[i_p,j_p]$ when $\ell_p$ does not contain an edge of $\mathcal{U}$. $i_p$ and $j_p$ in the figure represent the points $h^*_{i_p}$ and $h^*_{j_p}$, respectively.

Theorems & Definitions (9)

• Lemma 1
• proof
• Corollary 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 1