On Line-Separable Weighted Unit-Disk Coverage and Related Problems

TL;DR

This paper studies the line-separable, unit-disk coverage problem where all disks have the same radius and centers lie on one side of a separating line from the point set $P$. It introduces a dynamic-programming approach augmented with hierarchical cuttings to achieve a subquadratic $O(n\sqrt{m}\log^2 m +(n+m)\log(n+m))$ time algorithm for the weighted case, improving the previous $O(n^2\log n)$ bound, and derives corresponding improvements for related halfplane coverage and hitting-set problems via duality. The core ideas combine a left-to-right DP with two key operations, FindMinCost and ResetCost, implemented efficiently using a duality-based reduction to an interval-coverage problem and a hierarchical $(1/r)$-cutting structure. The results unify and extend prior work (e.g., Pedersen and Wang) by showing that the optimal line-separable unit-disk coverage can be solved in subquadratic time, with strong implications for weighted halfplane problems and dual hitting sets. The approach demonstrates how geometric data structures (cuttings) and duality can be orchestrated to yield provably efficient algorithms for challenging coverage and hitting-set problems in the plane.

Abstract

Given a set $P$ of $n$ points and a set $S$ of $n$ weighted disks in the plane, the disk coverage problem is to compute a subset of disks of smallest total weight such that the union of the disks in the subset covers all points of $P$. The problem is NP-hard. In this paper, we consider a line-separable unit-disk version of the problem where all disks have the same radius and their centers are separated from the points of $P$ by a line $\ell$. We present an $O(n^{3/2}\log^2 n)$ time algorithm for the problem. This improves the previously best work of $O(n^2\log n)$ time. Our result leads to an algorithm of $O(n^{{7}/{2}}\log^2 n)$ time for the halfplane coverage problem (i.e., using $n$ weighted halfplanes to cover $n$ points), an improvement over the previous $O(n^4\log n)$ time solution. If all halfplanes are lower ones, our algorithm runs in $O(n^{{3}/{2}}\log^2 n)$ time, while the previous best algorithm takes $O(n^2\log n)$ time. Using duality, the hitting set problems under the same settings can be solved with similar time complexities.

Figures (2)

• Figure 1: Illustrating the line-separable unit-disk case: All points of $P$ are above $\ell$ while the centers of all disks are below $\ell$.
• Figure 2: Illustrating a pseudo-trapezoid.

Theorems & Definitions (11)

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