A linear time algorithm to cover and hit a set of line segments optimally by two axis-parallel squares

TL;DR

This paper presents a linear-time, in-place algorithm for optimally covering and hitting a set of line segments with two axis-parallel squares, also providing a √2 approximation for disk coverage.

Contribution

It introduces the first linear-time algorithms for covering and hitting line segments with two squares, improving efficiency and space usage over previous methods.

Findings

Algorithms run in O(n) time and use O(1) space.

Optimal solutions achieved with only two passes over data.

Provides a √2 approximation for disk coverage problems.

Abstract

This paper discusses the problem of covering and hitting a set of line segments $\cal L$ in ${\mathbb R}^2$ by a pair of axis-parallel squares such that the side length of the larger of the two squares is minimized. We also discuss the restricted version of covering, where each line segment in $\cal L$ is to be covered completely by at least one square. The proposed algorithm for the covering problem reports the optimum result by executing only two passes of reading the input data sequentially. The algorithm proposed for the hitting and restricted covering problems produces optimum result in $O(n)$ time. All the proposed algorithms are in-place, and they use only $O(1)$ extra space. The solution of these problems also give a $\sqrt{2}$ approximation for covering and hitting those line segments $\cal L$ by two congruent disks of minimum radius with same computational complexity.