Constrained Two-Line Center Problems

TL;DR

This paper introduces faster algorithms for constrained two-line center problems, significantly improving the computational efficiency for various orientation constraints compared to the previous $O(n^2 ext{log}^2 n)$ algorithm.

Contribution

The paper presents new algorithms with improved time complexities for three variants of the constrained two-line center problem, addressing fixed orientations and fixed angles.

Findings

Algorithms run in $O(n ext{log} n)$, $O(n ext{log}^3 n)$, and $O(n^2 ext{alpha}(n) ext{log} n)$ time.

Significant speedup over the previous $O(n^2 ext{log}^2 n)$ algorithm.

Provides efficient solutions for practical applications with orientation constraints.

Abstract

Given a set P of n points in the plane, the two-line center problem asks to find two lines that minimize the maximum distance from each point in P to its closer one of the two resulting lines. The currently best algorithm for the problem takes $O(n^2\log^2n)$ time by Jaromczyk and Kowaluk in 1995. In this paper, we present faster algorithms for three variants of the two-line center problem in which the orientations of the resulting lines are constrained. Specifically, our algorithms solve the problem in $O(n \log n)$ time when the orientations of both lines are fixed; in $O(n \log^3 n)$ time when the orientation of one line is fixed; and in $O(n^2 \alpha(n) \log n)$ time when the angle between the two lines is fixed, where $\alpha(n)$ denotes the inverse Ackermann function.