Constrained Two-Line Center Problems

TL;DR

Three faster algorithms are presented for three variants of the two-line center problem in which the orientations of the resulting lines are constrained, where $\alpha(n)$ denotes the inverse Ackermann function.

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 $O(n^2 α(n) \log n)$ time when the angle between the two lines is fixed, where $α(n)$ denotes the inverse Ackermann function.

Figures (5)

• Figure 1: Illustration of $\sigma_{[\theta_1,\theta_2]}(S) = \mathrm{conv}(S\cup\{q_1, q_2\})$ (shaded in light gray) when $\theta_1 < \theta_2$.
• Figure 2: Illustrations to the proof of Lemma \ref{['lem:constrained_width2']}.
• Figure 3: Snapshots of the sweeping process: (a) $\overline{P}_j$ dominates $P_i$ and (b) $P_{i'}$ dominates $\overline{P}_{j'}$.
• Figure 4: A snapshot at $\theta$ of the rotational sweeping process with fixed width $\omega$ and pivot $p$.
• Figure 5: (a) Illustration for the mapping $\tau_\beta(\cdot; \ell)$ on line $\ell$ and (b) its dual representation.

Theorems & Definitions (26)

• Lemma 1
• Theorem 1
• Lemma 2: Agarwal and Sharir as-odmwpps-91
• Lemma 3
• Lemma 4: Chan c-fdapw-03
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Theorem 2