Bichromatic Line-Centers for Point Pairs

2026-09-20

AI Summary

研究了平面上点对的双色线中心问题，通过分配点到红蓝集合来最小化包含各集合的最小宽度条带，提供了针对不同方向约束的有效算法。

Abstract

We study the \emph{bichromatic line-center problem} for $n$ pairs of points in the plane. A feasible solution assigns one point from each pair to the red set $R$ and the other to the blue set $B$. The goal is to minimize $\max\{w^\circ(R),\,w^\circ(B)\}$, where $w^\circ(X)$ denotes the minimum width of a strip enclosing $X$; the midlines of the corresponding optimal strips define the line-centers of $R$ and $B$. We consider several variants induced by orientational constraints on line-centers and provide efficient algorithms for each. For one line-center, which consists of computing a minimum-width strip that contains at least one point from each pair, we give an $O(n^2)$-time algorithm. For two line-centers, we obtain an $O(n)$-time algorithm when both are horizontal, and $Θ(n\log n)$-time algorithms when the two centers are parallel or when both orientations are prescribed. When exactly one orientation is prescribed, we give an $O(n^2)$-time algorithm. Finally, for the unrestricted case, we present an $O(n^3\log n)$-time algorithm.

Problem

• bichromatic line-center problem
• minimum width strip
• point pairs

Innovation

• bichromatic line-center problem
• efficient algorithms
• orientational constraints
• minimum-width strip