Endpoint Covering of Axis-Parallel Segments:Bichromatic and Monochromatic One-Center

2026-09-24

AI Summary

This study addresses the minimum enclosing square single-center optimization problem for axis-parallel line segments based on endpoint 1-covering, encompassing both monochromatic and bichromatic settings. Methodologically, it proposes novel algorithms for the monochromatic case under segment constraints alongside a deterministic algorithm for the bichromatic scenario, while establishing matching theoretical lower bounds. Technically, the approach integrates computational geometry, the algebraic decision tree model, and combinatorial optimization. The primary contributions lie in achieving time complexities of O(n log n) for the monochromatic case and O(m + n log²n) for the bichromatic case. Furthermore, the authors prove that these bounds tightly match the established theoretical lower limits, thereby providing optimal solutions for this covering optimization problem.

Abstract

We study exact one-center optimization for axis-parallel segments using axis-parallel squares under endpoint-based coverage, where a segment is \emph{$1$-covered} if the square contains at least one of its endpoints. In the monochromatic problem, we seek a minimum-side-length square that $1$-covers all $n$ input segments. We obtain $O(n\log n)$-time algorithms for both unrestricted and segment-constrained centers. The unrestricted bound matches the known bound implied by the two-representative color-spanning-square problem, whereas the segment-constrained result is new. We also prove matching $Ω(n\log n)$ lower bounds for both center models in the fixed-order algebraic decision-tree model. In the bichromatic problem, an admissible square must fully contain all $m$ blue segments, minimize the number of red segments with an endpoint in its interior, and, subject to this minimum, maximize its side length within a prescribed bounding box. We give deterministic $O(m+n\log^2 n)$-time and $O(m+mn\log n)$-time algorithms for unrestricted and blue-segment-constrained centers, respectively.

Problem

• one-center optimization
• axis-parallel segments
• endpoint covering
• monochromatic problem
• bichromatic problem

Innovation

• one-center optimization
• axis-parallel segments
• endpoint covering
• bichromatic problem
• computational geometry