多边形中的高效k-可见性查询
Efficient K-Visibility Query in Polygons
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中文总结 AI 辅助
本文针对现有k-可见性查询空间预处理方法未表征最小拓扑边界集的问题，提出含主顶点地平线和次互临界铰链线的单元分解框架，实现O(n⁴)存储与最优查询时间，且可扩展至带孔洞的多边形。
中文摘要 AI 辅助
本文研究k-可见性，其中视线可穿透最多k个障碍物。虽然从单个查询点计算k-可见性多边形已得到充分研究，但现有空间预处理方法依赖于经过所有顶点对的完整O(n²)直线排列，未表征最小拓扑边界集。我们提出一种改进的单元分解框架，该框架分离了控制k-可见性的精确几何事件：主顶点地平线和次互临界铰链线。我们证明，这组最小划分线可生成Θ(n⁴)个单元的空间分解，在这些单元内k-可见性多边形的组合结构保持严格不变。通过利用跨单元边界的组合δ压缩方案，我们实现了O(n⁴)的总体存储复杂度，同时支持最优的O(log n + m)查询时间，以重建大小为m的显式k-可见性多边形。该框架自然可扩展至包含孔洞的多边形。
英文摘要
This paper investigates $k$-visibility, where a line of sight can penetrate up to $k$ obstacles. While computing the $k$-visibility polygon from a single query point is well-studied, existing spatial preprocessing approaches rely on full $O(n^2)$ line arrangements through all vertex pairs without characterizing the minimal set of topological boundaries. We present a refined cell decomposition framework that isolates the exact geometric events governing $k$-visibility: primary vertex horizon lines and secondary mutually critical hinge lines. We prove that this minimal set of partition lines yields a spatial decomposition of $Θ(n^4)$ cells within which the combinatorial structure of the $k$-visibility polygon remains strictly invariant. By leveraging a combinatorial $δ$-compression scheme across cell boundaries, we achieve an overall storage complexity of $\mathcal{O}(n^4)$ while supporting optimal $\mathcal{O}(\log n + m)$ query time to reconstruct explicit $k$-visibility polygons of size $m$. Our framework naturally extends to polygons containing holes.
发表机构
- Toronto Metropolitan University（多伦多都会大学）
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