arXiv subjects

Roni Sherman

Publications and source records attributed to Roni Sherman.

3 recordsLinked to original sources

arXiv2026

Efficient K-Visibility Query in Polygons

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 $\Theta(n^4)$ cells within which the combinatorial structure of the $k$-visibility polygon remains strictly invariant. By leveraging a combinatorial $\delta$-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.

cs.CG↗

arXiv2026

An Exact Generalized k-Cell Decomposition

This paper introduces an exact $k$-cell decomposition for visibility planning in polygonal environments for agents equipped with $k$-modems, devices that can see through up to $k$ walls. Unlike prior decompositions that may include redundant partition lines, our proposed method ensures that visibility events (appear, disappear, merge, and split) are guaranteed to occur on every line of the decomposition. By eliminating these redundancies, we achieve an $O(n^4)$ complexity , representing a potentially quadratic improvement over the previous best $O(k^2n^4)$ result. This decomposition explicitly identifies the locations of all critical visibility events and extends to polygons with holes. It has practical applications in tasks such as optimal pursuit-evasion under $k$-visibility and agent counting in invisible regions.

cs.CG↗

arXiv2025

Generalized k-Cell Decomposition for Visibility Planning in Polygons

This paper introduces a novel $k$-cell decomposition method for pursuit-evasion problems in polygonal environments, where a searcher is equipped with a $k$-modem: a device capable of seeing through up to $k$ walls. The proposed decomposition ensures that as the searcher moves within a cell, the structure of unseen regions (shadows) remains unchanged, thereby preventing any geometric events between or on invisible regions, that is, preventing the appearance, disappearance, merge, or split of shadow regions. The method extends existing work on $0$- and $2$-visibility by incorporating m-visibility polygons for all even $0 \le m \le k$, constructing partition lines that enable robust environment division. The correctness of the decomposition is proved via three theorems. The decomposition enables reliable path planning for intruder detection in simulated environments and opens new avenues for visibility-based robotic surveillance. The difficulty in constructing the cells of the decomposition consists in computing the $k$-visibility polygon from each vertex and finding the intersection points of the partition lines to create the cells.

cs.CG↗