An Exact Generalized k-Cell Decomposition

2026-07-05

AI Summary

This work addresses the visibility planning problem for k-modem agents in polygonal environments with holes, where existing decomposition approaches suffer from redundant subdivisions and fail to precisely capture all critical visibility events—such as appearances, disappearances, merges, and splits. To overcome these limitations, the paper introduces a generalized k-cell exact decomposition method that guarantees every partitioning line corresponds to a genuine visibility event, thereby eliminating redundancy entirely. The proposed technique reduces the time complexity from O(k²n⁴) to O(n⁴), achieving a potential quadratic speedup. Furthermore, it enables practical applications including optimal pursuit-evasion strategies and counting agents within invisible regions.

Abstract

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.

Problem

• k-cell decomposition
• visibility planning
• k-modem
• polygonal environments
• visibility events

Innovation

• k-cell decomposition
• visibility planning
• k-modem
• computational geometry
• pursuit-evasion