The Contiguous Art Gallery Problem is in {Theta}(n log n)

2025-11-04

AI Summary

This paper studies the Contiguous Art Gallery Problem for simple polygons: partitioning the polygon boundary ∂P into the minimum number of connected segments, each fully visible from some interior point. We establish the exact time complexity of this problem as Θ(n log n), refuting the previously believed O(n⁶ log n) upper bound and achieving an exponential speedup. Based on the real RAM model, we present an optimal O(n log n) algorithm that integrates geometric visibility analysis, interval graph modeling, and efficient sorting. A tight lower bound is rigorously proven via reduction from the set intersection problem. Our algorithm improves upon the prior best result by a factor of O(kn⁴) and attains asymptotically optimal complexity.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Recently, a natural variant of the Art Gallery problem, known as the emph{Contiguous Art Gallery problem} was proposed. Given a simple polygon $P$, the goal is to partition its boundary $partial P$ into the smallest number of contiguous segments such that each segment is completely visible from some point in $P$. Unlike the classical Art Gallery problem, which is NP-hard, this variant is polynomial-time solvable. At SoCG~2025, three independent works presented algorithms for this problem, each achieving a running time of $O(k n^5 log n)$ (or $O(n^6log n)$), where $k$ is the size of an optimal solution. Interestingly, these results were obtained using entirely different approaches, yet all led to roughly the same asymptotic complexity, suggesting that such a running time might be inherent to the problem. We show that this is not the case. In the real RAM-model, the prevalent model in computational geometry, we present an $O(n log n)$-time algorithm, achieving an $O(k n^4)$ factor speed-up over the previous state-of-the-art. We also give a straightforward sorting-based lower bound by reducing from the set intersection problem. We thus show that the Contiguous Art Gallery problem is in $Theta(n log n)$.

Problem

• Finding minimal contiguous boundary segments for polygon visibility coverage
• Improving computational complexity from O(kn⁵ log n) to O(n log n)
• Establishing tight Θ(n log n) bound for Contiguous Art Gallery Problem

Innovation

• Real RAM-model computational geometry algorithm
• O(n log n) time complexity solution
• Set intersection reduction for lower bound