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

TL;DR

The paper proves that the Contiguous Art Gallery problem can be solved in optimal rac{n \, log n}{n \, log n} time, significantly improving previous algorithms and establishing its rac{n \, log n}{n \, log n} complexity.

Contribution

It introduces an rac{n \, log n}{n \, log n} time algorithm for the problem, matching the lower bound and showing the problem's true complexity.

Findings

New rac{n \, log n}{n \, log n} algorithm for the problem

Established rac{n \, log n}{n \, log n} complexity as optimal

Provided a sorting-based lower bound for the problem

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 $O(n^6\log n)$), 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 $\Theta(n \log n)$.