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 $Θ(n \log n)$.

翻译：近期，美术馆问题的一个自然变体——\emph{连续美术馆问题}被提出。给定一个简单多边形$P$，目标是将其边界$\partial P$划分为最少数量的连续段，使得每一段都能从$P$内的某个点完全可见。与经典的NP难美术馆问题不同，该变体可在多项式时间内求解。在SoCG~2025上，三项独立研究工作提出了针对该问题的算法，各自达到了$O(k n^5 \log n)$（或$O(n^6\log n)$）的运行时间，其中$k$为最优解的大小。值得注意的是，这些结果通过完全不同的方法获得，却都得到了大致相同的渐近复杂度，暗示此类运行时间可能是该问题固有的。我们证明事实并非如此。在计算几何学的主流模型——实随机存取机模型中，我们提出了一种$O(n \log n)$时间算法，相比现有最优结果实现了$O(k n^4)$倍的加速。同时，通过从集合相交问题归约，我们给出了基于排序的简明下界证明。由此我们证实连续美术馆问题属于$Θ(n \log n)$复杂度类。