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：最近，艺术画廊问题的一个自然变体——连续画廊问题被提出。给定一个简单多边形 $P$，目标是将边界 $\partial P$ 划分为最少数量的连续线段，使得每条线段都能从 $P$ 中的某一点完全可见。与经典的NP难艺术画廊问题不同，该变体可在多项式时间内求解。在SoCG 2025会议上，三项独立工作提出了解决该问题的算法，运行时间均为 $O(k n^5 \log n)$（或 $O(n^6\log n)$），其中 $k$ 为最优解的大小。有趣的是，这些结果采用了完全不同的方法，却得到了大致相同的渐进复杂度，暗示该问题可能固有这样的运行时间。我们证明事实并非如此。在计算几何的主流模型——实数RAM模型中，我们提出了一种 $O(n \log n)$ 时间复杂度的算法，相比此前最优算法实现了 $O(k n^4)$ 倍的加速。此外，我们通过归约集合交集问题，给出了一个基于排序的简单下界。因此，我们证明了连续画廊问题的时间复杂度为 $Θ(n \log n)$。