The Contiguous Art Gallery Problem is in Θ(n log n)

TL;DR

This work resolves the Contiguous Art Gallery problem on simple polygons by achieving near-linear time in the real RAM model. It introduces a streamlined framework built around the next-function, a linear-size witness set X, and a sweeping, dominance-based decomposition of guards into good, bad, and ugly categories, enabling efficient computation of optimal guarding sequences. The authors establish a matching $\Omega(n\log n)$ lower bound for comparison-based models and present an $O(n\log n)$-time algorithm with linear space, substantially improving over prior high-degree polynomial-time solutions. The result provides the first near-linear-time algorithm for a natural Art Gallery variant, highlighting that such running times may be intrinsic when structural constraints are leveraged through sweep-line and geometric-data-structure techniques.

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 $Θ(n \log n)$.

Figures (9)

• Figure 1: A simple polygon $P$ and five contiguous guards that guard the entire boundary of $P$.
• Figure 2: The lower-bound construction. Black points belong to $A$, and red points to $B$.
• Figure 3: Flowchart illustrating how a guard is dominated. We prove the green block in \ref{['lem:singleDominator']} and the red block in \ref{['lem:dominated_by_baddom']}.
• Figure 4: The visibility core $\mathcal{E}[i-1,j+1]$ in green and the segment-intersecting dominator $(g,[u_\text{max},v_\text{max}])$ for $(i,j)$.
• Figure 5: The visibility core $\mathcal{E}[i-1,j+1]$ in green and the angle-maximizing dominator $(g,[u_\text{max},v_\text{max}])$ for $(i,j)$.

Theorems & Definitions (34)

• Definition 1
• Definition 2
• Lemma 2
• proof
• Lemma 3
• proof
• proof
• Definition 3
• Definition 4
• Lemma 5