We study a variant of a polygon partition problem, introduced by Chung, Iwama, Liao, and Ahn [ISAAC'25]. Given orthogonal unit vectors $\mathbf{u},\mathbf{v}\in \mathbb{R}^2$ and a polygon $P$ with $n$ vertices, we partition $P$ into connected pieces by cuts parallel to $\mathbf{v}$ such that each resulting subpolygon has width at most one in direction $\mathbf{u}$. We consider the value version, which asks for the minimum number of strips, and the reporting version, which outputs a compact encoding of the cuts in an optimal strip partition. We give efficient algorithms and lower bounds for both versions on three classes of polygons of increasing generality: convex, simple, and self-overlapping. For convex polygons, we solve the value version in $O(\log n)$ time and the reporting version in $O\!\left(h \log\left(1 + \frac{n}{h}\right)\right)$ time, where $h$ is the width of $P$ in direction $\mathbf{u}$. We prove matching lower bounds in the decision-tree model, showing that the reporting algorithm is input-sensitive optimal with respect to $h$. For simple polygons, we present $O(n \log n)$-time, $O(n)$-space algorithms for both versions and prove an $Ω(n)$ lower bound. For self-overlapping polygons, we extend the approach for simple polygons to obtain $O(n \log n)$-time, $O(n)$-space algorithms for both versions, and we prove a matching $Ω(n \log n)$ lower bound in the algebraic computation-tree model via a reduction from the $δ$-closeness problem. Our approach relies on a lattice-theoretic formulation of the problem. We represent strip partitions as antichains of intervals in the Clarke--Cormack--Burkowski lattice, originally developed for minimal-interval semantics in information retrieval. Within this lattice framework, we design a dynamic programming algorithm that uses the lattice operations of meet and join.

翻译：我们研究了由Chung、Iwama、Liao和Ahn [ISAAC'25]引入的多边形分割问题的一个变体。给定正交单位向量$\mathbf{u},\mathbf{v}\in \mathbb{R}^2$和一个具有$n$个顶点的多边形$P$，我们通过平行于$\mathbf{v}$的切割将$P$分割为连通片段，使得每个子多边形在$\mathbf{u}$方向上的宽度至多为1。我们考虑值版本（询问最少所需条带数）和报告版本（输出最优条带分割中切割的紧凑编码）。针对三类具有递增一般性的多边形（凸多边形、简单多边形和自交多边形），我们为两个版本设计了高效算法并给出了下界。对于凸多边形，我们在$O(\log n)$时间内解出值版本，在$O\!\left(h \log\left(1 + \frac{n}{h}\right)\right)$时间内解出报告版本，其中$h$是$P$在$\mathbf{u}$方向上的宽度。我们在决策树模型中证明了匹配的下界，表明报告算法相对于$h$是输入敏感最优的。对于简单多边形，我们为两个版本给出了$O(n \log n)$时间、$O(n)$空间的算法，并证明了$Ω(n)$下界。对于自交多边形，我们将简单多边形的方法扩展到两个版本的$O(n \log n)$时间、$O(n)$空间算法，并通过从δ-接近度问题的归约，在代数计算树模型中证明了匹配的$Ω(n \log n)$下界。我们的方法依赖于问题的格论形式化。我们将条带分割表示为Clarke-Cormack-Burkowski格中的区间反链，该格最初用于信息检索中的最小区间语义。在此格框架下，我们设计了一个利用交与并格运算的动态规划算法。