Given a rectangle $R$ with area $A$ and a set of areas $L=\{A_1,...,A_n\}$ with $\sum_{i=1}^n A_i = A$, we consider the problem of partitioning $R$ into $n$ sub-regions $R_1,...,R_n$ with areas $A_1,...,A_n$ in a way that the total perimeter of all sub-regions is minimized. The goal is to create square-like sub-regions, which are often more desired. We propose an efficient $1.203$--approximation algorithm for this problem based on a divide and conquer scheme that runs in $\mathcal{O}(n^2)$ time. For the special case when the aspect ratios of all rectangles are bounded from above by 3, the approximation factor is $2/\sqrt{3} \leq 1.1548$. We also present a modified version of out algorithm as a heuristic that achieves better average and best run times.

翻译：给定一个面积为$A$的矩形$R$和一个面积集合$L=\{A_1,...,A_n\}$，满足$\sum_{i=1}^n A_i = A$，我们考虑将$R$划分为$n$个子区域$R_1,...,R_n$，使得每个子区域的面积为$A_1,...,A_n$，且所有子区域的总周长最小化的问题。目标是创建更受欢迎的近似正方形子区域。针对该问题，我们提出了一种基于分治策略的高效$1.203$近似算法，其运行时间为$\mathcal{O}(n^2)$。当所有矩形长宽比的上界为3时，近似因子为$2/\sqrt{3} \leq 1.1548$。此外，我们还提供了一种改进算法作为启发式方法，能够实现更优的平均运行时间和最佳运行时间。