Motivated by the problem of redistricting, we study area-preserving reconfigurations of connected subdivisions of a simple polygon. A connected subdivision of a polygon $\mathcal{R}$, called a district map, is a set of interior disjoint connected polygons called districts whose union equals $\mathcal{R}$. We consider the recombination as the reconfiguration move which takes a subdivision and produces another by merging two adjacent districts, and by splitting them into two connected polygons of the same area as the original districts. The complexity of a map is the number of vertices in the boundaries of its districts. Given two maps with $k$ districts, with complexity $O(n)$, and a perfect matching between districts of the same area in the two maps, we show constructively that $(\log n)^{O(\log k)}$ recombination moves are sufficient to reconfigure one into the other. We also show that $\Omega(\log n)$ recombination moves are sometimes necessary even when $k=3$, thus providing a tight bound when $k=O(1)$.

翻译：受选区重划问题启发，我们研究了简单多边形上连通剖分的面积保持重组。多边形的连通剖分称为区域图，是由一组内部不相交的连通多边形（称为区域）组成的集合，其并集等于该多边形。我们考虑将重组作为重新配置操作：该操作通过合并两个相邻区域，再将其分割为与原区域面积相等的两个连通多边形，从而产生新的剖分。图的复杂度是其区域边界上的顶点数量。给定两个具有k个区域且复杂度为O(n)的图，以及两图中相同面积区域之间的完美匹配，我们构造性地证明最多需要$(\log n)^{O(\log k)}$次重组操作即可将一个图重新配置为另一个图。我们还证明，即使当k=3时，有时也需要$\Omega(\log n)$次重组操作，从而在k=O(1)时给出了紧界。