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Motivated by the problem of redistricting, we study area-preservingreconfigurations of connected subdivisions of a simple polygon. A connectedsubdivision of a polygon $\mathcal{R}$, called a district map, is a set ofinterior disjoint connected polygons called districts whose union equals$\mathcal{R}$. We consider the recombination as the reconfiguration move whichtakes a subdivision and produces another by merging two adjacent districts, andby splitting them into two connected polygons of the same area as the originaldistricts. The complexity of a map is the number of vertices in the boundariesof 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, weshow constructively that $(\log n)^{O(\log k)}$ recombination moves aresufficient to reconfigure one into the other. We also show that $\Omega(\logn)$ recombination moves are sometimes necessary even when $k=3$, thus providinga tight bound when $k=O(1)$.

Reconfiguration of Polygonal Subdivisions via Recombination