When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2023.6
URN: urn:nbn:de:0030-drops-186598
URL: https://drops.dagstuhl.de/opus/volltexte/2023/18659/
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### Reconfiguration of Polygonal Subdivisions via Recombination

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### Abstract

Motivated by the problem of redistricting, we study area-preserving reconfigurations of connected subdivisions of a simple polygon. A connected subdivision of a polygon ℛ, called a district map, is a set of interior disjoint connected polygons called districts whose union equals 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 Ω(log n) recombination moves are sometimes necessary even when k = 3, thus providing a tight bound when k = 3.

### BibTeX - Entry

```@InProceedings{a.akitaya_et_al:LIPIcs.ESA.2023.6,
author =	{A. Akitaya, Hugo and Gonczi, Andrei and Souvaine, Diane L. and T\'{o}th, Csaba D. and Weighill, Thomas},
title =	{{Reconfiguration of Polygonal Subdivisions via Recombination}},
booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
pages =	{6:1--6:16},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-295-2},
ISSN =	{1868-8969},
year =	{2023},
volume =	{274},
editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},