The Flip Diameter of Rectangulations and Convex Subdivisions - Ackerman, Eyal; Allen, Michelle M.; Barequet, Gill; Löffler, Maarten; Mermelstein, Joshua; Souvaine, Diane L.; Tóth, Csaba D. (Discrete mathematics and theoretical computer science, 2016)
We study the configuration space of rectangulations and convex subdivisions of $n$ points in the plane. It is shown that a sequence of $O(n\log n)$ elementary flip and rotate operations can transform any rectangulation to any other rectangulation on the same set of $n$ points. This bound is the best possible for some point sets, while $Θ(n)$ operations are sufficient and necessary for others. Some of our bounds generalize to convex subdivisions of $n$ points in the plane.