Combinatorial generation via permutation languages. III. Rectangulations

TL;DR

This paper introduces a versatile, efficient algorithmic framework for exhaustively generating various classes of generic rectangulations, including pattern-avoiding types, using permutation encodings and Gray codes.

Contribution

It presents a unified, efficient generation method for multiple rectangulation classes, including pattern-avoiding variants, with cyclic Gray codes and optimality properties.

Findings

Algorithms achieve constant amortized time per rectangulation.

Generation algorithms are often loopless or have constant worst-case time.

Gray codes are cyclic and sometimes Hamilton cycles on polytope skeletons.

Abstract

A generic rectangulation is a partition of a rectangle into finitely many interior-disjoint rectangles, such that no four rectangles meet in a point. In this work we present a versatile algorithmic framework for exhaustively generating a large variety of different classes of generic rectangulations. Our algorithms work under very mild assumptions, and apply to a large number of rectangulation classes known from the literature, such as generic rectangulations, diagonal rectangulations, 1-sided/area-universal, block-aligned rectangulations, and their guillotine variants, including aspect-ratio-universal rectangulations. They also apply to classes of rectangulations that are characterized by avoiding certain patterns, and in this work we initiate a systematic investigation of pattern avoidance in rectangulations. Our generation algorithms are efficient, in some cases even loopless or constant amortized time, i.e., each new rectangulation is generated in constant time in the worst case or on average, respectively. Moreover, the Gray codes we obtain are cyclic, and sometimes provably optimal, in the sense that they correspond to a Hamilton cycle on the skeleton of an underlying polytope. These results are obtained by encoding rectangulations as permutations, and by applying our recently developed permutation language framework.