Gray codes for reflectable languages

Li, Yue; Sawada, Joe

doi:10.1016/j.ipl.2008.11.007

Cited by 15 publications

(8 citation statements)

References 9 publications

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“…Li and Sawada [LS09] considered another tree-based approach for generating so-called reflectable languages, yielding Gray codes for k-ary strings and trees, restricted growth strings, and open meandric systems (see also [XCU10]). Ruskey, Sawada, and Williams [RSW12,SW12] proposed a generation framework based on binary strings with a fixed numbers of 1s, called bubble languages, which allows to generate e.g.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial generation via permutation languages

Hartung

Hoang

Mütze

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an n-element set by adjacent transpositions; the binary reflected Gray code to generate all n-bit strings by flipping a single bit in each step; the Gray code for generating all n-vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an n-element ground set by element exchanges due to Kaye.The first main application of our framework are permutation patterns, yielding new Gray codes for different pattern-avoiding permutations, such as vexillary, skew-merged, separable, Baxter and twisted Baxter permutations, 2-stack sortable permutations, geometric grid classes, and many others. We also obtain new Gray codes for many combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into n rectangles subject to different restrictions.The second main application of our framework are lattice congruences of the weak order on the symmetric group Sn. Recently, Pilaud and Santos realized all those lattice congruences as (n − 1)-dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian. 2 COMBINATORIAL GENERATION VIA PERMUTATION LANGUAGES. I. FUNDAMENTALS for specific classes of objects, and many of these algorithms are covered in depth in the most recent volume of Knuth's seminal series 'The Art of Computer Programming' [Knu11].1.1. Overview of our results. The main contribution of this paper is a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, which provides a unified view on many known results and allows us to prove many new ones. The basic idea is to encode a particular set of objects as a set of permutations L n ⊆ S n , where S n denotes all permutations of [n] := {1, 2, . . . , n}, and to use a simple greedy algorithm to generate those permutations by cyclic rotations of substrings, an operation we call a jump. This works under very mild assumptions on the set L n , and allows us to generate more than double-exponentially (in n) many distinct sets L n . Moreover, the jump orderings obtained from our algorithm translate into listings of combinatorial objects where consecutive objects differ by small changes, i.e., we obtain Gray codes [Sav97], and...

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Section: Introductionmentioning

confidence: 99%

Combinatorial generation via permutation languages

Hartung

Hoang

Mütze

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…One major application of the greedy method are the Gray codes for zigzag languages of permutations introduced by Hartung, Hoang, Mütze and Williams [HHMW21], which include many classes of pattern-avoiding permutations (recall Section 5.4), and which encode several classes of rectangulations [MM21] (recall Section 8.3), and elimination trees of chordal graphs [CMM21] (recall Section 9.4). These Gray codes derived from the greedy approach all have an underlying hierarchical generation tree, instances of which can be found in several predecessor papers, including Hurtado and Noy's tree of triangulations [HN99], Takaoka's twisted lexico tree [Tak99b], the production trees of the ECO method [BBGP04], and in Li and Sawada's reflectable languages [LS09] (see also [XCU10]).…”

Section: Greedy Gray Codesmentioning

confidence: 99%

Combinatorial Gray codes-an updated survey

Mütze¹

2022

Preprint

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A combinatorial Gray code for a class of objects is a listing that contains each object from the class exactly once such that any two consecutive objects in the list differ only by a 'small change'. Such listings are known for many different combinatorial objects, including bitstrings, combinations, permutations, partitions, triangulations, but also for objects defined with respect to a fixed graph, such as spanning trees, perfect matchings or vertex colorings. This survey provides a comprehensive picture of the state-of-the-art of the research on combinatorial Gray codes. In particular, it gives an update on Savage's influential survey [C. D. Savage. A survey of combinatorial Gray codes. SIAM Rev., 39(4):605-629, 1997.], incorporating many more recent developments. We also elaborate on the connections to closely related problems in graph theory, algebra, order theory, geometry and algorithms, which embeds this research area into a broader context. Lastly, we collect and propose a number of challenging research problems, thus stimulating new research endeavors.

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“…By Theorem 5, the sets A 9 5 (31000) and A 9 5 (24000) listed in ⊳ order are 4-close 3-Gray codes. However, it is easy to check that in particular, A 9 5 (31000) is a 3-close 3-Gray code, and A 9 5 (24000) is 4-close 2-Gray code. For example:…”

Section: Factors Inducing Zero Periodicitymentioning

confidence: 99%

“…Although in [9] it is proved that the set of words avoiding a given factor is 'reflectable' under some conditions on the alphabet cardinality and the forbidden factor, our construction yields Gray codes for any alphabet and forbidden factor, and has a natural algorithmic implementation.…”

Section: Introductionmentioning

confidence: 99%

Gray code orders for $$q$$ q -ary words avoiding a given factor

et al. 2015

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Based on BRGC inspired order relations we define Gray codes and give a generating algorithm for q-ary words avoiding a prescribed factor. These generalize an early 2001 result and a very recent one published by some of the present authors, and can be seen as an alternative to those of Squire published in 1996. Among the involved tools, we make use of generalized BRGC order relations, ultimate periodicity of infinite words, and word matching techniques.

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Gray codes for reflectable languages

Cited by 15 publications

References 9 publications

Combinatorial generation via permutation languages

Combinatorial generation via permutation languages

Combinatorial Gray codes-an updated survey

Gray code orders for $$q$$ q -ary words avoiding a given factor

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