A general exhaustive generation algorithm for Gray structures

Bernini, Antonio; Grazzini, Elisabetta; Pergola, Elisa; Pinzani, Renzo

doi:10.1007/s00236-007-0053-0

Cited by 11 publications

(13 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The main tool is an infinite tree with integer node labels, and a set of production rules for creating the children of a node based on its label. Bacchelli, Barcucci, Grazzini, and Pergola [BBGP04] also used ECO for exhaustive generation, deriving an efficient algorithm for generating the corresponding root-tonode label sequences in the ECO tree in lexicographic order, which was later turned into a Gray code [BGPP07]. Dukes, Flanagan, Mansour, and Vajnovszki [DFMV08], Baril [Bar09], and Do, Tran and Vajnovszki [DTV19] used ECO for deriving Gray codes for different classes of patternavoiding permutations, which works under certain regularity assumptions on the production rules.…”

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

View full text Add to dashboard Cite

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...

show abstract

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

View full text Add to dashboard Cite

show abstract

“…Similarly to what has been done for classical rules [4], one can try to develop general exhaustive generation algorithms based on mixed succession rules, maybe finding a new way of defining general Gray codes depending only on the form of the mixed succession rule under consideration.…”

Section: Final Remarksmentioning

confidence: 99%

“…The variety of problems in which the ECO method has shown its soundness ranges from enumerative and bijective combinatorics to random [2] and exhaustive [4] generation.…”

Section: Introductionmentioning

confidence: 99%

Mixed succession rules: The commutative case

Bacchelli¹,

Ferrari²,

Pinzani³

et al. 2010

Journal of Combinatorial Theory, Series A

Self Cite

View full text Add to dashboard Cite

We begin a systematic study of the enumerative combinatorics of mixed succession rules, i.e. succession rules such that, in the associated generating tree, nodes are allowed to produce sons at several different levels according to different production rules.Here we deal with a specific case, namely that of two different production rules whose rule operators commute. In this situation, we are able to give a general formula expressing the sequence associated with the mixed succession rule in terms of the sequences associated with the component production rules. We end by providing examples illustrating our approach.

show abstract

“…Our work is different from similar work for combinatorial classes having the same counting sequence, see for instance [6,22]. Indeed, as Savage [21,Section 7] points out: 'Since bijections are known between most members of the Catalan family, a Gray code for one member of the family gives implicitly a listing scheme for every other member of the family.…”

Section: Introductionmentioning

confidence: 95%

Combinatorial Gray codes for classes of pattern avoiding permutations

Dukes¹,

Flanagan²,

Mansour³

et al. 2008

Theoretical Computer Science

View full text Add to dashboard Cite

The past decade has seen a flurry of research into pattern avoiding permutations but little of it is concerned with their exhaustive generation. Many applications call for exhaustive generation of permutations subject to various constraints or imposing a particular generating order. In this paper we present generating algorithms and combinatorial Gray codes for several families of pattern avoiding permutations. Among the families under consideration are those counted by Catalan, large Schröder, Pell, even-index Fibonacci numbers and the central binomial coefficients. We thus provide Gray codes for the set of all permutations of {1, . . . , n} avoiding the pattern τ for all τ ∈ S 3 and the Gray codes we obtain have distances 4 or 5.

show abstract

A general exhaustive generation algorithm for Gray structures

Cited by 11 publications

References 20 publications

Combinatorial generation via permutation languages

Combinatorial generation via permutation languages

Mixed succession rules: The commutative case

Combinatorial Gray codes for classes of pattern avoiding permutations

Contact Info

Product

Resources

About