Linear time construction of a compressed Gray code

Dimitrov, Darko; Dvořák, Tomáš; Gregor, Petr; Škrekovski, Riste

doi:10.1016/j.ejc.2012.07.015

Cited by 5 publications

(3 citation statements)

References 8 publications

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“…Moreover, Suparta [Sup17] describes a K k,n−k -code for any 1 ⩽ k ⩽ n − 1. Dimitrov, Dvořák, Gregor, and Škrekovski [DDGŠ13] constructed an n-bit G-code such that G is an induced subgraph of the ⌈log 2 (n)⌉-cube. In particular, if n is a power of 2, then this is a Q log 2 (n) -code; see Figure 2 (i').…”

Section: Transition Graphsmentioning

confidence: 99%

Combinatorial Gray Codes — an Updated Survey

Mütze

2023

Electron. J. Combin.

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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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Section: Transition Graphsmentioning

confidence: 99%

Combinatorial Gray Codes — an Updated Survey

Mütze

2023

Electron. J. Combin.

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“…Goddyn and Gvozdjak constructed an n-bit Gray code in which any two flips of the same bit are almost n steps apart [7], which is best possible. These are only two examples of a large body of work on possible Gray code transition sequences; see also [3,5,34]. Savage and Winkler constructed a Gray code that generates all 2 n bitstrings such that all bitstrings with Hamming weight k appear before all bitstrings with weight k + 2, for each 0 ≤ k ≤ n − 2 [29], where the Hamming weight of a bitstring is the number of its 1-bits.…”

Section: Introductionmentioning

confidence: 99%

Gray codes and symmetric chains

Gregor

Jäger

Mütze

et al. 2022

Journal of Combinatorial Theory, Series B

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“…Goddyn and Gvozdjak constructed an n-bit Gray code in which any two flips of the same bit are almost n steps apart [GG03], which is best possible. These are only two examples of a large body of work on possible Gray code transition sequences; see also [BR96,SvZ08,DDGŠ13]. Savage 2 n bitstrings such that all bitstrings with Hamming weight k appear before all bitstrings with weight k + 2, for each 0 ≤ k ≤ n − 2 [SW95], where the Hamming weight of a bitstring is the number of its 1-bits.…”

Section: Introductionmentioning

confidence: 99%

Gray codes and symmetric chains

Gregor,

Jäger,

Mütze

et al. 2018

Preprint

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We consider the problem of constructing a cyclic listing of all bitstrings of length 2n+1 with Hamming weights in the interval [n+1− , n+ ], where 1 ≤ ≤ n+1, by flipping a single bit in each step. This is a far-ranging generalization of the wellknown middle two levels problem (the case = 1). We provide a solution for the case = 2 and solve a relaxed version of the problem for general values of , by constructing cycle factors for those instances. Our proof uses symmetric chain decompositions of the hypercube, a concept known from the theory of posets, and we present several new constructions of such decompositions. In particular, we construct four pairwise edgedisjoint symmetric chain decompositions of the n-dimensional hypercube for any n ≥ 12.

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Linear time construction of a compressed Gray code

Cited by 5 publications

References 8 publications

Combinatorial Gray Codes — an Updated Survey

Combinatorial Gray Codes — an Updated Survey

Gray codes and symmetric chains

Gray codes and symmetric chains

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