scholarly journals Note on Gray codes for permutation lists

2005 ◽  
Vol 78 (92) ◽  
pp. 87-92
Author(s):  
Seymour Lipschutz ◽  
Jie Gao ◽  
Wang Dianjun
Keyword(s):  
Gray Code ◽  
Gray Codes ◽  
Classical Algorithm

Robert Sedgewick [5] lists various Gray codes for the permutations in Sn including the classical algorithm by Johnson and Trotter. Here we give an algorithm which constructs many families of Gray codes for Sn, which closely follows the construction of the Binary Reflexive Gray Code for the n-cube Qn.

scholarly journals Permutational labelling of constant weight Gray codes

2002 ◽  
Vol 65 (3) ◽  
pp. 399-406
Author(s):  
Inessa Levi ◽  
Steve Seif
Keyword(s):  
Gray Code ◽  
Gray Codes ◽  
Constant Weight ◽  
Single Exception ◽  
Positive Integers

We prove that for positive integers n and r satisfying 1 < r < n, with the single exception of n = 4 and r = 2, there exists a constant weight Gray code of r-sets of Xn = {1, 2, …, n} that admits an orthogonal labelling by distinct partitions, with each subsequent partition obtained from the previous one by an application of a permutation of the underlying set. Specifically, an r-set A and a partition π of Xn are said to be orthogonal if every class of π meets A in exactly one element. We prove that for all n and r as stated, and taken modulo , there exists a list of the distinct r-sets of Xn with |Ai ∩ Ai+1| = r − 1 and a list of distinct partitions such that πi is orthogonal to both Ai and Ai+1, and πi+1 = πiλi for a suitable permutation λi of Xn.

On the exhaustive generation of generalized ballot sequences in lexicographic and Gray code order

2019 ◽  
Vol 28 (1) ◽  
pp. 109-119
Author(s):  
Ahmad Sabri ◽  
Vincent Vajnovszki
Keyword(s):  
Positive Integer ◽  
Gray Code ◽  
Lexicographic Order ◽  
Gray Codes ◽  
Fixed Length ◽  
Generating Algorithms

Abstract A generalized (resp. p-ary) ballot sequence is a sequence over the set of non-negative integers (resp. integers less than p) where in any of its prefixes each positive integer i occurs at most as often as any integer less than i. We show that the Reected Gray Code order induces a cyclic 3-adjacent Gray code on both, the set of fixed length generalized ballot sequences and p-ary ballot sequences when p is even, that is, ordered list where consecutive sequences (regarding the list cyclically) differ in at most 3 adjacent positions. Non-trivial efficient generating algorithms for these ballot sequences, in lexicographic order and for the obtained Gray codes, are also presented.

scholarly journals Binary Gray Codes with Long Bit Runs

10.37236/1720 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Luis Goddyn ◽  
Pavol Gvozdjak
Keyword(s):  
Hamming Distance ◽  
Gray Code ◽  
Gray Codes

We show that there exists an $n$-bit cyclic binary Gray code all of whose bit runs have length at least $n - 3\log_2 n$. That is, there exists a cyclic ordering of $\{0,1\}^n$ such that adjacent words differ in exactly one (coordinate) bit, and such that no bit changes its value twice in any subsequence of $n-3\log_2 n$ consecutive words. Such Gray codes are 'locally distance preserving' in that Hamming distance equals index separation for nearby words in the sequence.

scholarly journals A Less Complex Algorithmic Procedure for Computing Gray Codes

2009 ◽  
Vol 6 (2) ◽  
pp. 12 ◽  
Author(s):  
Afaq Ahmad ◽  
Mohammed M. Bait Suwailam
Keyword(s):  
Boolean Function ◽  
Gray Code ◽  
Gray Codes ◽  
Time And Space ◽  
Design And Implementation ◽  
Algorithmic Procedure

 The purpose of this paper is to present a new and faster algorithmic procedure for generating the n bi Gray codes. Thereby, through this paper we have presented the derivation, design and implementation of a newly developed algorithm for the generation of an n-bit binary reflected Gray code sequences. The developed algorithm is stemmed from the fact of generating and properly placing the min-terms from the universal set of all the possible min-terms [m0 m1 m2 …. mN] of Boolean function of n variables, where, 0 < N <  2n-1. The resulting algorithm is in concise form and trivial to implement. Furthermore, the developed algorithm is equipped with added attributes of optimizing of time and space while executed. 

GENERATING n-ARY REFLECTED GRAY CODES ON A LINEAR ARRAY OF PROCESSORS

1996 ◽  
Vol 06 (01) ◽  
pp. 27-34 ◽  
Author(s):  
IVAN STOJMENOVIC
Keyword(s):  
Parallel Algorithm ◽  
Arbitrary Number ◽  
Linear Array ◽  
Gray Code ◽  
Gray Codes ◽  
Model Of Computation ◽  
Constant Size

We present a cost-optimal parallel algorithm for generating n-ary reflected Gray codes, i.e. variations of m elements out of {0, 1,…, n–1} in a Gray code order. It uses a linear array of m processors, each having constant size memory and each being responsible for producing one part of a given variation. The algorithm is simple and uses a weaker model of computation than a recently published algorithm. In addition, it can be made adaptive (i.e. to run on a linear array with an arbitrary number of processors) and can be generalized to produce variations out of an arbitrary set of elements.

scholarly journals Gray codes and overlap cycles for restricted weight words

2014 ◽  
Vol 06 (04) ◽  
pp. 1450062
Author(s):  
Victoria Horan ◽  
Glenn Hurlbert
Keyword(s):  
Gray Code ◽  
Existence Results ◽  
Minimal Change ◽  
Gray Codes ◽  
Weight Range ◽  
Combinatorial Objects ◽  
The Past ◽  
Fixed Weight ◽  
New Type ◽  
Overlap Cycles

A Gray code is a listing structure for a set of combinatorial objects such that some consistent (usually minimal) change property is maintained throughout adjacent elements in the list. While Gray codes for m-ary strings have been considered in the past, we provide a new, simple Gray code for fixed-weight m-ary strings. In addition, we consider a relatively new type of Gray code known as overlap cycles and prove basic existence results concerning overlap cycles for fixed-weight and weight-range m-ary words.

scholarly journals Properties of Gray and Binary Representations

2004 ◽  
Vol 12 (1) ◽  
pp. 47-76 ◽  
Author(s):  
Jonathan Rowe ◽  
Darrell Whitley ◽  
Laura Barbulescu ◽  
Jean-Paul Watson
Keyword(s):  
Genetic Algorithms ◽  
Optimization Problems ◽  
Search Algorithm ◽  
Search Space ◽  
Gray Code ◽  
Global Optimum ◽  
Gray Codes ◽  
Neighborhood Structure ◽  
Local Optima ◽  
Vertex Set

Representations are formalized as encodings that map the search space to the vertex set of a graph. We define the notion of bit equivalent encodings and show that for such encodings the corresponding Walsh coefficients are also conserved. We focus on Gray codes as particular types of encoding and present a review of properties related to the use of Gray codes. Gray codes are widely used in conjunction with genetic algorithms and bit-climbing algorithms for parameter optimization problems. We present new convergence proofs for a special class of unimodal functions; the proofs show that a steepest ascent bit climber using any reflected Gray code representation reaches the global optimum in a number of steps that is linear with respect to the encoding size. There are in fact many different Gray codes.Shifting is defined as a mechanism for dynamically switching from one Gray code representation to another in order to escape local optima. Theoretical results that substantially improve our understanding of the Gray codes and the shifting mechanism are presented. New proofs also shed light on the number of unique Gray code neighborhoods accessible via shifting and on how neighborhood structure changes during shifting. We show that shifting can improve the performance of both a local search algorithm as well as one of the best genetic algorithms currently available.

scholarly journals Transition Restricted Gray Codes

10.37236/1235 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Bette Bultena ◽  
Frank Ruskey
Keyword(s):  
Undirected Graph ◽  
Gray Code ◽  
Hamilton Cycle ◽  
Gray Codes ◽  
Hamilton Path ◽  
Vertex Set ◽  
G Code

A Gray code is a Hamilton path $H$ on the $n$-cube, $Q_n$. By labeling each edge of $Q_n$ with the dimension that changes between its incident vertices, a Gray code can be thought of as a sequence $H = t_1,t_2,\ldots,t_{N-1}$ (with $N = 2^n$ and each $t_i$ satisfying $1 \le t_i \le n$). The sequence $H$ defines an (undirected) graph of transitions, $G_H$, whose vertex set is $\{1,2,\ldots,n\}$ and whose edge set $E(G_H) = \{ [t_i,t_{i+1}] \mid 1 \le i \le N-1 \}$. A $G$-code is a Hamilton path $H$ whose graph of transitions is a subgraph of $G$; if $H$ is a Hamilton cycle then it is a cyclic $G$-code. The classic binary reflected Gray code is a cyclic $K_{1,n}$-code. We prove that every tree $T$ of diameter 4 has a $T$-code, and that no tree $T$ of diameter 3 has a $T$-code.

scholarly journals Gray Codes for A-Free Strings

10.37236/1241 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Matthew B. Squire
Keyword(s):  
Positive Result ◽  
Infinite Number ◽  
Gray Code ◽  
Combinatorial Theory ◽  
Gray Codes ◽  
Parity Problem

For any $q \geq 2$, let $\Sigma_{q}=\{0,\ldots,q\!-\!1\}$, and fix a string $A$ over $\Sigma_{q}$. The $A$-free strings of length $n$ are the strings in $\Sigma_{q}^n$ which do not contain $A$ as a contiguous substring. In this paper, we investigate the possibility of listing the $A$-free strings of length $n$ so that successive strings differ in only one position, and by $\pm 1$ in that position. Such a listing is a Gray code for the $A$-free strings of length $n$. We identify those $q$ and $A$ such that, for infinitely many $n \geq 0$, a Gray code for the $A$-free strings of length $n$ is prohibited by a parity problem. Our parity argument uses techniques similar to those of Guibas and Odlyzko (Journal of Combinatorial Theory A 30 (1981) pp. 183–208) who enumerated the $A$-free strings of length $n$. When $q$ is even, we also give the complementary positive result: for those $A$ for which an infinite number of parity problems do not exist, we construct a Gray code for the $A$-free strings of length $n$ for all $n \geq 0$.

scholarly journals Gray codes avoiding matchings

2009 ◽  
Vol Vol. 11 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Darko Dimitrov ◽  
Tomáš Dvořák ◽  
Petr Gregor ◽  
Riste Škrekovski
Keyword(s):  
Hamiltonian Cycle ◽  
Perfect Matching ◽  
Hamiltonian Path ◽  
Gray Code ◽  
Gray Codes ◽  
Similar Characterization ◽  
International Audience ◽  
Forbidden Configurations ◽  
Np Complete ◽  
Necessary And Sufficient

Graphs and Algorithms International audience A (cyclic) n-bit Gray code is a (cyclic) ordering of all 2(n) binary strings of length n such that consecutive strings differ in a single bit. Equivalently, an n-bit Gray code can be viewed as a Hamiltonian path of the n-dimensional hypercube Q(n), and a cyclic Gray code as a Hamiltonian cycle of Q(n). In this paper we study (cyclic) Gray codes avoiding a given set of faulty edges that form a matching. Given a matching M and two vertices u, v of Q(n), n >= 4, our main result provides a necessary and sufficient condition, expressed in terms of forbidden configurations for M, for the existence of a Gray code between u and v that avoids M. As a corollary. we obtain a similar characterization for a cyclic Gray code avoiding M. In particular, in the case that M is a perfect matching, Q(n) has a (cyclic) Gray code that avoids M if and only if Q(n) - M is a connected graph. This complements a recent result of Fink, who proved that every perfect matching of Q(n) can be extended to a Hamiltonian cycle. Furthermore, our results imply that the problem of Hamilionicity of Q(n) with faulty edges, which is NP-complete in general, becomes polynomial for up to 2(n-1) edges provided they form a matching.

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