Combinatorial Gray Code Research Articles

Combinatorial Gray Codes — an Updated Survey

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.

Generating a Gray code for prefix normal words in amortized polylogarithmic time per word

A prefix normal word is a binary word with the property that no substring has more 1s than the prefix of the same length. By proving that the set of prefix normal words is a bubble language, we can exhaustively list all prefix normal words of length n as a combinatorial Gray code, where successive strings differ by at most two swaps or bit flips. This Gray code can be generated in O(log2⁡n) amortized time per word, while the best generation algorithm hitherto has O(n) running time per word. We also present a membership tester for prefix normal words, as well as a novel characterization of bubble languages.

To Compose a Lead: Structures Related to Change Ringing

To Compose a Lead: Structures Related to Change Ringing

Gray codes and lexicographical combinatorial generation for nonnesting and sparse nonnesting set partitions

We present combinatorial Gray codes and explicit designs of efficient algorithms for lexicographical combinatorial generation of the sets of nonnesting and sparse nonnesting set partitions of length n.

More restrictive Gray codes for some classes of pattern avoiding permutations

In a recent article [W.M.B. Dukes, M.F. Flanagan, T. Mansour, V. Vajnovszki, Combinatorial Gray codes for classes of pattern avoiding permutations, Theoret. Comput. Sci. 396 (2008) 35–49], Dukes, Flanagan, Mansour and Vajnovszki present Gray codes for several families of pattern avoiding permutations. In their Gray codes two consecutive objects differ in at most four or five positions, which is not optimal. In this paper, we present a unified construction in order to refine their results (or to find other Gray codes). In particular, we obtain more restrictive Gray codes for the two Wilf classes of Catalan permutations of length n; two consecutive objects differ in at most two or three positions which is optimal for n odd. Other refinements have been found for permutation sets enumerated by the numbers of Schröder, Pell, even index Fibonacci numbers and the central binomial coefficients. A general efficient generating algorithm is also given.

On combinatorial Gray codes with distance 3

We suggest a construction of the cyclic binary combinatorial Gray codes with distance 3 and dimension n = 2k – 1, where k = 3, 4, . . . We give a method of construction of Hamiltonian cycles in the graphs of minimum distances of binary Hamming codes. For all admissible lengths n ≥ 15, we give nonlinear perfect binary codes whose graphs of minimum distances contain a Hamiltonian cycle.

Gray Codes, Loopless Algorithm and Partitions

The generation of efficient Gray codes and combinatorial algorithms that list all the members of a combinatorial object has received a lot of attention in the last few years. Knuth gave a code for the set of all partitions of [n] = {1,2,...,n}. Ruskey presented a modified version of Knuth’s algorithm with distance 2. Ehrlich introduced a looplees algorithm for the set of the partitions of [n]; Ruskey and Savage generalized Ehrlich’s results and introduced two Gray codes for the set of partitions of [n]. In this paper, we give another combinatorial Gray code for the set of the partitions of [n] which differs from the aforementioned Gray codes. Also, we construct a different loopless algorithm for generating the set of all partitions of [n] which gives a constant time between successive partitions in the construction process.

Computing Binary Combinatorial Gray Codes Via Exhaustive Search With SAT Solvers

The term binary combinatorial Gray code refers to a list of binary words such that the Hamming distance between two neighboring words is one and the list satisfies some additional properties that are of interest to a particular application, e.g., circuit testing, data compression, and computational biology. New distance-preserving and circuit codes are presented along with a complete list of equivalence classes of the coil-in-the-box codes for codeword length with respect to symmetry transformations of hypercubes. A Gray-ordered code composed of all necklaces of the length is presented, improving the known result with length .

Combinatorial Gray codes for classes of pattern avoiding permutations

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.

Constant Time Generation of Rectangular Drawings with Exactly n Faces

A plane drawing of a plane graph is called a rectangular drawing if every face including the outer face is a rectangle. A based rectangular drawing is a rectangular drawing with a designated base line segment the outer face. An algorithm to generate all based rectangular drawings having exactly n faces is known. The algorithm generates each based rectangular drawing having exactly n faces constant time on average. In this paper, we give another simple algorithm to generate all based rectangular drawings having exactly n faces. Our algorithm generates each based rectangular drawing having exactly n faces constant time in the worst case. Our algorithm generates each based rectangular drawing so that it can be obtained from the preceding one by at most three operations and does not output entire drawings but the difference from the preceding one. Therefore the derived sequence of based rectangular drawings is a kind of combinatorial Gray code for based rectangular drawings.

A loopless Gray code for rooted trees

Beyer and Hedetniemi [1980] gave the first constant average-time algorithm for the generation of all rooted trees with n nodes. This article presents the first combinatorial Gray code for these trees and a loopless algorithm for its generation.

Generating Chordal Graphs Included in Given Graphs

A chordal graph is a graph which contains no chordless cycle of at least four edges as an induced subgraph. The class of chordal graphs contains many famous graph classes such as trees, interval graphs, and split graphs, and is also a subclass of perfect graphs. In this paper, we address the problem of enumerating all labeled chordal graphs included in a given graph. We think of some variations of this problem. First we introduce an algorithm to enumerate all connected labeled chordal graphs in a complete graph of n vertices. Next, we extend the algorithm to an algorithm to enumerate all labeled chordal graphs in a n-vertices complete graph. Then, we show that we can use, with small changes, these algorithms to generate all (connected or not necessarily connected) labeled chordal graphs in arbitrary graph. All our algorithms are based on reverse search method, and time complexities to generate a chordal graph are O(1), and also O(1) delay. Additionally, we present an algorithm to generate every clique of a given chordal graph in constant time. Using these algorithms we obtain combinatorial Gray code like sequences for these graph structures in which the differences between two consecutive graphs are bounded by a constant size.

On the low hamming weight discrete logarithm problem for nonadjacent representations

So-called nonadjacent representations are commonly used in elliptic curve cryptography to facilitate computing a scalar multiple of a point on an elliptic curve. A nonadjacent representation having few non-zero coefficients would further speed up the computations. However, any attempt to use these techniques must also consider the impact on the security of the cryptosystem. The security is studied by examining a related discrete logarithm problem, the topic of this paper. We describe an algorithm to solve the relevant discrete logarithm problem in time that is approximately the square root of the search space. This algorithm is of the familiar ``baby-step giant-step'' type. In developing our algorithm we use two tools of independent interest; namely, a combinatorial set system called a ``splitting system'' and a new type of combinatorial Gray code.

Efficient algorithms on distributive lattices

We present several efficient algorithms on distributive lattices. They are based on a compact representation of the lattice, called the ideal tree. This allows us to exploit regularities in the structure of distributive lattices. The algorithms include a linear-time algorithm to reconstruct the covering graph of a distributive lattice from its ideal tree, a linear-time incremental algorithm for building the ideal lattice of a poset and a new incremental algorithm for listing the ideals of a poset in a combinatorial Gray code manner (in an H(1,2) code.)

A Note on a Question of C. D. Savage

Given a graph G and an orientation $\sigma$ of some of its edges, consider the graph $AO_\sigma(G)$ which is defined as follows: The vertices are the acyclic orientations of G which agree with $\sigma$, and two of these are adjacent if they differ only by the reversal of a single edge. $AO_\sigma(G)$ is easily seen to be bipartite. The purpose of this note is to show that it need not contain a Hamilton path even if both partite sets have the same cardinality. This answers a question of C. D. Savage [SIAM Rev., 39 (1997), pp. 605--629] and sheds new light onto two well-known open questions in the field of combinatorial Gray codes.

O(1) Time Algorithms for Combinatorial Generation by Tree Traversal

There are algorithms for generating combinatorial objects such as combinations, permutations and well formed parenthesis strings in O(1) time per object, in the worst case. Those known algorithms are designed based on the intrinsic nature of each problem, causing difficulty in applying a method in one area to the other. On the other hand, there are many results on combinatorial generation with minimal change order, in which just a few changes, one or two, are allowed from object to object. These results are classified in a general framework of a combinatorial Gray code, many of which are based on a recursive algorithm, causing O(n) time from object to object. To derive O(1) time algorithms for combinatorial generation systematically, we formalize the idea of combinatorial Gray code by a twisted lexico tree, which is obtained from the lexicographic tree for the given set of combinatorial objects by twisting branches depending on the parity of the nodes. An iterative algorithm which traverses this tree will generate the given set of combinatorial objects in O(1) time as well as with a fixed number of changes from the present combinatorial object to the next. Although the idea of a twisted lexico tree is not new, the mechanisms of tree traversal and computation of changing places in this paper are new. As examples of this approach, we present new algorithms for generating well formed parenthesis strings and combinations in O(1) time per object. The generation of combinations is done ‘in-place’, that is, taking O(n) space to generate combinations of n elements out of r elements. Previous algorithms take O(r) space to represent a combination by a binary vector of size r.

A Survey of Combinatorial Gray Codes

The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing n-bit binary numbers so that successive numbers differ in exactly one bit position, as well as work in the 1960s and 1970s on minimal change listings for other combinatorial families, including permutations and combinations. The area of combinatorial Gray codes was popularized by Herbert Wilf in his invited address at the SIAM Conference on Discrete Mathematics in 1988 and his subsequent SIAM monograph [Combinatorial Algorithms: An Update, 1989] in which he posed some open problems and variations on the theme. This resulted in much recent activity in the area, and most of the problems posed by Wilf are now solved. In this paper, we survey the area of combinatorial Gray codes, describe recent results, variations, and trends, and highlight some open problems.

Combinatorial Gray Codes

We consider families $\{ {\bf C}(n,k):O \leqq k \leqq n\} $ where each ${\bf C}(n,k)$ is a set of combinatorial objects, $C(n,k) = |{\bf C}(n,k)|$ satisfies a recursion $C(n,k)= a_{n,k}C(n - 1,k - 1) + b_{n,k} C(n - 1,k)$, and each object in ${\bf C}(n,k)$ is represented by an n-vector. We study “loop-free” or “uniformly bounded transition” algorithms, i.e., algorithms which yield linear orders on the sets ${\bf C}(n,k)$ so that the vectors representing consecutive objects are “close to each other” (combinatorial Gray codes).Key words. listing algorithms, uniformly bounded operations, uniformly bounded transition algorithms, loop-free algorithms, binary reflected Gray codes, combinatorial Gray codes, binomial grids