Gray Code Order Research Articles

Generating 2-Gray codes for ballot sequences in constant amortized time

We present a simple algorithm that generates a cyclic 2-Gray code for ballot sequences. The algorithm generates each ballot sequence in constant amortized time using a linear amount of space. This is the first known cyclic 2-Gray code for ballot sequences that achieves this time bound. In addition, the algorithm can be easily modified to output ballot sequences in binary reflected Gray code order in constant amortized time per string using a linear amount of space.

Flip-swap languages in binary reflected Gray code order

A flip-swap language is a set S of binary strings of length n such that S∪{0n} is closed under two operations (when applicable): (1) Flip the leftmost 1; and (2) Swap the leftmost 1 with the bit to its right. Flip-swap languages model many combinatorial objects including necklaces, Lyndon words, prefix normal words, left factors of k-ary Dyck words, lattice paths with at most k flaws, and feasible solutions to 0-1 knapsack problems. We prove that any flip-swap language forms a cyclic 2-Gray code when listed in binary reflected Gray code (BRGC) order. Furthermore, a generic successor rule computes the next string when provided with a membership tester. The rule generates each string in the aforementioned flip-swap languages in O(n)-amortized per string, except for prefix normal words of length n which require O(n1.864)-amortized per string. Our work generalizes results on necklaces and Lyndon words by Vajnovszki (2008) [35].

Improving Hamiltonian encodings with the Gray code

Due to the limitations of present-day quantum hardware, it is especially critical to design algorithms that make the best possible use of available resources. When simulating quantum many-body systems on a quantum computer, straightforward encodings that transform many-body Hamiltonians into qubit Hamiltonians use $N$ of the available basis states of an $N$-qubit system, whereas ${2}^{N}$ are in theory available. We explore an efficient encoding that uses the entire set of basis states, where terms in the Hamiltonian are mapped to qubit operators with a Hamiltonian that acts on the basis states in Gray code order. This encoding is applied to the commonly studied problem of finding the ground-state energy of a deuteron with a simulated variational quantum eigensolver (VQE). It is compared to a standard ``one-hot'' encoding, and various trade-offs that arise are analyzed. The energy distribution of VQE solutions has smaller variance than the one obtained by the one-hot encoding even in the presence of simulated hardware noise, despite an increase in the number of measurements. The reduced number of qubits and a shorter-depth variational Ansatz enables the encoding of larger problems on current-generation machines. This encoding also demonstrates improvements for simulating time evolution of the same system, producing circuits for the evolution operators with reduced depth and roughly half the number of gates compared to a one-hot encoding.

A Deep-Network Piecewise Linear Approximation Formula.

The mathematical foundation of deep learning is the theorem that any continuous function can be approximated within any specified accuracy by using a neural network with certain non-linear activation functions. However, this theorem does not tell us what the network architecture should be and what the values of the weights are. One must train the network to estimate the weights. There is no guarantee that the optimal weights can be reached after training. This paper develops an explicit architecture of a universal deep network by using the Gray code order and develops an explicit formula for the weights of this deep network. This architecture is target function independent. Once the target function is known, the weights are calculated by the proposed formula, and no training is required. There is no concern whether the training may or may not reach the optimal weights. This deep network gives the same result as the shallow piecewise linear interpolation function for an arbitrary target function.

A loopless algorithm for generating (k, m)-ary trees in Gray code order

A family of (k, m)-ary trees was firstly introduced by Du and Liu when they studied hook length polynomial for plane trees. Recently, Amani and Nowzari-Dalini presented a generation algorithm to produce (k, m)-ary trees of order n encoding by Z-sequences in reverse lexicographic order. In this paper, we propose a loopless algorithm to generate all such Z-sequences in Gray code order. Hence, the worst-case time complexity of generating one Z-sequence is $${\mathcal {O}}(1)$$ , and the space requirement of our algorithm is $$2n+{\mathcal {O}}(1)$$ . Moreover, based on this ordering, we also provide ranking and unranking algorithms. The ranking algorithm can be run in $${\mathcal {O}}(\max \{kmn,n^2\})$$ time and $${\mathcal {O}}(kmn)$$ space, whereas the unranking algorithm requires $${\mathcal {O}}(kmn^2)$$ time and space.

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

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.

Necklaces and Lyndon words in colexicographic and binary reflected Gray code order

We present the first efficient algorithm to list necklaces, Lyndon words, or pseudo-necklaces of length n in colexicographic order. The algorithm has two interesting properties. First, it can be applied to construct a de Bruijn sequence of order n in O(1)-time per bit. Second, it can easily be modified to efficiently list necklaces, Lyndon words, or pseudo-necklaces of length n in binary reflected Gray code order.

More Restricted Growth Functions: Gray Codes and Exhaustive Generation

A Gray code for a combinatorial class is a method for listing the objects in the class so that successive objects differ in some prespecified, small way, typically expressed as a bounded Hamming distance. In a previous work, the authors of the present paper showed, among other things, that the m-ary Reflected Gray Code Order yields a Gray code for the set of restricted growth functions. Here we further investigate variations of this order relation, and give the first Gray codes and efficient generating algorithms for bounded restricted growth functions.

Differential Spatial Modulation With Gray Coded Antenna Activation Order

Differential spatial modulation (DSM) was recently proposed to overcome the challenge of channel estimation in spatial modulation (SM). In this letter, we propose a gray code order of antenna index permutations for DSM. To facilitate the implementation, the well-known Trotter-Johnson ranking and unranking algorithms are adopted, which results in similar computational complexity to the existing DSM that uses the lexicographic order. The coding gain achieved by the proposed gray code order over the existing lexicographic order is also analyzed and verified via simulations, which reveals a maximum of about 1.2dB for the case of four transmit antennas. Based on the gray coding framework, we further propose a diversity-enhancing scheme named intersected gray (I-gray) code order for DSM, where the permutations of active antenna indices are selected directly from the odd (even) positions of the full permutations in the gray code order. From analysis and simulations, it is shown that the I-gray code order can harvests an additional diversity order at the expense of only one information bit loss for each transmission with respect to the gray code order.

A loopless algorithm for generating multiple binary tree sequences simultaneously

Pallo and Wu et al. respectively introduced the left-weight sequences (LW-sequences) and right-weight sequences (RW-sequences) for representing binary trees. In this paper, we introduce two new types of binary tree sequences called the left-child sequences (LC-sequences) and right-child sequences (RC-sequences). Next, we propose a loopless algorithm associated with rotations of binary trees for generating LW-, RW-, LC-, and RC-sequences simultaneously. Moreover, we show that LW- and RW-sequences are generated in Gray-code order, and LC- and RC-sequences are generated so that each sequence can be obtained from its predecessor by changing at most two digits. Our algorithm is shown to be more efficient in both space and time than the existing known algorithms.

Ranking and Unranking of Non-regular Trees in Gray-Code Order

A non-regular tree T with a prescribed branching sequence (s 1 , s 2 ,..., s n ) is a rooted and ordered tree such that its internal nodes are numbered from 1 to n in preorder and every internal node i in T has s i children. Recently, Wu et al. (2010) introduced a concise representation called RD-sequences to represent all non-regular trees and proposed a loopless algorithm for generating all non-regular trees in a Gray-code order. In this paper, based on such a Gray-code order, we present efficient ranking and unranking algorithms of non-regular trees with n internal nodes. Moreover, we show that the ranking algorithm and the unranking algorithm can be run in O(n 2 ) time and O(n 2 + nS n―1 ) time, respectively, provided a preprocessing takes O(n 2 S n―1 ) time and space in advance, where S n―1 = Σ n―1 i=1 (s i ―1).

Ranking and Unranking t-ary Trees in a Gray-Code Order

Ranking and Unranking t-ary Trees in a Gray-Code Order

Ranking and Unranking Algorithms for k-ary Trees in Gray Code Order

In this paper, we present two new ranking and unranking algorithms for k-ary trees represented by x-sequences in Gray code order. These algorithms are based on a gray code generation algorithm developed by Ahrabian et al.. In mentioned paper, a recursive backtracking generation algorithm for x-sequences corresponding to k-ary trees in Gray code was presented. This generation algorithm is based on Vajnovszki’s algorithm for generating binary trees in Gray code ordering. Up to our knowledge no ranking and unranking algorithms were given for x-sequences in this ordering. we present ranking and unranking algorithms with O(kn) time complexity for x-sequences in this Gray code ordering . Keywords—k-ary Tree Generation, Ranking, Unranking, Gray Code.

Efficient generation of the ideals of a poset in gray code order, part II

Pruesse and Ruskey presented an algorithm for generating their Gray code for the ideals of a poset, where two adjacent ideals differ by one or two elements. Their algorithm takes O(n) amortized time per ideal, where n is the number of elements in the poset. Squire presented a recurrence for the ideals of a poset that enabled him to find an algorithm for generating these ideals in O(logn) amortized time per ideal, but not in Gray code order. In a recent article, we used Squire’s recurrence to find a Gray code for the ideals of a poset, where two adjacent ideals differ by one or two elements. Our algorithm, presented in a recently published article, has the same complexity as that of Pruesse and Ruskey, but it generates a Gray code that is different from the one generated by their algorithm. In this article, we generalize Squire’s recurrence. Squire’s recurrence includes a condition that enables his algorithm to run in O(logn) amortized time per ideal. We propose a less restrictive condition than his. This condition, along with our generalization of Squire’s recurrence, enabled us to improve the complexity of our algorithm. We implemented our algorithm and tested it on several examples which together illustrate most of the possible cases. We found that the running time of our program is almost the same as that of our program of Squire’s algorithm for all those posets. For most of those posets, the running time of Ruskey’s program is close to that of each of the other two programs, whereas for some posets it is much greater than that of either of the other two programs.

Efficient Oracles for Generating Binary Bubble Languages

A simple meta-algorithm is provided to efficiently generate a wide variety of combinatorial objects that can be represented by binary strings with a fixed number of 1s. Such objects include: $k$-ary Dyck words, connected unit interval graphs, binary strings lexicographically larger than $\omega$, those avoiding $10^k$ for fixed $k$, reversible strings and feasible solutions to knapsack problems. Each object requires only a very simple object-specific subroutine (oracle) that plugs into the generic cool-lex framework introduced by Williams. The result is that each object can be generated in amortized $O(1)$-time. Moreover, the strings can be listed in either a conventional co-lexicographic order, or in the cool-lex Gray code order.

A Gray code for fixed-density necklaces and Lyndon words in constant amortized time

This paper develops a constant amortized time algorithm to produce a cyclic cool-lex Gray code for fixed-density binary necklaces, Lyndon words, and pseudo-necklaces. It is the first Gray code for these objects that achieves this time bound. The algorithm is applied: (i) to develop a constant amortized time cyclic Gray code for necklaces, Lyndon words, and pseudo-necklaces ordered by density and (ii) to obtain a fixed-density de Bruijn sequence using constant time per n bits on average. In addition to Gray code order, the algorithms can be easily modified to output the strings in co-lex order.

A Ranking Algorithm of Non-Regular Trees in Gray-Code Order

A non-regular tree T with a prescribed branching sequence is an ordered tree whose internal nodes are numbered from 1 to n in preorder such that every node in T has a prescribed number of children. Recently, Wu et al. (2010) introduced a concise representation called RD-sequences to represent all non-regular trees and proposed a loopless algorithm for generating all non-regular trees in a Gray-code order. In this paper, based on such a Gray-code order, we show that a ranking algorithm can be done in quadratic time provided a preprocessing in advance.

Binary bubble languages and cool-lex order

A bubble language is a set of binary strings with a simple closure property: The first 01 of any string can be replaced by 10 to obtain another string in the set. Natural representations of many combinatorial objects are bubble languages. Examples include binary string representations of k-ary trees, unit interval graphs, linear-extensions of B-posets, binary necklaces and Lyndon words, and feasible solutions to knapsack problems. In co-lexicographic order, fixed-weight binary strings are ordered so that their suffixes of the form 10 i occur (recursively) in the order i = max , max − 1 , … , min + 1 , min for some values of max and min. In cool-lex order the suffixes occur (recursively) in the order max − 1 , … , min + 1 , min , max . This small change has significant consequences. We prove that the strings in any bubble language appear in a Gray code order when listed in cool-lex order. This Gray code may be viewed from two different perspectives. On one hand, successive binary strings differ by one or two transpositions, and on the other hand, they differ by a shift of some substring one position to the right. This article also provides the theoretical foundation for many efficient generation algorithms, as well as the first construction of fixed-weight binary de Bruijn sequences; results that will appear in subsequent articles.

Efficient generation of the ideals of a poset in Gray code order

Preusse and Ruskey [G. Pruesse, F. Ruskey, Gray codes from antimatroids, Order 10 (1993) 239–252] presented an algorithm for generating a Gray code for the ideals of a poset, where two adjacent ideals differ by one or two elements. Their algorithm takes O ( n ) amortized time per ideal. Squire [M. Squire, Enumerating the ideals of a poset, http://citeseer.ist.psu.edu/465417.html] presented a recurrence for the ideals of a poset that enabled him to find an algorithm for generating these ideals in O ( log n ) amortized time per ideal, but not in Gray code order. We use Squire's recurrence to find a Gray code for the ideals of a poset, where two adjacent ideals differ by one or two elements. In the worst case our algorithm has the same complexity as that of Pruesse and Ruskey and in the other cases its complexity is better and approaches that of Squire's algorithm.

Gray codes for reflectable languages

We classify a type of language called a reflectable language. We then develop a generic algorithm that can be used to list all strings of length n for any reflectable language in Gray code order. The algorithm generalizes Gray code algorithms developed independently for k-ary strings, restricted growth strings, and k-ary trees, as each of these objects can be represented by a reflectable language. Finally, we apply the algorithm to open meandric systems which can also be represented by a reflectable language.