Generation of Well-Formed Parenthesis Strings in Constant Worst-Case Time

doi:10.1006/jagm.1998.0960

Journal of Algorithms

Volume 29, Issue 1, October 1998, Pages 165-173

https://doi.org/10.1006/jagm.1998.0960 Get rights and content

Abstract

Proskurowski and Ruskey (J. Algorithms11(1990), 68–84) published a recursive algorithm for generating well-formed parenthesis strings of length 2nand challenged the reader to find a loop-free version of their algorithm. We present two nonrecursive versions of their algorithm, one of which generates each string inO(n) worst-case time and requires space for onlyO(1) extra integer variables, and the other generates each string inO(1) worst-case time and usesO(n) extra space.

References (9)

J.M. Lucas et al.
On rotations and the generation of binary trees
J. Algorithms
(1993)
D.R. van Baronaigien
A loopless algorithm for generating binary tree sequences
Inform. Process. Lett.
(1991)
J.R. Bitner et al.
Efficient generation of the binary reflected Gray code and its applications
Comm. ACM
(1976)
P.J. Chase
Combination generation and graylex ordering, Proceedings of the 18th Manitoba Conference on Numerical Mathematics and Computing, Winnipeg, 1988
Congressus Numerantium
(1989)

There are more references available in the full text version of this article.

Cited by (14)

A minimum-change version of the Chung–Feller theorem for Dyck paths
2018, European Journal of Combinatorics
A Dyck path of length $2 k$ with $e$ flaws is a path in the integer lattice that starts at the origin and consists of $k$ many $↗$ -steps and $k$ many $↘$ -steps that change the current coordinate by $(1, 1)$ or $(1, - 1)$ , respectively, and that has exactly $e$ many $↘$ -steps below the line $y = 0$ . Denoting by $D_{2 k}^{e}$ the set of Dyck paths of length $2 k$ with $e$ flaws, the well-known Chung–Feller theorem asserts that the sets $D_{2 k}^{0}, D_{2 k}^{1}, \dots, D_{2 k}^{k}$ all have the same cardinality $\frac{1}{k + 1} (\binom{2 k}{k}) = C_{k}$ , the $k$ th Catalan number. The standard combinatorial proof of this classical result establishes a bijection $f^{'}$ between $D_{2 k}^{e}$ and $D_{2 k}^{e + 1}$ that swaps certain parts of the given Dyck path $x$ , with the effect that $x$ and $f^{'} (x)$ may differ in many positions. In this paper we strengthen the Chung–Feller theorem by presenting a simple bijection $f$ between $D_{2 k}^{e}$ and $D_{2 k}^{e + 1}$ which has the additional feature that $x$ and $f (x)$ differ in only two positions (the least possible number). We also present an algorithm that allows to compute a sequence of applications of $f$ in constant time per generated Dyck path. As an application, we use our minimum-change bijection $f$ to construct cycle-factors in the odd graph $O_{2 k + 1}$ and the middle levels graph $M_{2 k + 1}$ – two intensively studied families of vertex-transitive graphs – that consist of $C_{k}$ many cycles of the same length.
A minimum-change version of the Chung-Feller theorem for Dyck paths
2017, Electronic Notes in Discrete Mathematics
A Dyck path of length 2k with e flaws is a path in the integer lattice that starts at the origin and consists of k many ↗-steps and k many ↘-steps that change the current coordinate by (1, 1) or (1, −1), respectively, and that has exactly e many ↘-steps below the line $y = 0$ . Denoting by $D_{2 k}^{e}$ the set of Dyck paths of length 2k with e flaws, the well-known Chung-Feller theorem asserts that the sets $D_{2 k}^{0}, D_{2 k}^{1}, \dots, D_{2 k}^{k}$ all have the same cardinality $\frac{1}{k + 1} (\begin{matrix} 2 k \\ k \end{matrix}) = C_{k}$ , the k-th Catalan number. The standard combinatorial proof of this classical result establishes a bijection $f^{'}$ between $D_{2 k}^{e}$ and $D_{2 k}^{e + 1}$ that swaps certain parts of the given Dyck path x, with the effect that x and $f^{'} (x)$ may differ in many positions. In this work we strengthen the Chung-Feller theorem by presenting a simple bijection f between $D_{2 k}^{e}$ and $D_{2 k}^{e + 1}$ which has the additional feature that x and f(x) differ in only two positions (the least possible number). We also present an algorithm that allows to compute a sequence of applications of f in constant time per generated Dyck path. As an application, we use our minimum-change bijection f to construct cycle-factors in the odd graph $O_{2 k + 1}$ and the middle levels graph $M_{2 k + 1}$ — two intensively studied families of vertex-transitive graphs — that consist of $C_{k}$ many cycles of the same length.
A loop-free two-close Gray-code algorithm for listing k-ary Dyck words
2006, Journal of Discrete Algorithms
Citation Excerpt :
The terms minimal, homogeneous and 2-close will mean the same thing for Dyck words and their generalizations as they do for combinations. A minimal Gray code for Dyck words was published by Ruskey and A. Proskurowski [16]; a non-recursive description and a loop-free implementation of this Gray code appears in [20]. A homogeneous Gray code for Dyck words was discovered by B. Bultena and Ruskey [2].
P. Chase and F. Ruskey each published a Gray code for length n binary strings with m occurrences of 1, coding m-combinations of n objects, which is two-close—that is, in passing from one binary string to its successor a single 1 exchanges positions with a 0 which is either adjacent to the 1 or separated from it by a single 0. If we impose the restriction that any suffix of a string contains at least $k - 1$ times as many 0's as 1's, we obtain k-suffixes: suffixes of k-ary Dyck words. Combinations are retrieved as special case by setting $k = 1$ and k-ary Dyck words are retrieved as a special case by imposing the additional condition that the entire string has exactly $k - 1$ times as many 0's as 1's. We generalize Ruskey's Gray code to the first two-close Gray code for k-suffixes and we provide a loop-free implementation for $k ⩾ 2$ . For $k = 1$ we use a simplified version of Chase's loop-free algorithm for generating his two-close Gray code for combinations. These results are optimal in the sense that there does not always exist a Gray code, either for combinations or Dyck words, in which the 1 and the 0 that exchange positions are adjacent.
A Loopless Gray-Code Algorithm for Listing k-ary Trees
2000, Journal of Algorithms
The bit sequence representation for k-ary trees is a sequence b₁, b₂,…,b_nk + 1 of bits that is formed by doing a preorder traversal of the k-ary tree and writing a 1 when the visited subtree is not empty and a zero when the visited subtree is empty. The representation is well known and in the case of binary trees, the bit sequence representation also represents well-formed parenthesis strings. This paper presents the first loopless algorithm for listing the bit sequence representations of k-ary trees in a Gray-code order. The algorithm is simpler than existing loop free algorithms for both binary and k-ary trees. The algorithm is also the only known algorithm that generates the bit sequences by homogeneous transpositions.
Greedy Gray Codes for Dyck Words and Ballot Sequences
2024, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Combinatorial Gray codes-an updated survey
2022, arXiv

View all citing articles on Scopus

View full text

Regular ArticleGeneration of Well-Formed Parenthesis Strings in Constant Worst-Case Time

Abstract

J. Algorithms

Inform. Process. Lett.

Efficient generation of the binary reflected Gray code and its applications

Comm. ACM

Combination generation and graylex ordering, Proceedings of the 18th Manitoba Conference on Numerical Mathematics and Computing, Winnipeg, 1988

Congressus Numerantium

Regular Article
Generation of Well-Formed Parenthesis Strings in Constant Worst-Case Time