Binary bubble languages and cool-lex order

Sawada, Joe; Williams, Aaron

doi:10.1016/j.jcta.2011.07.005

Cited by 43 publications

(79 citation statements)

References 20 publications

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“…In Section 4 we show how the Gray code can be efficiently ranked and unranked. With respect to the existing literature these results include -the first prefix-shift Gray code for k-ary Dyck words [6], -the first loopless algorithm for generating k-ary Dyck words that uses O (1) additional index variables (when k is constant), -a simpler loopless algorithm for generating k-ary Dyck words using 1/3 the if-statements and additional arrays as [9] (when k is arbitrary), and -the first order of k-ary Dyck words that has a loopless generation algorithm as well as efficient ranking and unranking algorithms.…”

Section: New Resultsmentioning

confidence: 99%

“…This definition comes from Ruskey, Sawada, and Williams who showed that bubble languages generalize many combinatorial objects including binary necklaces and solutions to knapsack problems [6]. They substantially generalized Theorem 1 by proving that cool-lex order provides a cyclic Gray code for any bubble language.…”

Section: Bubble Languages and Cool-lex Ordermentioning

confidence: 99%

“…Table 2. Rules for generating a bubble language L from [6]. Strings change according to (2a)-(2e) by the specified shift or equivalent swap(s).…”

Section: Bubble Languages and Cool-lex Ordermentioning

confidence: 99%

“…Theorem 2 ( [6]). The strings in any bubble language are generated in cool-lex order by the shift (or equivalent swap(s)) in Table 2.…”

Section: Bubble Languages and Cool-lex Ordermentioning

confidence: 99%

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Ranking and Loopless Generation of k-ary Dyck Words in Cool-lex Order

Durocher

Mondal

et al. 2011

Lecture Notes in Computer Science

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Abstract. A binary string B of length n = kt is a k-ary Dyck word if it contains t copies of 1, and the number of 0s in every prefix of B is at most k−1 times the number of 1s. We provide two loopless algorithms for generating k-ary Dyck words in cool-lex order: (1) The first requires two index variables and assumes k is a constant; (2) The second requires t index variables and works for any k. We also efficiently rank k-ary Dyck words in cool-lex order. Our results generalize the "coolCat" algorithm by Ruskey and Williams (Generating balanced parentheses and binary trees by prefix shifts in CATS 2008) and provide the first loopless and ranking applications of the general cool-lex Gray code by Ruskey, Sawada, and Williams (Binary bubble languages and cool-lex order under review). Background k-ary Dyck WordsLet B(n, t) be the set of binary strings of length n containing t copies of 1. A string B ∈ B(kt, t) is a k-ary Dyck word if the number of 0s in each prefix is at most k−1 times the number of 1s. Let D k (t) be the set of k-ary Dyck words of length kt. For example, the k-ary Dyck words with k = t = 3 are given below D 3 (3) = {111000000, 110100000, 101100000, 110010000, 101010000, 100110000, 110001000, 101001000, 100101000, 110000100, 101000100, 100100100}.The k-ary Dyck words of length kt have simple bijections with a number of combinatorial objects including k-ary trees with t internal nodes [2] [3]. The 2-ary Dyck words are known as balanced parentheses when 1 and 0 are replaced by '(' and ')' respectively, and the cardinality of D 2 (t) is the tth Catalan number. A simple property of k-ary Dyck words is that they can be "separated" according to the following remark. We let αβ denote the concatenation of the binary strings α and β, and we say that α and β have the same content if they have equal length and an equal number of 1s. Remark 1. If αβ, γδ ∈ D k (t) and α and γ have the same content, then αδ, βγ ∈ D k (t). In other words, prefixes (or suffixes) of k-ary Dyck words with the same content can be separated and recombined.

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Section: New Resultsmentioning

confidence: 99%

Section: Bubble Languages and Cool-lex Ordermentioning

confidence: 99%

“…Table 2. Rules for generating a bubble language L from [6]. Strings change according to (2a)-(2e) by the specified shift or equivalent swap(s).…”

Section: Bubble Languages and Cool-lex Ordermentioning

confidence: 99%

“…Theorem 2 ( [6]). The strings in any bubble language are generated in cool-lex order by the shift (or equivalent swap(s)) in Table 2.…”

Section: Bubble Languages and Cool-lex Ordermentioning

confidence: 99%

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Ranking and Loopless Generation of k-ary Dyck Words in Cool-lex Order

Durocher

Mondal

et al. 2011

Lecture Notes in Computer Science

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“…List the necklaces of length n + 1 and weight d in reverse cool-lex order [21], 2. Append the aperiodic prefixes of the necklaces together to get the string UC d d−1 .…”

Section: Even Weight-rangementioning

confidence: 99%

Universal Cycles for Weight-Range Binary Strings

Sawada

Williams

Wong

2013

Lecture Notes in Computer Science

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Abstract. We present an efficient universal cycle construction for the set of binary strings of length n whose weight (number of 1s) are in the range c, c+1, . . . , d where 0 ≤ c < d ≤ n. The construction can be implemented to generate each character in constant amortized time using O(n) space which based on a simple lemma for gluing universal cycles together. The Gluing lemma can also be applied to construct universal cycles for other combinatorial objects including passwords and labeled graphs.

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Combinatorial Gray Code

Ruskey

2016

Encyclopedia of Algorithms

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Binary bubble languages and cool-lex order

Cited by 43 publications

References 20 publications

Ranking and Loopless Generation of k-ary Dyck Words in Cool-lex Order

Ranking and Loopless Generation of k-ary Dyck Words in Cool-lex Order

Universal Cycles for Weight-Range Binary Strings

Combinatorial Gray Code

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