Generating Combinations by Three Basic Operations

Cheng, Yongxi

doi:10.1007/s11390-007-9094-7

Cited by 4 publications

(2 citation statements)

References 12 publications

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“…For example, see Table 2.1 (iv). It was proven that σ 2 and σ n cannot be used to create a Gray code for B w (n) by Cheng [3]. The sufficiency of σ n−1 and σ n , and the insufficiency of σ 2 and σ n , are two special cases of a general question asked in [21]: Which sets of σ i are necessary and sufficient for generating a (cyclic) Gray code for B w (n)?…”

Section: The Set B Wmentioning

confidence: 99%

“…The circular string in Figure 1.2 is a fixed-weight de Bruijn sequence for B 3 (5). Its substrings of length four include each string in B 3 2 (4) exactly once; appending the 'missing' bit extends each substring to a unique string in B 3 (5). In general, the shorthand sequence of a fixed-weight de Bruijn sequence for B w (n) is its circular sequence of substrings of length n − 1 and the longhand sequence is obtained by appending the missing bit to each string in the shorthand sequence so that each resulting string has weight w.…”

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De Bruijn Sequences for Fixed-Weight Binary Strings

Sawada¹,

Williams²

2012

SIAM J. Discrete Math.

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De Bruijn sequences are circular strings of length 2 n whose length n substrings are the binary strings of length n. Our focus is on creating circular strings of length n w for the binary strings of length n with weight (number of 1s) equal to w. In this case, each fixed-weight string can be encoded by its first n−1 bits since the final bit is redundant. For this reason, we construct circular strings of length n−1 w + n−1 w−1 whose length n−1 substrings are the binary strings of length n−1 with weight w or w−1. Our construction is reminiscent of the construction for the lexicographically least de Bruijn sequence, except the underlying algorithm is applied to cool-lex order instead of lexicographic order. The construction can be efficiently implemented so that successive blocks of n bits are generated in constant amortized time (CAT) while using O(n log n)-space. This article's results were also used to create de Bruijn sequences for binary strings of length n with a specified maximum weight.

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Section: The Set B Wmentioning

confidence: 99%

mentioning

confidence: 99%

De Bruijn Sequences for Fixed-Weight Binary Strings

Sawada¹,

Williams²

2012

SIAM J. Discrete Math.

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Constructing the first (and coolest) fixed-content universal cycle

Sawada

Williams

2022

Algorithmica

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Solving the Sigma-Tau Problem

Sawada

Williams

2019

ACM Trans. Algorithms

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Knuth assigned the following open problem a difficulty rating of 48/50 in The Art of Computer Programming Volume 4A : For odd n ≥ 3, can the permutations of { 1,2,… , n } be ordered in a cyclic list so that each permutation is transformed into the next by applying either the operation σ, a rotation to the left, or τ, a transposition of the first two symbols? The Sigma-Tau problem is equivalent to finding a Hamilton cycle in the directed Cayley graph generated by σ = (1 2 ⋅ n ) and τ = (1 2). In this article, we solve the Sigma-Tau problem by providing a simple O ( n )-time successor rule to generate successive permutations of a Hamilton cycle in the aforementioned Cayley graph.

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Generating Combinations by Three Basic Operations

Cited by 4 publications

References 12 publications

De Bruijn Sequences for Fixed-Weight Binary Strings

De Bruijn Sequences for Fixed-Weight Binary Strings

Constructing the first (and coolest) fixed-content universal cycle

Solving the Sigma-Tau Problem

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