Cut-down de Bruijn sequences

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2024-08-14 DOI:10.1016/j.disc.2024.114204

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引用次数: 0

Abstract

A cut-down de Bruijn sequence is a cyclic string of length L, where $1 \leq L \leq k^{n}$ , such that every substring of length n appears at most once. Etzion [Theor. Comp. Sci 44 (1986)] introduced an algorithm to construct binary cut-down de Bruijn sequences requiring $o (n)$ simple n-bit operations per symbol generated. In this paper, we simplify the algorithm and improve the running time to $O (n)$ time per symbol generated using $O (n)$ space. Additionally, we develop the first successor-rule approach for constructing a binary cut-down de Bruijn sequence by leveraging recent ranking/unranking algorithms for fixed-density Lyndon words. Finally, we develop an algorithm to generate cut-down de Bruijn sequences for $k > 2$ that runs in $O (n)$ time per symbol using $O (n)$ space after some initialization.

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删减的德布鲁因序列

删减德布鲁因序列是长度为 L（其中 1≤L≤kn ）的循环字符串，长度为 n 的每个子串最多出现一次。Etzion [Theor. Comp. Sci 44 (1986)]介绍了一种构建二进制削减德布鲁因序列的算法，每个符号的生成需要 o(n) 个简单的 n 位运算。在本文中，我们简化了该算法，并将运行时间改进为使用 O(n) 空间生成每个符号需要 O(n) 时间。此外，我们利用最近针对固定密度 Lyndon 字的排序/解排序算法，开发了第一种用于构建二进制删减 de Bruijn 序列的后继规则方法。最后，我们开发了一种生成 k>2 cut-down de Bruijn 序列的算法，该算法在初始化后使用 O(n) 空间，每个符号只需运行 O(n) 时间。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.