Prefer-same与Prefer-opposite de Bruijn序列的高效构造
Efficient constructions of the Prefer-same and Prefer-opposite de Bruijn sequences
AI总结：
本研究揭示Prefer-same de Bruijn序列的底层结构，提出每比特O(n)时间、O(n)空间的高效构造算法解决长期公开问题，并以类似方法实现Prefer-opposite de Bruijn序列的高效构造。
摘要：
贪心式Prefer-same de Bruijn序列构造法最早由Eldert等人提出[AIEE Transactions 77 (1958)]。作为一种贪心算法，它有一个主要缺点：需要指数级空间来存储长度为$2^n$的de Bruijn序列。尽管de Bruijn序列在过去60年里得到了广泛研究，但寻找Prefer-same de Bruijn序列的高效构造方法仍是一个极具吸引力的公开问题。本文揭示了Prefer-same de Bruijn序列的底层结构，并通过提出一种每比特仅需$O(n)$时间、仅占$O(n)$空间的高效构造算法解决了该公开问题。采用类似方法，我们还提出了一种构造Prefer-opposite de Bruijn序列的高效算法。
英文摘要：
The greedy Prefer-same de Bruijn sequence construction was first presented by Eldert et al.[AIEE Transactions 77 (1958)]. As a greedy algorithm, it has one major downside: it requires an exponential amount of space to store the length $2^n$ de Bruijn sequence. Though de Bruijn sequences have been heavily studied over the last 60 years, finding an efficient construction for the Prefer-same de Bruijn sequence has remained a tantalizing open problem. In this paper, we unveil the underlying structure of the Prefer-same de Bruijn sequence and solve the open problem by presenting an efficient algorithm to construct it using $O(n)$ time per bit and only $O(n)$ space. Following a similar approach, we also present an efficient algorithm to construct the Prefer-opposite de Bruijn sequence.
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