arXiv 2007.071642020-07math.COcs.DM

关于Knuth的一个组合生成问题

On a combinatorial generation problem of Knuth

Arturo Merino, Ondřej Mička, Torsten Mütze

摘要：

著名的中间层猜想断言，对于每一个整数$n\geq 1$，所有长度为$2(n+1)$且恰好有$n+1$多个0和1的二进制字符串都可以循环排序，这样任何两个连续的字符串在交换第一个比特时都会有所不同，而在后面的某个位置会有一个互补比特。在他的著作《计算机程序设计的艺术》（The Art of Computer Programming Vol. 4A）中，克努特提出了这个猜想的一个更强的形式（第7章第2.1.3节中的问题56），它要求在这种排序的每一步中交换第一位的位置序列具有相同长度的$2n+1$块，并且每个块都是通过将$s=1$（模$2n+1$）添加到前一个块中获得的。在这项工作中，我们证明了高德纳猜想在一个更一般的形式，允许任意移位$s\geq 1$是互质的$2n+1$。我们还提出了一个算法来计算这种排序，生成每个新的位串在$\mathcal{O}(n)$的时间，使用$\mathcal{O}(n)$的内存。

英文摘要：

The well-known middle levels conjecture asserts that for every integer $n\geq 1$, all binary strings of length $2(n+1)$ with exactly $n+1$ many 0s and 1s can be ordered cyclically so that any two consecutive strings differ in swapping the first bit with a complementary bit at some later position. In his book `The Art of Computer Programming Vol. 4A' Knuth raised a stronger form of this conjecture (Problem 56 in Chapter 7, Section 2.1.3), which requires that the sequence of positions with which the first bit is swapped in each step of such an ordering has $2n+1$ blocks of the same length, and each block is obtained by adding $s=1$ (modulo $2n+1$) to the previous block. In this work, we prove Knuth's conjecture in a more general form, allowing for arbitrary shifts $s\geq 1$ that are coprime to $2n+1$. We also present an algorithm to compute this ordering, generating each new bitstring in $\mathcal{O}(n)$ time, using $\mathcal{O}(n)$ memory in total.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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