Efficient computation of middle levels Gray codes

TL;DR

This paper presents the first efficient algorithm for generating middle levels Gray codes, which are cyclic sequences of specific bitstrings, with optimal average time complexity per generated bitstring.

Contribution

It introduces the first efficient algorithm to compute middle levels Gray codes, improving the computational efficiency of generating these sequences.

Findings

Algorithm computes the next  bitstrings in  time  per bitstring for large 

Achieves  average time complexity per bitstring when  

Provides practical method for generating middle levels Gray codes efficiently

Abstract

For any integer $n\geq 1$ a middle levels Gray code is a cyclic listing of all bitstrings of length $2n+1$ that have either $n$ or $n+1$ entries equal to 1 such that any two consecutive bitstrings in the list differ in exactly one bit. The question whether such a Gray code exists for every $n\geq 1$ has been the subject of intensive research during the last 30 years, and has been answered affirmatively only recently [T. M\"utze. Proof of the middle levels conjecture. Proc. London Math. Soc., 112(4):677--713, 2016]. In this work we provide the first efficient algorithm to compute a middle levels Gray code. For a given bitstring, our algorithm computes the next $\ell$ bitstrings in the Gray code in time $\mathcal{O}(n\ell(1+\frac{n}{\ell}))$, which is $\mathcal{O}(n)$ on average per bitstring provided that $\ell=\Omega(n)$.