Efficient Computation of Middle Levels Gray Codes

For any integer n ≥ 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 ≥ 1 has been the subject of intensive research during the last 30 years, and has been answered affirmatively only recently [T. Mütze. Proof of the middle levels conjecture. arXiv:1404.4442, 2014]. 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 ` bitstrings in the Gray code in time O(n`(1 + n` )), which is O(n) on average per bitstring provided that ` = Ω(n).