A constant-time algorithm for middle levels Gray codes
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by
Torsten Mütze, Jerri Nummenpalo
2016
Abstract
For any integer n≥ 1 a middle levels Gray code is a cyclic listing of
all n-element and (n+1)-element subsets of {1,2,...,2n+1} such that
any two consecutive subsets differ in adding or removing a single element. The
question whether such a Gray code exists for any 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.
Proc. London Math. Soc., 112(4):677--713, 2016]. In a follow-up paper [T.
Mütze and J. Nummenpalo. An efficient algorithm for computing a middle levels
Gray code. To appear in ACM Transactions on Algorithms, 2018] this existence
proof was turned into an algorithm that computes each new set in the Gray code
in time O(n) on average. In this work we present an algorithm for
computing a middle levels Gray code in optimal time and space: each new set is
generated in time O(1), and the required space is O(n).
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