Is the following Greedy algorithm to generate Gray Codes always correct?

Question

I recently solved the basic problem of generating a n-bit Gray Code. The solution I used involved building larger-bit Gray Codes from smaller ones recursively (the solution I've seen on most websites).

However, I then had an idea for a Greedy Gray Code Algorithm: Start the Gray Code at 0 and at every step, find and change the single smallest bit such that we get a new, not-visited-before number. Keep on repeating this until we have exhausted all the 2^N possible numbers with N bits.

I was sure that this approach wouldn't work for some values of N, but so far, it has worked for all N from 1 to 30. This was quite surprising for me, especially as I haven't seen this Greedy approach mentioned anywhere as a solution to the Gray Code Problem.

So, my question is, does this Greedy algorithm for generating N-bit Gray Codes always work? Is there a proof of its correctness? Otherwise, what value of N causes this algorithm to not be able to generate a correct Gray Code?

Answer 1

Your construction results in the standard binary-reflected Gray code, a recursive construction of which is described on Wikipedia, for example. We can also describe it in a different way: to generate the $m$'th codeword from the $(m-1)$'th, flip the least significant 1 bit of $m$. For example, when $n=3$, we have: $$ \begin{array}{c|c} 000 & 000 \\ 00\textbf{1} & 00\textbf{1} \\ 0\textbf{1}1 & 0\textbf{1}0 \\ 01\textbf{0} & 01\textbf{1} \\ \textbf{1}10 & \textbf{1}00 \\ 11\textbf{1} & 10\textbf{1} \\ 1\textbf{0}1 & 1\textbf{1}0 \\ 10\textbf{0} & 11\textbf{1} \end{array} $$