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I'm looking for an algorithm that generates all k-combinations of a set, such that each successive combination generated differs as much as possible (or in practice, a lot) from all previous combinations, or alternatively, the combinations appeared to be in a random order (actual randomness is not required).

For example, given the set {1,2,3,4,5,6}, I might get something along the lines of:

(Max-Difference)
{1, 2, 3}
{4, 5, 6}
{1, 3, 5}
{2, 4, 6}
{1, 2, 4}
...

or 

(Random)
{1, 2, 5}
{2, 3, 4}
{1, 5, 6}
{1, 4, 6}
...

I'm aware that using Gray codes would effectively give me the opposite of what I want... unfortunately, it's not obvious to me how I would be able to reverse the Gray code algorithm to produce something similar to my desired output.

Is there a known algorithm that would suit my needs?

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  • $\begingroup$ "all k-combinations of a set". Since (n, k) Gray codes is about permutations, can you confirm that you are talking about subset here? "Each successive combination generated differs as much as possible". Can you give a numerical definition of the difference between two combinations? $\endgroup$ – Apass.Jack Jan 4 at 16:26
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if you have a set that has n elements, you can dedicate the numbers from 0 to $2^n-1$ to your set in binary form.for example if you have a set consists of 3 elements,you have the numbers $$000,001,010,011,100,101,110,111$$ that these are the numbers 0 to $2^n-1$ in binary form.then dedicate each digit in a number to the K'th element in your set.for example if the number is 011 and your set consists of {1,2,3},then in this combination you have the subset:{2,3}. now if you want to have max-differing order in your combination,you can use n bit gray code if you have n elements in the set and generate all the combinations by dedicating each digit of a gray code to an element respectively like the binary form I had show above.

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