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Given a ring of size $n = 2^m$, starting with element $0$ to element $n-1$, what general formula gives the distance between two arbitrary elements $i$ and $j$? Note that the distance between the elements $0$ and $n-1$ is $1$.

There are a few cases...

  • Assume for instance that $i>j$ and $|i-j|\leq\frac{n}{2}=2^{m-1}$, we then have $distance_{n}(i,j) = |i-j|$

  • But for $\frac{n}{2} < |i-j| \leq n-1$, we need another formula. For e.g. let $n = 8$, then the distance between $1$ and $7$ is $1$, not $|7-1|=6$

What is the general, one-line formula for this problem?

Thank you!

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    $\begingroup$ Would $\min ( |i-j|,n+1-|i-j|)$ work? $\endgroup$ – coffeemath Feb 15 '15 at 11:34
  • $\begingroup$ @coffemath Yes, this is what I needed. Actually it is $\min ( |i-j|,n-|i-j|)$. You added 1 because I incorrectly gave you the distance between 1 and 7 as 2 while it is 1; you took it into account. Thank you! $\endgroup$ – Symeof Feb 15 '15 at 14:36
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Answer: $distance_{n}(i,j) = \min ( |i-j|,n-|i-j|)$

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