## How to pair socks from a pile efficiently?

Yesterday I was pairing the socks from the clean laundry and figured out the way I was doing it is not very efficient. I was doing a naive search — picking one sock and "iterating" the pile in order to find its pair. This requires iterating over n/2 * n/4 = n^{2}/8 socks on average.

As a computer scientist I was thinking what I could do? Sorting (according to size/color/...) of course came to mind to achieve an O(NlogN) solution.

Hashing or other not-in-place solutions are not an option, because I am not able to duplicate my socks (though it could be nice if I could).

**So, the question is basically:**

Given a pile of n pairs of socks, containing 2n elements (assume each sock has exactly one matching pair), what is the best way to pair them up efficiently with up to logarithmic extra space? (I believe I can remember that amount of info if needed.)

I will appreciate an answer that addresses the following aspects:

- A general
*theoretical*solution for a huge number of socks. - The actual number of socks is not that large, I don't believe my spouse and I have more than 30 pairs. (And it is fairly easy to distinguish between my socks and hers; can this be used as well?)
- Is it equivalent to the element distinctness problem?