How can I pair socks from a pile efficiently?

Question

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

Answer 1

This question is actually deeply philosophical. At heart it's about whether the power of people to solve problems (the "wetware" of our brains) is equivalent to what can be accomplished by algorithms.

An obvious algorithm for sock sorting is:

Let N be the set of socks that are still unpaired, initially empty
for each sock s taken from the dryer
  if s matches a sock t in N
    remove t from N, bundle s and t together, and throw them in the basket
  else
    add s to N

Now the computer science in this problem is all about the steps

1. "if s pairs with a sock t in N". How quickly can we "remember" what we've seen so far?
2. "remove t from N" and "add s to N". How expensive is keeping track of what we've seen so far?

Human beings will use various strategies to effect these. Human memory is associative, something like a hash table where feature sets of stored values are paired with the corresponding values themselves. For example, the concept of "red car" maps to all the red cars a person is capable of remembering. Someone with a perfect memory has a perfect mapping. Most people are imperfect in this regard (and most others). The associative map has a limited capacity. Mappings may bleep out of existence under various circumstances (one beer too many), be recorded in error ("I though her name was Betty, not Nettie"), or never be overwritten even though we observe that the truth has changed ("dad's car" evokes "orange Firebird" when we actually knew he'd traded that in for the red Camaro).

In the case of socks, perfect recall means looking at a sock s always produces the memory of its sibling t, including enough information (where it is on the ironing board) to locate t in constant time. A person with photographic memory accomplishes both 1 and 2 in constant time without fail.

Someone with less than perfect memory might use a few commonsense equivalence classes based on features within his capability to track: size (papa, mama, baby), color (greenish, redish, etc.), pattern (argyle, plain, etc.), style (footie, knee-high, etc.). So the ironing board would be divided into sections for the categories. This usually allows the category to be located in constant time by memory, but then a linear search through the category "bucket" is needed.

Someone with no memory or imagination at all (sorry) will just keep the socks in one pile and do a linear search of the whole pile.

A neat freak might use numeric labels for pairs as someone suggested. This opens the door to a total ordering, which allows the human to use exactly the same algorithms we might with a CPU: binary search, trees, hashes, etc.

So the "best" algorithm depends on the qualities of the wetware/hardware/software that is running it and our willingness to "cheat" by imposing a total order on pairs. Certainly a "best" meta-algorithm is to hire the worlds best sock-sorter: a person or machine that can aquire and quickly store a huge set N of sock attribute sets in a 1-1 associative memory with constant time lookup, insert, and delete. Both people and machines like this can be procured. If you have one, you can pair all the socks in O(N) time for N pairs, which is optimal. The total order tags allow you to use standard hashing to get the same result with either a human or hardware computer.