Here are the operations that I'd like to perform on a hypothetical collection data structure that holds sets as its elements:
All the sets in question are subsets of a known finite set, say {0..10^4}.
Is there a way to do this efficiently?
Here is a recent paper on this problem: http://research.google.com/pubs/pub36974.html
Briefly, you cannot do much better than quadratic time in the worst case; but there are some tricks to speed it up in practice.
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