Whats the best algorithm for Non Disjoint Set Union?

Lets say there are two (non disjoint) sets of points (c开发者_如何转开发artesian space), what is the best case complexity algorithm to perform the union of the two sets ?

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Since the point coordinates are arbitrary and there is no special relation between them, I don't see this problem as a geometric specific problem. It is the generic problem of efficiently merging S1 and S2 into a new set S.

I know about two options:

1) When the sets are stored in a hash table (actually a hash set), the union takes O(|S1|+|S2|) in average.

2) If you store the structures in a balanced search tree, you can achieve a worst case time of O(|S1| * Log(|S1|)), assuming that |S1|>|S2|.

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Actually, if the sets are in a balanced search tree, I think you can do it in O(|S2| (1+log (|S1|/|S2|)), where |S1|>=|S2|. This is optimal in the worst case.