A partition of a set S is a separation of the set into an arbitrary number of no

Question

A partition of a set S is a separation of the set into an arbitrary number of non-empty, pairwise disjoint subsets whose union is exactly S. What manner of a data structure should be used to represent a partition of a set if the following methods need to be optimized:

moving an element from one part to another, possibly an entirely new one, and
iterating over the parts of the partition.

A naive way of prioritizing 1 would be a hash/tree/whatever mapping from set elements to "part labels", but iterating over the parts would require O(N) for first constructing the actual parts from the labels. 2 is naively prioritized as a hash/tree/whatever set of hash/tree/whatever sets, but then moving elements around, especially to new subsets, incurs that memory management overhead.

Is there a way to get the best of both worlds? The implementation I need is Python but I imagine this is a cross-language question.

Explanation / Answer

I'm not familiar with the topic, so all I can do is give a few pointers. Manipulating partitions of finite sets is known by several names:

union-find when you start with singletons and progressively merge sets;
partition refinement when you start with a single set and progressively split it;
union-split-find when sets can be both merged and split as in your case.

Katherine J. Lai's discusses the union-split-find problem and proposes an algorithm using B-trees that double as finger trees to represent each set. Each element is represented by a distinct integer; using consecutive values for elements that are likely to remain together is more efficient. To merge two sets, split them into non-overlapping intervals that are entirely contained in one of the sets, and join the pieces together. See her thesis for details.

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