Union-copy structures and dynamic segment trees

The union-copy structure, a new data structure that generalizes the well-known union-find structure, is introduced. Besides the usual union and find operations, the new structure also supports a copy operation that generates a duplicate of a given set. The structure can enumerate a given set, find all sets that contain a given element, and insert and delete elements. All these operations can be performed very efficiently. The structure can be tuned as to obtain different trade-offs in the efficiency of the different operations. As an application of the union-copy structure, a dynamic version of the segment tree is given. Contrary to the classical semi-dynamic segment trees, the dynamic segment tree is not restricted to a fixed universe, from which the endpoints of the segments must be chosen. The tree allows for insertions, splits, and concatenations in O(log n)-time each. Deletions can be performed in slightly more time.