Link/cut tree
A link/cut tree is a data structure for representing a forest, a set of rooted trees, and offers the following operations:
Add a tree consisting of a single node to the forest.
Given a node in one of the trees, disconnect it (and its subtree) from the tree of which it is part.
Attach a node to another node as its child.
Given a node, find the root of the tree to which it belongs. By doing this operation on two distinct nodes, one can check whether they belong to the same tree.
The represented forest may consist of very deep trees, so if we represent the forest as a plain collection of parent pointer trees, it might take us a long time to find the root of a given node. However, if we represent each tree in the forest as a link/cut tree, we can find which tree an element belongs to in O(log(n)) amortized time. Moreover, we can quickly adjust the collection of link/cut trees to changes in the represented forest. In particular, we can adjust it to merge (link) and split (cut) in O(log(n)) amortized time.
