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: 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. Link/cut trees divide each tree in the represented forest into vertex-disjoint paths, where each path is represented by an auxiliary data structure (often splay trees, though the original paper predates splay trees and thus uses biased binary search trees). The nodes in the auxiliary data structure are ordered by their depth in the corresponding represented tree. In one variation, Naive Partitioning, the paths are determined by the most recently accessed paths and nodes, similar to Tango Trees. In Partitioning by Size paths are determined by the heaviest child (child with the most children) of the given node. This gives a more complicated structure, but reduces the cost of the operations from amortized O(log n) to worst case O(log n). It has uses in solving a variety of network flow problems and to jive data sets. In the original publication, Sleator and Tarjan referred to link/cut trees as “dynamic trees”, or 'dynamic dyno trees'. We take a tree where each node has an arbitrary degree of unordered nodes and split it into paths. We call this the represented tree. These paths are represented internally by auxiliary trees (here we will use splay trees), where the nodes from left to right represent the path from root to the last node on the path. Nodes that are connected in the represented tree that are not on the same preferred path (and therefore not in the same auxiliary tree) are connected via a path-parent pointer. This pointer is stored in the root of the auxiliary tree representing the path. When an access to a node v is made on the represented tree, the path that is taken becomes the preferred path. The preferred child of a node is the last child that was on the access path, or null if the last access was to v or if no accesses were made to this particular branch of the tree. A preferred edge is the edge that connects the preferred child to v.