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. 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".
Structure 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.
Preferred paths 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. In an alternate version, preferred paths are determined by the heaviest child.
Operations The operations we are interested in are FindRoot(Node v), Cut(Node v), Link(Node v, Node w), and Path(Node v). Every operation is implemented using the Access(Node v) subroutine. When we access a vertex v, the preferred path of the represented tree is changed to a path from the root R of the represented tree to the node v. If a node on the access path previously had a preferred child u, and the path now goes to child w, the old preferred edge is deleted (changed to a path-parent pointer), and the new path now goes through w.
Access After performing an access to node v, it will no longer have any preferred children, and will be at the end of the path. Since nodes in the auxiliary tree are keyed by depth, this means that any nodes to the right of v in the auxiliary tree must be disconnected. In a splay tree this is a relatively simple procedure; we splay at v, which brings v to the root of the auxiliary tree. We then disconnect the right subtree of v, which is every node that came below it on the previous preferred path. The root of the disconnected tree will have a path-parent pointer, which we point to v. We now walk up the represented tree to the root R, breaking and resetting the preferred path where necessary. To do this we follow the path-parent pointer from v (since v is now the root, we have direct access to the path-parent pointer). If the path that v is on already contains the root R (since the nodes are keyed by depth, it would be the left most node in the auxiliary tree), the path-parent pointer will be null, and we are done the access. Otherwise we follow the pointer to some node on another path w. We want to break the old preferred path of w and reconnect it to the path v is on. To do this we splay at w, and disconnect its right subtree, setting its path-parent pointer to w. Since all nodes are keyed by depth, and every node in the path of v is deeper than every node in the path of w (since they are children of w in the represented tree), we simply connect the tree of v as the right child of w. We splay at v again, which, since v is a child of the root w, simply rotates v to root. We repeat this entire process until the path-parent pointer of v is null, at which point it is on the same preferred path as the root of the represented tree R.
FindRoot FindRoot refers to finding the root of the represented tree that contains the node v. Since the access subroutine puts v on the preferred path, we first execute an access. Now the node v is on the same preferred path, and thus the same auxiliary tree as the root R. Since the auxiliary trees are keyed by depth, the root R will be the leftmost node of the auxiliary tree. So we simply choose the left child of v recursively until we can go no further, and this node is the root R. The root may be linearly deep (which is worst case for a splay tree), we therefore splay it so that the next access will be quick.
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