Introduction to Link-Cut Tree – GeeksforGeeks

Link Cut Trees (LCT) is a data structure that allows for efficient dynamic maintenance of trees. It is a type of self-adjusting data structure that allows for efficient manipulation of trees, such as link and cut operations, find-root, and access.

• The implementation of an LCT typically consists of a set of nodes, each representing a tree or a subtree, and a set of pointers linking the nodes together. 
• Each node contains two pointers, one pointing to its parent and one pointing to its child, and a value associated with the node.

The basic operations that can be performed on an LCT include:

1. Link(u, v): This operation creates a new edge between two nodes u and v, making u the parent of v.

2. Cut(u, v): This operation removes the edge between two nodes u and v, disconnecting v from its parent u.

3. Find-root(u): This operation finds the root of the tree that contains the node u.

3. Access(u): This operation returns the value associated with the node u, and also updates all the necessary tree information.

Characteristics of Link-Cut Trees:

• LCTs are useful in many algorithms such as dynamic connectivity, lowest common ancestor, and dynamic trees.
• The above is just a brief implementation of LCT with examples, But LCT is a bit more complex than that, it’s recommended to use a library or a pre-built class for LCT.
• In Python, there are several libraries available for implementing Link Cut Trees, such as the “lct” library and the “linkcuttree” library.
• The “lct” library is a small and simple library that provides basic LCT functionality, such as link, cut, find-root, and access operations.

Here’s an example of how to use the “lct” library to implement a LCT:

Another library “linkcut tree” is a more advanced library that provides additional functionality such as subtree size and path sum.

It’s worth noting that LCT is a complex data structure and it’s recommended to use a library or pre-built class to avoid errors and bugs, these libraries are available with clear documentation and examples.

Conclusion:

Link Cut Trees is a powerful data structure that allows for the efficient manipulation of trees. It is based on a set of nodes and pointers linking them together. It supports basic operations like link, cut, find-root, and access, but it’s a bit more complex than that, and a library or a pre-built class is recommended to use.