Union-Find Algorithm with Path Compression

Understanding the Union-Find Algorithm's Path Compression

A common approach for dividing a set of elements into disjoint subgroups is called the Union-Find algorithm. The method keeps track of which items belong to which subset and runs on a forest, which is a collection of trees. Path compression is one of the improvements that may be used on the Union-Find algorithm with the aim of lowering the height of the forest's trees. We will get into the specifics of path compression and how it impacts the Union-Find algorithm's characteristics in this post.

What is Path Compression?

The Union-Find algorithm employs a method called path compression to lower the height of the forest's trees. It is accomplished during the find procedure by re-attaching a node to the root of its related tree. In other words, while conducting a search operation, the algorithm will reattach all of the nodes along the path from the node to the root to the root itself in addition to returning the tree's root.

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As a result, there is a shorter distance between the node and the root, which lowers the height of the related tree.

How Does the Union-Find Algorithm's Behavior React to Path Compression?

Path compression has an impact on a number of Union-Find algorithm features. Let's examine each one in greater detail:

Rank of a Node

The rank of a node is no longer always equal to the height of the tree rooted at that node, to start. The rank is still a limit on the height of the matching tree, though.

Take the following instance into consideration to demonstrate. Assume we have a tree with nodes 2, 3, and 6 with a root that is 5. Each node's rank is initially 0. The method will reattach node 6 to the root when we do a find operation on it, producing the tree shown below: 5, 3, 1. The rank of the root is still equal to 2, but the height of the three nodes is now equal to 1.

Subtree Size

The fact that the matching subtree of any root node I of rank k of the Union-Find method has at least 2k elements is another crucial characteristic. When path compression is applied, this condition remains true. This is due to the fact that no root nodes are impacted by path compression. No matter how many times the find operation is applied to nodes in this subtree, if a node is a root and has a rank of k, then its related subtree has at least 2k elements.

Number of Nodes of Rank k

It is also true that there can only be n/2k nodes of level k in our forest. This characteristic results from the fact that merging two nodes of rank k-1 results in the creation of a new node of rank k. If there are too many nodes of rank k, the total number of elements in the forest would surpass n, which is illogical because each node of level k has at least 2k items.

Rank Increase When Going Up

Finally, it is also evident that the rank strictly rises as we climb the forest's trees. Prior to path compression, this property was true, and it is still true after path compression.

In order to minimize the height of the trees in the forest, the Union-Find algorithm employs a technique called path compression. Despite changing a number of algorithmic parameters, it nonetheless maintains the crucial characteristics that make the Union-Find algorithm a valuable one.