Lecture 380:- Disjoint Subsets

Disjoint subsets, also known as disjoint sets or disjoint collections, are sets that have no elements in common. In other words, two or more subsets are disjoint if they do not share any elements.

In the context of data structures and algorithms, disjoint sets are often used to represent a partition of a set into non-overlapping subsets. This data structure is useful for solving problems that involve grouping elements into distinct sets and performing operations like merging two sets or finding the parent set of an element.

The disjoint-set data structure supports two main operations:

1. Union: Given two elements, it merges the sets to which these elements belong. It ensures that the merged set contains all the elements from both sets.
2. Find: Given an element, it returns the representative (root) element of the set to which the element belongs. The representative element is used to identify the entire set.

To efficiently implement these operations, various algorithms and data structures can be used, such as Union-Find (also known as Disjoint-Set Union) data structure, which utilizes techniques like path compression and union by rank to achieve nearly constant time complexity for each operation on average.

Disjoint subsets find applications in various algorithms and data structures, including Kruskal's algorithm for finding the Minimum Spanning Tree, cycle detection in graphs, and more. They play a crucial role in solving problems that involve grouping elements into disjoint sets and maintaining these sets under union and find operations.