• To represent network connectivity

In computer networks, disjoint sets are frequently used to illustrate network connectedness. Each network node or device is viewed as an element in this application, and the sets represent interconnected parts. When there is a link between the matching nodes, the 'Union' operation is used to combine two components, and the 'Find' operation is used to determine whether two nodes are part of the same component. Detecting network outages, locating related network components, and examining the general network structure are all made possible due to this.

• Image processing

Disjoint Sets can be utilized for picture segmentation in image processing. The sets represent areas or segments of the image, and each pixel in the image is handled as an element. In order to combine adjacent pixels with comparable characteristics, the 'Union' operation is used, and the 'Find' operation aids in determining to which segment a specific pixel belongs. This is helpful for many different things, including edge detection, object recognition, and image compression.

• To find least common ancestor

The least common ancestor (LCA) in a tree can be quickly found by using disjoint sets. Each node is handled as an element in this application, and the sets serve as the nodes' ancestors. In order to connect nodes to their parents, the 'Union' operation is employed, while the 'Find' operation aids in determining the LCA of two nodes by following their respective roots. This is helpful for many tree-based algorithms and data structures, including Binary Lifting and LCA tree queries.

• To define equivalence of finite state automata

In a Finite State Automaton (FSA), equivalence classes of states can be defined using disjoint sets. The 'Union' procedure is used to combine states that are equivalent between the sets, which represent various classes. To ascertain the representative state of each class, utilize the 'Find' operation. This is crucial in situations when two automata must recognize the same language, such as in language equivalence applications.

• Kruskal’s minimum spanning tree algorithm (graph theory)

A popular approach for determining the Minimum Spanning Tree (MST) of a graph is Kruskal's algorithm. Disjoint Sets are utilized in this approach to keep track of related parts. Each vertex is initially a distinct set. When edges are added to the MST, the 'Union' operation combines sets, and the 'Find' operation is used to determine whether adding an edge between two vertices will result in a cycle. The approach effectively identifies the minimum-weight edges that connect all vertices without forming a cycle, ensuring that the MST stays acyclic.

• In game algorithms

Different game algorithms can effectively handle and analyze game states by using disjoint sets. Disjoint Sets, for instance, can represent many game states in a two-player game where players take turns, and the 'Union' operation can be used to combine states with the same properties. The optimal course of action or strategy for the current situation can then be ascertained using the 'Find' operation. These methods are particularly helpful in games that incorporate state space exploration and sophisticated decision trees.