DISJOINT SETS
Disjoint sets, also known as partitions, are a useful and versatile data structure used in computer science. Disjoint sets are used to group together objects that are different, but related in some way. The main idea behind disjoint sets is to divide a set of objects into smaller sets that have no elements in common. This article will discuss the concept of disjoint sets, their applications, and the various algorithms used to manipulate them.
A disjoint set is defined as a collection of sets that are disjoint, meaning they have no elements in common. Each set in a disjoint set is called a class and the elements in each class are called members. Disjoint sets can be represented in a variety of ways, including lists, arrays, linked lists, and trees. For example, a set of four classes can be represented as:
Class 1: {a, b, c}
Class 2: {d, e, f}
Class 3: {g, h, i}
Class 4: {j, k, l}
Disjoint sets can be used to solve a variety of problems. One common application is in graph theory, where disjoint sets are used to represent the vertices and edges of a graph. Disjoint sets can also be used to solve problems involving the partitioning of data, such as image segmentation or clustering. In addition, disjoint sets can be used to represent hierarchical data, such as a family tree.
There are several algorithms that can be used to manipulate disjoint sets. These include union-find, path compression, and rank. The union-find algorithm is used to combine two classes in a disjoint set. Path compression is used to reduce the path length between two nodes in a graph. Lastly, rank is used to determine the relative size of two classes in a disjoint set.
In conclusion, disjoint sets are a powerful and versatile data structure used in many computer science applications. They can be represented in a variety of ways and manipulated with several algorithms. Disjoint sets can be used to represent graphs, partition data, and represent hierarchical data.
References
Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to algorithms. Cambridge, MA: MIT Press.
Kumar, S. (2018). Algorithms for disjoint sets. Retrieved from https://www.geeksforgeeks.org/algorithms-for-disjoint-set-data-structure/
Levin, S. (2016). A Disjoint Set Data Structure. Retrieved from https://www.cs.cmu.edu/~scandal/alg/set.html
Tims, J. (2019). Disjoint Sets. Retrieved from https://www.tutorialspoint.com/data_structures_algorithms/disjoint_sets_algorithm.htm