Is Disjoint Set a Data Structure?


Angela Bailey

Is Disjoint Set a Data Structure?

A disjoint set is a data structure that keeps track of a set of elements partitioned into several disjoint subsets. It is commonly used to solve problems involving connectivity, such as finding connected components in a graph or checking if two elements belong to the same set.


In computer science, a data structure is a way of organizing and storing data in a computer’s memory so that it can be accessed and manipulated efficiently. Commonly used data structures include arrays, linked lists, stacks, queues, trees, and graphs. Each data structure has its own strengths and weaknesses and is suitable for solving specific types of problems.

Disjoint Set Data Structure

The disjoint set data structure provides an efficient way to perform operations on disjoint sets. It supports two main operations:

  • MakeSet(x): Creates a new set containing the element x.
  • Find(x): Returns the representative element of the set containing x.

The representative element of each set is used to determine whether two elements belong to the same set or not. If two elements have the same representative element, they are in the same set; otherwise, they are in different sets.

The disjoint set data structure also supports an additional operation:

  • Union(x, y): Merges the sets containing elements x and y into a single set.

This operation modifies the underlying partitioning of elements into sets by merging two sets into one. The representative element of one of the sets becomes the new representative element for all elements in both sets.


The disjoint set data structure has various applications, including:

  • Connected Components in Graphs: It can be used to find connected components in an undirected graph efficiently. Each connected component corresponds to a separate set in the disjoint set data structure.
  • Dynamic Connectivity: It can determine whether two elements are connected or not in a dynamic graph where edges can be added or removed.
  • Maze Generation: It can be used to generate random mazes by creating sets for each cell and merging adjacent cells as paths are carved.


The disjoint set data structure can be implemented using various techniques, such as:

  • Array Representation: Each element is stored as an index in an array, and the value at each index represents the parent of that element. The representative element of a set is the root of its corresponding tree.
  • Rank-Based Union by Rank and Path Compression: This optimization technique improves the efficiency of the union operation by always merging smaller trees under larger ones. Path compression is applied during the find operation to flatten the tree structure.


The disjoint set data structure is an essential tool for solving problems involving connectivity. Its efficient operations make it suitable for various applications, such as finding connected components in graphs, determining dynamic connectivity, and generating mazes. By understanding its implementation techniques, you can leverage the power of this data structure to solve complex problems efficiently.

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