What Is Disjoint-Set Data Structure Operation?


Larry Thompson

What Is Disjoint-Set Data Structure Operation?

A disjoint-set data structure, also known as a union-find data structure, is a data structure that keeps track of a collection of disjoint (non-overlapping) sets. It provides efficient operations to perform union (merge) and find operations on these sets.

Union Operation

The union operation combines two sets into a single set by connecting the roots of the trees representing these sets. In other words, it merges two disjoint sets together.


  1. Find the root of the first set.
  2. Find the root of the second set.
  3. Connect one root to the other, making one set a subset of the other.


Consider two disjoint sets: Set A = {1, 2, 3} and Set B = {4, 5}. After performing the union operation on these sets, they will be merged into a single set: {1, 2, 3, 4, 5}.

Find Operation

The find operation determines which set an element belongs to. It returns the representative (root) element of the set that contains the given element.


  1. If the element is its own root, return it as the representative of its set.
  2. If not, recursively find its root until reaching an element whose parent is itself (root).


Assume the disjoint set has the following elements: {1, 2, 3, 4, 5}. The find operation on element 3 will return its representative (root) element, which is also 3.


The disjoint-set data structure has various applications in computer science:

  • It is used in image segmentation algorithms to group pixels with similar characteristics together.
  • It can be applied to find connected components in a graph.
  • The disjoint-set data structure is also used in Kruskal’s algorithm for finding the minimum spanning tree of a graph.


The disjoint-set data structure is a powerful tool for managing collections of disjoint sets. It offers efficient union and find operations, making it valuable in various algorithms and applications. By understanding its operations and implementing them correctly, you can leverage this data structure to solve complex problems effectively.

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