Union Find

Rahul Pandey | January 31, 2017

The Union Find (also called Disjoint Sets) data structure is a data structure to merge disjoint sets efficiently. The context is that you have a large number of items partitioned into several smaller sets. All items are part of exactly 1 set, and there is no overlap between any 2 sets. The supported operations are:


Two implementations of the data structures optimize for the efficiency of Find vs Union. With QuickFind, find is constant time, while union is linear time. This can be implemented simply with an array. Each item is labelled as an index in the array, and the value in the array is the set that the item belongs to. This allows constant time lookup for which set an item belongs to, but linear time to combine two sets, since we have to walk through every item in the array to (potentially) update the value.

The more interesting implementation is QuickUnion, in which union is constant time, while find is log(number unions) done. The usual way to implement this is using a (non-binary) tree. Each set is a separate tree, and the set is identified by the root (“canonical node”). To union two sets, the smaller tree becomes a subtree of the other, which makes it an O(1) operation.

Find can be done in O(log u) time, where u is the number of unions. To find an item in this implementation, represent the tree as an array where positive numbers indicate the parent node, and negative numbers represent the size of the set rooted at that node. (Path compression is an optimization.) This grows proportional to the inverse Ackermann function, which is practically a constant ~3. However, that is a topic for another blog post.

Real World Application

Disjoint-set data structures are helpful when we want to track partitions of a set. For example, given a graph, we can use the Union Find algorithm to determine whether two vertices belong to the same component. It is also used to implement Kruskal’s algorithm to find the minimum spanning tree of a graph.