A Disjoint-Set data structure (aka Union-Find w/ Rank)

What is Union-Find?

Suppose you have a collection S of elements e1, e2, ..., en, and wish to group them into different collections using operations:

• "put ei and ej into the same group" (union),
• "give me a representative of the group ei belongs to" (find).

Then a Union-Find data structure helps to store the underlying groups very efficiently and implements this API.

Note: The variant implemented uses Path Compression to further improve the performance.

(Some) Applications

• Detect Cycles in Graph: Given a graph G, we can put the endpoints of edges into the same group (same connected component) unless there is a pair of endpoints (ei, ej) that share a group representative. If that happens, there was already a path existing between them, and adding this edge will add multiple paths, which cannot be the case for acyclic graphs.

• Number of connected components in Graph: Given a graph G, put the endpoints of edges into the same group (same connected component). Once all nodes are exhausted, the number of groups formed is the number of connected components in G.

Some interesting lecture notes regarding Union-Find.

Usage

Setup

Using any of the package installers, install pyreunion from the PyPi.

For example, pip install pyreunion.

API

Example 1

Task: Create a UnionFind data structure of arbitrary size that contains usize at its elements. Then, union a few elements and capture the state of the data structure after that.

Solution:

import pyreunion


def main():

    # Create an empty UnionFind data structure.
    uf = pyreunion.UnionFind()

    print("Initial state:", uf.str())
    print("All elements form their own group (singletons).")
    print(uf.subsets())

    uf.union(2, 1)
    print("After combining the groups that contains 2 and 1:", uf.str())

    uf.union(4, 3)
    print("After combining the groups that contains 4 and 3:", uf.str())

    uf.union(6, 5)
    print("After combining the groups that contains 6 and 5:", uf.str())

    hs1 = {1, 2}
    hs2 = {3, 4}
    hs3 = {5, 6}

    subsets = uf.subsets()
    assert (len(subsets) == 3)

    assert (hs1 in subsets)
    assert (hs2 in subsets)
    assert (hs3 in subsets)

    uf.union(1, 5)

    print("After combining the groups that contains 1 and 5", uf.str())

    subsets = uf.subsets()
    assert (len(subsets) == 2)

    for x in hs1:
        hs3.add(x)

    assert (hs3 in subsets)
    assert (hs2 in subsets)

    # It is possible to iterate over the subsets.
    for partition in uf.subsets():
        print(partition)


if __name__ == '__main__':
    main()

Performance

The underlying implementation uses Path Compression and is written in Rust. The implementation and some performance statistics are available here.