How do I implement the Union-Find algorithm?

Can you explain what the Union-Find algorithm is?

Sure! The Union-Find algorithm, also known as Disjoint Set Union (DSU), is a data structure that keeps track of a partition of a set into disjoint subsets. It supports two main operations: union and find.

How do I implement it in Python?

You can implement it using two main arrays: one for the parent pointers and another for the rank. Would you like to see a code example?

Yes, please provide a code example.

Here's a simple implementation:

class UnionFind:
    def __init__(self, size):
        self.parent = list(range(size))
        self.rank = [1] * size

    def find(self, p):
        if self.parent[p] != p:
            self.parent[p] = self.find(self.parent[p])
        return self.parent[p]

    def union(self, p, q):
        rootP = self.find(p)
        rootQ = self.find(q)
        if rootP != rootQ:
            if self.rank[rootP] > self.rank[rootQ]:
                self.parent[rootQ] = rootP
            elif self.rank[rootP] < self.rank[rootQ]:
                self.parent[rootP] = rootQ
            else:
                self.parent[rootQ] = rootP
                self.rank[rootP] += 1

What are the time complexities for the operations?

The time complexity for both the find and union operations is nearly constant, specifically O(α(n)), where α is the inverse Ackermann function, which grows very slowly.