How to create a union-find data structure, in Java?

A union-find data structure, also known as a disjoint-set data structure, is a data structure that keeps track of a set of elements partitioned into a number of disjoint (non-overlapping) subsets. It supports two main operations:

1. find: determine which subset a particular element is in.
2. union: join two subsets into a single subset.

To create a union-find data structure in Java, you can define a class that implements the UnionFind interface, which defines the find and union operations.

Here is an example of how you can create a union-find data structure in Java:

import java.util.Arrays;

public class UnionFind {
  private int[] parents;  // Parent of each element
  private int[] ranks;  // Rank of each element

  // Initialize the union-find data structure with n elements
  public UnionFind(int n) {
    parents = new int[n];
    ranks = new int[n];
    for (int i = 0; i < n; i++) {
      parents[i] = i;  // Set the parent of each element to itself
      ranks[i] = 0;  // Set the rank of each element to 0
    }
  }

  // Find the parent of element x
  public int find(int x) {
    if (parents[x] != x) {
      // Set the parent of x to the parent of its parent
      parents[x] = find(parents[x]);
    }
    return parents[x];
  }

  // Join the sets containing element x and element y
  public void union(int x, int y) {
    int xParent = find(x);
    int yParent = find(y);
    if (xParent == yParent) {
      // x and y are already in the same set
      return;
    }

    // Set the parent of y to x if x has a higher rank
    // Otherwise, set the parent of x to y
    if (ranks[xParent] < ranks[yParent]) {
      parents[xParent] = yParent;
    } else if (ranks[xParent] > ranks[yParent]) {
      parents[yParent] = xParent;
    } else {
      // x and y have the same rank, so set the parent of y to x
      // and increment the rank of x
      parents[yParent] = xParent;
      ranks[xParent]++;
    }
  }

  // Check if element x is in the same set as element y
  public boolean isConnected(int x, int y) {
    return find(x) == find(y);
  }

  @Override
  public String toString() {
    return Arrays.toString(parents);
  }
}

This class defines a union-find data structure with n elements, where n is the number of elements passed to the constructor. It stores the parent of each element in an array called parents, and the rank of each element in an array called ranks. The find method returns the parent of an element by recursively calling itself on the parent until it reaches the root element, which is an element whose parent is itself.