content: // https://leetcode.com/problems/number-of-connected-components-in-an-undirected-graph
  // T: O(V + E * alpha(V)) alpha = inverse ackerman function
  // S: O(V)
  
  public class NumberOfConnectedComponentsInAnUndirectedGraph {
      private static final class DisjointSet {
          private final int[] root, rank;
  
          public DisjointSet(int size) {
              root = new int[size];
              rank = new int[size];
              for (int i = 0 ; i < size ; i++) {
                  root[i] = i;
                  rank[i] = 1;
              }
          }
  
          public int find(int num) {
              if (num == root[num]) {
                  return num;
              }
              return root[num] = find(root[num]);
          }
  
          public boolean areConnected(int x, int y) {
              return find(x) == find(y);
          }
  
          public void union(int x, int y) {
              final int rootX = find(x), rootY = find(y);
              if (rootX == rootY) {
                  return;
              }
              if (rank[rootX] < rank[rootY]) {
                  root[rootX] = rootY;
              } else if (rank[rootX] > rank[rootY]) {
                  root[rootY] = rootX;
              } else {
                  root[rootY] = rootX;
                  rank[rootX]++;
              }
          }
      }
  
      public int countComponents(int n, int[][] edges) {
          final DisjointSet disjointSet = new DisjointSet(n);
          for (int[] edge : edges) {
              if (!disjointSet.areConnected(edge[0], edge[1])) {
                  disjointSet.union(edge[0], edge[1]);
                  n--;
              }
          }
          return n;
      }
  }
  