Disjoint Set Union

Union–Find, watched step by step

A live trace of the classic DSU with path compression and union by rank. Every find lights its path to the root, then flattens it; every unite hangs the shorter tree under the taller one. Watch the parent[] and rnk[] arrays move in lockstep with the forest.

Forest 1 set
set root on current find path same fill = same set arrow → points to parent
State arrays
parent[i] — who i points at (i itself ⇒ root)
rnk[i] — rank; only meaningful at roots
Operations
1×

Tip: unite the same pair twice — the second call returns false.

Console
C++ source › 
struct DSU {
  vector<int> parent, rnk;
  DSU(int n) : parent(n), rnk(n, 0) {
    iota(parent.begin(), parent.end(), 0);
  }
  int find(int x) {
    return parent[x] == x ? x
         : parent[x] = find(parent[x]);  // path compression
  }
  bool unite(int a, int b) {   // false if already joined
    a = find(a); b = find(b);
    if (a == b) return false;
    if (rnk[a] < rnk[b]) swap(a, b);
    parent[b] = a;
    if (rnk[a] == rnk[b]) rnk[a]++;
    return true;
  }
  bool connected(int a, int b) {
    return find(a) == find(b);
  }
};