// check optimality conditions (takes time proportional to E V lg* V)
  private boolean check(EdgeWeightedGraph G) {

    // check total weight
    double total = 0.0;
    for (Edge e : edges()) {
      total += e.weight();
    }
    double EPSILON = 1E-12;
    if (Math.abs(total - weight()) > EPSILON) {
      System.err.printf("Weight of edges does not equal weight(): %f vs. %f\n", total, weight());
      return false;
    }

    // check that it is acyclic
    UF uf = new UF(G.V());
    for (Edge e : edges()) {
      int v = e.either(), w = e.other(v);
      if (uf.connected(v, w)) {
        System.err.println("Not a forest");
        return false;
      }
      uf.union(v, w);
    }

    // check that it is a spanning forest
    for (Edge e : edges()) {
      int v = e.either(), w = e.other(v);
      if (!uf.connected(v, w)) {
        System.err.println("Not a spanning forest");
        return false;
      }
    }

    // check that it is a minimal spanning forest (cut optimality conditions)
    for (Edge e : edges()) {
      int v = e.either(), w = e.other(v);

      // all edges in MST except e
      uf = new UF(G.V());
      for (Edge f : mst) {
        int x = f.either(), y = f.other(x);
        if (f != e) uf.union(x, y);
      }

      // check that e is min weight edge in crossing cut
      for (Edge f : G.edges()) {
        int x = f.either(), y = f.other(x);
        if (!uf.connected(x, y)) {
          if (f.weight() < e.weight()) {
            System.err.println("Edge " + f + " violates cut optimality conditions");
            return false;
          }
        }
      }
    }

    return true;
  }
  // Kruskal's algorithm
  public KruskalMST(EdgeWeightedGraph G) {
    // more efficient to build heap by passing array of edges
    MinPQ<Edge> pq = new MinPQ<Edge>();
    for (Edge e : G.edges()) {
      pq.insert(e);
    }

    // run greedy algorithm
    UF uf = new UF(G.V());
    while (!pq.isEmpty() && mst.size() < G.V() - 1) {
      Edge e = pq.delMin();
      int v = e.either();
      int w = e.other(v);
      if (!uf.connected(v, w)) { // v-w does not create a cycle
        uf.union(v, w); // merge v and w components
        mst.enqueue(e); // add edge e to mst
        weight += e.weight();
      }
    }

    // check optimality conditions
    assert check(G);
  }