// based on: https://algs4.cs.princeton.edu/33balanced/RedBlackBST.java.html

sclass IntPairTreeSet extends AbstractSet<IntPair> {

  private static final boolean RED   = true;

  private static final boolean BLACK = false;

  private Node root;     // root of the BST

  int size; // size of tree set

  // BST helper node data type

  private sclass Node {

    private long val;          // associated data

    private Node left, right;  // links to left and right subtrees

    private bool color;        // color of parent link

    public Node(long val, boolean color) {

      this.val = val;

      this.color = color;

    }

  }

  *() {}

  *(Cl<IntPair> cl) { addAll(cl); }

  // is node x red; false if x is null ?

  private boolean isRed(Node x) {

      if (x == null) return false;

      return x.color == RED;

  }

  public int size() {

    ret size;

  }

  public bool isEmpty() {

    return root == null;

  }

  @Override public bool add(IntPair val) { ret add(intPairToLong(val)); }

  public bool add(long val) {

    int oldSize = size;

    root = put(root, val);

    root.color = BLACK;

    ifdef CompactTreeSet_debug assertTrue(check()); endifdef

    ret size > oldSize;

  }

  // insert the value in the subtree rooted at h

  private Node put(Node h, long val) { 

    if (h == null) { ++size; ret new Node(val, RED); }

    int cmp = compare(val, h.val);

    if      (cmp < 0) h.left  = put(h.left,  val); 

    else if (cmp > 0) h.right = put(h.right, val); 

    else              { /*h.val = val;*/ } // no overwriting

    // fix-up any right-leaning links

    if (isRed(h.right) && !isRed(h.left))      h = rotateLeft(h);

    if (isRed(h.left)  &&  isRed(h.left.left)) h = rotateRight(h);

    if (isRed(h.left)  &&  isRed(h.right))     flipColors(h);

    ret h;

  }

  // override if you want custom sorting

  int compare(long a, long b) {

    ret cmp(a, b);

  }

  @Override public bool remove(O key) { ret remove(intPairToLong((IntPair) key)); }

  public bool remove(long key) { 

    if (!contains(key)) false;

    // if both children of root are black, set root to red

    if (!isRed(root.left) && !isRed(root.right))

        root.color = RED;

    root = delete(root, key);

    if (!isEmpty()) root.color = BLACK;

    // assert check();

    true;

  }

  // delete the key-value pair with the given key rooted at h

  private Node delete(Node h, long key) { 

    // assert get(h, key) != null;

    if (compare(key, h.val) < 0)  {

      if (!isRed(h.left) && !isRed(h.left.left))

          h = moveRedLeft(h);

      h.left = delete(h.left, key);

    }

    else {

      if (isRed(h.left))

          h = rotateRight(h);

      if (compare(key, h.val) == 0 && (h.right == null)) {

          --size; null;

      } if (!isRed(h.right) && !isRed(h.right.left))

          h = moveRedRight(h);

      if (compare(key, h.val) == 0) {

          --size;

          Node x = min(h.right);

          h.val = x.val;

          // h.val = get(h.right, min(h.right).val);

          // h.val = min(h.right).val;

          h.right = deleteMin(h.right);

      }

      else h.right = delete(h.right, key);

    }

    return balance(h);

  }

  // make a left-leaning link lean to the right

  private Node rotateRight(Node h) {

      // assert (h != null) && isRed(h.left);

      Node x = h.left;

      h.left = x.right;

      x.right = h;

      x.color = x.right.color;

      x.right.color = RED;

      return x;

  }

  // make a right-leaning link lean to the left

  private Node rotateLeft(Node h) {

      // assert (h != null) && isRed(h.right);

      Node x = h.right;

      h.right = x.left;

      x.left = h;

      x.color = x.left.color;

      x.left.color = RED;

      return x;

  }

  // flip the colors of a node and its two children

  private void flipColors(Node h) {

      // h must have opposite color of its two children

      // assert (h != null) && (h.left != null) && (h.right != null);

      // assert (!isRed(h) &&  isRed(h.left) &&  isRed(h.right))

      //    || (isRed(h)  && !isRed(h.left) && !isRed(h.right));

      h.color = !h.color;

      h.left.color = !h.left.color;

      h.right.color = !h.right.color;

  }

  // Assuming that h is red and both h.left and h.left.left

  // are black, make h.left or one of its children red.

  private Node moveRedLeft(Node h) {

      // assert (h != null);

      // assert isRed(h) && !isRed(h.left) && !isRed(h.left.left);

      flipColors(h);

      if (isRed(h.right.left)) { 

          h.right = rotateRight(h.right);

          h = rotateLeft(h);

          flipColors(h);

      }

      return h;

  }

  // Assuming that h is red and both h.right and h.right.left

  // are black, make h.right or one of its children red.

  private Node moveRedRight(Node h) {

      // assert (h != null);

      // assert isRed(h) && !isRed(h.right) && !isRed(h.right.left);

      flipColors(h);

      if (isRed(h.left.left)) { 

          h = rotateRight(h);

          flipColors(h);

      }

      return h;

  }

  // restore red-black tree invariant

  private Node balance(Node h) {

      // assert (h != null);

      if (isRed(h.right))                      h = rotateLeft(h);

      if (isRed(h.left) && isRed(h.left.left)) h = rotateRight(h);

      if (isRed(h.left) && isRed(h.right))     flipColors(h);

      return h;

  }

  /**

   * Returns the height of the BST (for debugging).

   * @return the height of the BST (a 1-node tree has height 0)

   */

  public int height() {

      return height(root);

  }

  private int height(Node x) {

      if (x == null) return -1;

      return 1 + Math.max(height(x.left), height(x.right));

  }

  @Override public bool contains(O val) { ret contains(intPairToLong((IntPair) val)); }

  public bool contains(long val) {

    ret find(root, val) != null;

  }

  Node find(Node x, long key) {

    while (x != null) {

      int cmp = compare(key, x.val);

      if      (cmp < 0) x = x.left;

      else if (cmp > 0) x = x.right;

      else              ret x;

    }

    null;

  }

  private boolean check() {

      if (!is23())             println("Not a 2-3 tree");

      if (!isBalanced())       println("Not balanced");

      return is23() && isBalanced();

  }

  // Does the tree have no red right links, and at most one (left)

  // red links in a row on any path?

  private boolean is23() { return is23(root); }

  private boolean is23(Node x) {

      if (x == null) return true;

      if (isRed(x.right)) return false;

      if (x != root && isRed(x) && isRed(x.left))

          return false;

      return is23(x.left) && is23(x.right);

  } 

  // do all paths from root to leaf have same number of black edges?

  private boolean isBalanced() { 

      int black = 0;     // number of black links on path from root to min

      Node x = root;

      while (x != null) {

          if (!isRed(x)) black++;

          x = x.left;

      }

      return isBalanced(root, black);

  }

  // does every path from the root to a leaf have the given number of black links?

  private boolean isBalanced(Node x, int black) {

      if (x == null) return black == 0;

      if (!isRed(x)) black--;

      return isBalanced(x.left, black) && isBalanced(x.right, black);

  }

  public void clear() { root = null; size = 0; }

  // the smallest key in subtree rooted at x; null if no such key

  private Node min(Node x) { 

    // assert x != null;

    while (x.left != null) x = x.left;

    ret x;

  }

  private Node deleteMin(Node h) { 

    if (h.left == null)

      return null;

    if (!isRed(h.left) && !isRed(h.left.left))

      h = moveRedLeft(h);

    h.left = deleteMin(h.left);

    ret balance(h);

  }

  public Iterator<IntPair> iterator() {

    ret new MyIterator;

  }

  class MyIterator extends ItIt<IntPair> {

    new L<Node> path;

    *() {

      fetch(root);

    }

    void fetch(Node node) {

      while (node != null) {

        path.add(node);

        node = node.left;

      }

    }

    public bool hasNext() { ret !path.isEmpty(); }

    public IntPair next() {

      if (path.isEmpty()) fail("no more elements");

      Node node = popLast(path);

      // last node is always a leaf, so left is null

      // so proceed to fetch right branch

      fetch(node.right);

      ret longToIntPair(node.val);

    }

  }

  // Returns the smallest key in the symbol table greater than or equal to {@code key}.

  public IntPair ceiling(long key) {

    Node x = ceiling(root, key);

    ret x == null ? null : longToIntPair(x.val);

  }

  // the smallest key in the subtree rooted at x greater than or equal to the given key

  Node ceiling(Node x, long key) {  

    if (x == null) null;

    int cmp = compare(key, x.val);

    if (cmp == 0) ret x;

    if (cmp > 0)  ret ceiling(x.right, key);

    Node t = ceiling(x.left, key);

    if (t != null) ret t; 

    else           ret x;

  }

  public IntPair floor(long key) {

    Node x = floor(root, key);

    ret x == null ? null : longToIntPair(x.val);

  }    

  // the largest key in the subtree rooted at x less than or equal to the given key

  Node floor(Node x, long key) {

    if (x == null) null;

    int cmp = compare(key, x.val);

    if (cmp == 0) ret x;

    if (cmp < 0)  ret floor(x.left, key);

    Node t = floor(x.right, key);

    if (t != null) ret t; 

    else           ret x;

  }

}