// Textbook fragment 08.14
  /** Returns but does not remove an entry with minimum key */
  public Entry<K,V> min() throws EmptyPriorityQueueException {
    if (isEmpty())
      throw new EmptyPriorityQueueException("Priority queue is empty");
    return heap.root().element();
  }
  /** Inserts a key-value pair and returns the entry created */
  public Entry<K,V> insert(K k, V x) throws InvalidKeyException {
    checkKey(k);  // may throw an InvalidKeyException
    Entry<K,V> entry = new MyEntry<K,V>(k,x);
    upHeap(heap.add(entry));
    return entry;
  }
  /** Removes and returns an entry with minimum key */
  public Entry<K,V> removeMin() throws EmptyPriorityQueueException {
    if (isEmpty())
      throw new EmptyPriorityQueueException("Priority queue is empty");
    Entry<K,V> min = heap.root().element();
    if (size() == 1)
      heap.remove();
    else {
      heap.replace(heap.root(), heap.remove());
      downHeap(heap.root());
    }
    return min;
  }
  /** Determines whether a given key is valid */
  protected void checkKey(K key) throws InvalidKeyException {
    try {
      comp.compare(key,key);
    }
    catch(Exception e) {
      throw new InvalidKeyException("Invalid key");
    }
  }