1. Use a k-d tree to implement deleteMin. What would you expect the  average running time to be for a random tree?

2. Use a k-d heap to implement a double-ended queue that also supports deleteMin.

3. Implement the pairing heap with a nullNode sentinel.

4. Show that the amortized cost of each operation is O(log N) for the pairing heap algorithm in the text.

5. An alternative method for combineSiblings is to place all of the siblings on a queue, and repeatedly dequeue and merge the ?rst two items on the queue, placing the result at the end of the queue. Implement this variation.