1. When do M consecutive insertions into a binomial queue take less than 2M time units?

2. Suppose a binomial queue of N = 2k - 1 elements is built. Alternately perform M insert and deleteMin pairs. Clearly, each operation takes O(log N) time. Why does this not contradict the amortized bound of O(1) for insertion?

3. Show that the amortized bound  of  O(log N)  for  the  skew  heap  operations described in the text cannot be converted to a worst-case bound by giving a sequence of operations that lead to a merge requiring 8(N) time.

4. Show how to merge two skew heaps with one top-down pass and reduce the merge cost to O(1) amortized time.

5. Extend skew heaps to support the decreaseKey operation in O(log N) amortized time.