2. (10 marks) Consider our binary tree HEAP data structure. Recall that it supports INSERT and EX- TRACT MAX in O(log n) worst case time, where n is the number of elements in the PRIORITY QUEUE. (a) Give a potential function such that the amortized cost of INSERT is O(log n) and the amor. tized cost of EXTRACT-MAX is 0(1) with respect to . Justify your answer. (c) Consider a sequence of n EXTRACT_MAX operations performed on a heap H that initially contains n elements. Does the fact that the amortized cost of each EXTRACT MAX operation is O(1) mean that the entire sequence can be processed in O(n) time? Justify your