a. Show how to implement a queue with two ordinary stacks so that the amortized cost of each Enqueue and Dequeue operation is O(1). b. Consider an ordinary min-heap supporting operations Insert and Extract-min that each costs O(lg n) worst-case time complexity when there are n items in the heap. Give a potential function Φ such that the amortized cost of Insert is O(lg n) and the amortized cost of Extract-min is O(1). Show Φ yields these amortized time bounds (n is the number of items currently in the heap and the size of the heap is unknown). Note: “Show” means to provide a proof with logical and precise arguments and/or derivations.

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A potential method of analysis

The complexity of the algorithm model emphasizes the potential method. This method is useful for examining a data structure's amortized time and space complexity.