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D) Solve the recurrence, and prove that your solution is correct. ] (e) Prove that the number of swaps executed by STOOGESORT is at most n 2 . 9. Describe an algorithm to compute the median of an array A[1 .. 5] of distinct numbers using at most 6 comparisons. ” and each leaf is an index into the array. A[1]:A[2] < A[1]:A[3] < A[2]:A[3] < A[2] > A[1] > > A[1]:A[3] < A[1] > A[2]:A[3] < A[3] A[3] > A[2] Finding the median of a 3-element array using at most 3 comparisons 10. Suppose you are given a stack of n pancakes of different sizes.

For example, there are six local minima in the following array: 9 7 7 2 1 3 7 5 4 7 3 3 4 8 6 9 We can obviously find a local minimum in O(n) time by scanning through the array. Describe and analyze an algorithm that finds a local minimum in O(log n) time. [Hint: With the given boundary conditions, the array must have at least one local minimum. ] 15 Lecture 1: Recursion [Fa’10] Algorithms 12. ” You are presented with an m × n grid of boxes, each containing a unique number. It costs $100 to open a box.

In fact, when n is a power of two, this algorithm is exactly optimal. However, there are slightly faster methods for other values of n. For example, our divide-and-conquer algorithm computes a15 in six multiplications (a15 = a7 · a7 · a; a7 = a3 · a3 · a; a3 = a · a · a), but only five multiplications are necessary (a → a2 → a3 → a5 → a10 → a15 ). 9 Exercises 1. (a) Professor George O’Jungle has a 27-node binary tree, in which every node is labeled with a unique letter of the Roman alphabet or the character &.

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