Homework 4

Background: Ch. 10, 7.

Due October 2nd:

• 4.1: Given a list L of n numbers, we want to compute . Prove that setting x equal to the median of L minimizes the expression. [Extra Credit!]
• 4.2: Given a list L of n numbers, we want to compute . Prove that setting x equal to the mean of L minimizes the expression. [Extra Credit!]
• 4.3: Give an algorithm for efficiently computing the mode of a list of numbers (not necessarily sorted).
• 4.4: Give an algorithm for efficiently computing the rank of GPA x both when the list L of GPAs is sorted and when it is not. Give tight worst-case running-time bounds.
• 4.5: Show that the expected running time of SELECT is O(n). Hint: Modify the quicksort recurrence.
• 4.6: CLR Exercise 10.3-3 (pg. 192).
• 4.7: CLR Exercise 10.3-7 (pg. 192).
• 4.8: Say we have two heaps, A and B, each of size n, and we want to create a third heap C. Here are two ways to do that. Method 1 runs through the elements of A and INSERTs each one into B. Method 2 simply copies all the elements of A and B into a new array C and calls ``build-heap'' on the result. Give asymptotic upper bounds on the running time of these two choices. Which do you prefer? (Note that there are alternate implementations of heaps that support this ``merge heap'' operation in time!)
• 4.9: We've got a set of n distinct positive integers, e.g., (7, 10, 9, 19, 8, 6). There are subsets. Here's one: (7, 10, 19, 8). We define the sum of a subset to be the sum of the integers in that subset (59 and 44 for the two subsets here). Give an efficient algorithm to list the n subsets with the largest sums. Hint: use a priority queue to store the subsets that might have the largest sums. Give a bound on the running time of your algorithm.
• 4.10: CLR Exercise 7.5-2 (pg. 150).
• 4.11: CLR Exercise 7.5-5 (pg. 151).
• 4.12: CLR Exercise 13.2-1 (pg. 249).