Lecture 10

Learning objectives

After this class, you should be able to:

1. Give algorithms to determine the minimum and maximum of a set of numbers, and determine the maximum number of comparisons performed.
2. Given an array of n numbers, an integer i, 1 < i < n, and a sequence of random numbers in (0, 1), show the steps performed by the randomized expected time algorithm in determining the i th smallest element in the array.
3. Derive the expected time complexity of the randomized selection algorithm discussed in class.

Reading assignment

1. Lecture 10 slides.
2. CLR: Chapter 9, up to (and including) section 9.2.
3. CLR: Chapter 15, page 371.

Exercises and review questions

• Questions on current lecture's material
  1. Write C/C++ code implementing the algorithm given in section 9.1 for finding the minimum and maximum simultaneously.
  2. Give a divide and conquer algorithm to find the minimum and maximum simultaneously, and determine the worst case number of comparisons.
  3. (Post your solution on the discussion board) Give a sequence of random numbers in (0, 1), and show how the randomized selection algorithm of section 9.2 will use it to determine the 3 rd smallest number in the array: 8, 6, 3, 2, 9, 0, 4, 1, 7, 5 . Your random number sequence should be chosen such that it does not yield the answer after just one partitioning.

• Questions on next lecture's material
  1. (Post your answer on the discussion board) Give an example of a 3 x 3 matrix and a 3 x 2 matrix, and give their product.