# jan23 - Illinois Institute of Technology Department of...

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Unformatted text preview: Illinois Institute of Technology Department of Computer Science Lecture 1: January 23, 2008 CS 330 Discrete Structures Spring Semester, 2008 1 Why Study Mathematics and Theoretical Computer Science? This question is best answered by an example. Consider yourself an employee of a factory. Your boss comes up to you with the following dilemma: “Because of the tough economic times, we need to operate at peak efficiency. We need you to determine what our peak efficiency is at the lowest cost possible.” After further discussion with your employer, you determine the following facts: • The function relating efficiency to the number of employees is unimodal . That is, efficiency will increase to a maximum, then decrease as the number of employees increases. • The only way to determine efficiency is to build a pilot plant. • Pilot plants are expensive to build, and your boss will be unhappy with any solution that requires the construction of more pilot plants than necessary. There is an obvious solution to this problem: you could build a plant for one employee then find the efficiency of that plant, then build a plant with two employees and find its efficiency, then build a plant for three employees, and so on, until the efficiency goes down. You will then have the peak. 1.1 Finding a Better Solution Unfortunately, this is not a very efficient solution. It runs in linear time—that is, time proportional to the number of employees. (Consider that the peak may be at a few thousand employees. Your boss won’t be happy building thousands of pilot plants if he can do better.) But, as you are familiar with the binary search algorithm and that it is a faster way to search for items. You may even remember that this processand that it is a faster way to search for items....
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jan23 - Illinois Institute of Technology Department of...

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