UMBC CMSC641, Design & Analysis of Algorithms, Spring 2001

Homework 2

Due 02/13/01

• Exercise 18.2-3, page 363.
  

• Problem 18-2, page 376.
  

• In this question, we consider a problem in car ownership. As a car gets older, the maintenance costs (fuel costs, repair costs, insurance costs, etc) for the car may increase to the extent that it would be advantageous to sell the current car and buy a new car. The difficulty in this problem is that the prices of new cars change from year to year, the maintenance costs of cars purchased in different years may be different and the resale value of a car can change from year to year as well.

  For this question, you are given the following information:

  • p(i) = the price of a new car in year i, for 1 <= i <= n
  • v(i,k) = the resale value of a car purchased in year i and sold in year k, for 1 <= i < k <= n+1.
  • m(i,k) = the maintenance cost during year k of a car purchased in year i, for 1 <= i <= k <= n.

  The problem is to determine the years y₁, y₂, ..., y_r when you would purchase new cars such that the total cost of car ownership from years 1 through n is minimized. The total cost is the sum of the maintenance costs for years 1 through n plus the price of each car purchased minus the resale value of each car sold.

  For example, if n = 10, y₁ = 1, y₂ = 5 and y₃ = 7, then this solution states that you should purchase a new car in year 1, buy the second car in year 5 and buy the third car in year 7. (You would also sell the first car in year 5, the second car in year 7 and the third car at the beginning of year 11.) In addition, you should make the following assumptions.

  • You don't have a car before year 1. (Since you have to buy a new car in year 1, y₁ = 1.)
  • There is only 1 type of car for sale each year.
  • You are always able to sell your old car for v(i,k) dollars.
  • At the end of n years, you sell your last car for v(y_r,n+1) dollars.

  Describe in some detail a dynamic programming algorithm which finds the optimal solution to this problem. Be sure to include a description of the content and size of the memoization table. Analyze and justify the running time of your algorithm.

Back up to Spring 2001 CMSC 641 Homepage