CSE 2320 Section 501/571 Fall 1999
Due: November 9, 1999, in class (November 10, 1999, 5:00pm for -10%)
- Consider the problem of using rental cars to drive along a highway
from one location to another. There are n rental car agencies along the
highway. At any of the agencies you can rent a car to be returned at any
other agency down the road. You cannot backtrack. For each possible
departure point i and each possible arrival point j the cost of a
rental from i to j is known. However, the cost of renting from i to
j may be higher than the total cost of a series of shorter rentals. In
this case you can return the first car at some agency k between i and
j and continue your trip in a second car. There is no extra charge for
changing cars in this way.
- Let c(i,j) be the optimal cost of driving from agency i to
agency j. Assuming the optimal trip from i to j consists of
trips from i to k and from k to j, give an expression for
c(i,j) in terms of c(i,k) and c(k,j).
- Prove that the driving problem exhibits optimal substructure.
- Define a recursive solution for computing c(i,j) and write
pseudocode for a divide-and-conquer algorithm implementing your solution.
- Give a recurrence T(n) for the running time of your recursive
solution in part c, where n=j-i+1, and show that
- Demonstrate that the recursive solution has overlapping subproblems
and compute the number of unique subproblems.
- Write pseudocode for a O(n3) bottom-up, dynamic programming
solution to the driving problem. Justify the O(n3) running time of your
solution. Input to your procedure will be i, j and the cost matrix,
and the output will be the optimal-cost sequence of stops along the way.
- Consider a greedy algorithm for the driving problem where the greedy
choice is to choose the lowest cost single car trip from your current
location (originally i) to some other location k along the way to j,
and then continue with the same greedy choice from k. Again, no
backtracking is allowed. Give a counterexample showing that this greedy
choice does not satisfy the greedy choice property, and explain why.
- Problem 17-1a, page 353.