CS 1510 Midterm 2
Fall 2003
1. (20 points) We consider the problem of computing the longest common subsequence of two
sequences A=a1,...,amand B=b1,...,bn. Let T(i, j) be the length of the longest common
subsequence of a1,...,aiand b1,...,bj.
(a) Write a recursive function to compute T(i, j ) in the naive way. Don’t forget the base
case(s).
(b) Show that if you implement this recursion directly in say the C programming language,
that the program could use time that is exponential in n.
(c) Write iterative array based code to compute T(m, n) that runs in O(n2) time.
(d) Write code to actually find the longest common subsequence from your array.
2. (20 points) Consider the problem where the input is a collection of ntrain trips within
Germany. For the ith trip Tiyou are given the date diof that trip, and the non-discounted
fare fifor that trip. For simplicity we will assume that dates are nonnegative integers. The
German railway system sells a class A Bahncard for YEuros that entitles you to a 50% fare
reduction on all train travel within Germany within Adays of purchase. The German railway
system also sells a class B Bahncard for MEuros that entitles you to a 60% fare reduction
on all train travel within Germany within Bdays of purchase. You can apply at most one
Bahncard discount to a particular trip. The problem is to determine the least you can spend
on your travel.
In this paragraph we give an example. Assume that Y= 500, A= 365, and M= 60 and
B= 30. Further assume, d1= 1, f1= 20 Euros, d2= 40, f2= 200 Euros, d3= 80, f3= 200
Euros, d4= 120, f4= 100 Euros, d5= 200, f5= 200 Euros, and d6= 400, and f6= 600
Euros. Then you might buy a class B Bahncard on day 40, and a class A Bahncard on day
200. This results in a total cost of
20 + 60 + (.4)(200) + 200 + 100 + 500 + (.5)(200 ) + (.5)(600)
Euros.
Give an O(n3) time algorithm for problem. Significant partial credit will be given for any
polynomial time algorithm.
