Assignment 8
Computer Science 231
Spring 2009
Due: Friday, April 17
Reading. CLRS Sections 16.1, 16.2, 16.3, pages 370 -- 392
CLR Sections 17.1, 17.2, 17.3, pages 330 -- 344
Note. To complete Exercises 16.2-1, 16.2-2, and 16.2-3, you will need to read section
16.2 of CLRS (section 17.2 of CLR).
Exercise 16.2-1. Prove that the fractional knapsack problem has the greedy-choice
property.
Exercise 16.2-2. Give a dynamic-programming solution to the 0-1 knapsack problem that
runs in O(n W) time, where n is the number of items and W is the maximum weight of the
items that the thief can put in his knapsack.
Exercise 16.2-3. Suppose that in a 0-1 knapsack problem, the order of the items when
sorted by increasing weight is the same as their order when sorted by decreasing value.
Give an efficient algorithm to find an optimal solution to this variant of the knapsack
problem, and argue that your algorithm is correct.
Exercise 16.2-4. Professor Midas drives an automobile from Newark to Reno along
Interstate 80. His car's gas tank, when full holds enough gas to travel n miles, and his map
gives the distance between gas stations on his route. The processor wishes to make as few
gas stops as possible along the way. Give an efficient method by which Professor Midas
can determine at which gas stations he should stop, and prove that your strategy yields an
optimal solution.
Exercise 8-5. (Adapted from Abelson and Sussman, Structure and Interpretation of
Computer Programs, p. 125).
Here we consider various approaches to compressing the following 1950s style song
lyrics:
get a job>
sha na na na na na na na na>
get a job>
sha na na na na na na na na>
wah yip yip yip yip yip yip yip yip>
sha boom>
For simplicity, the only characters are lower-case letters, spaces, and line-ending carriage
returns (written >). Here are some useful facts about the lyrics:
Total number of characters: 121
Character frequency table:
a
b
e
g
h
i
j
m
n
o
p
s
t
w
y
>
22
3
2
2
4
8
2
1
16
4
8
3
2
1
8
29
6
