Description
Problem 1.
Write a concrete example of the knapsack problem where you specify a set of at least 5 objects,
their dollar values (i.e., benefits) and their weights, as well as the weight of the knapsack, denoted
W. Now, consider the greedy approach of sorting items based on decreasing benefit/weight ratios
and picking items from the beginning of the list. In the context of your example, show that
Problem 1.a. (2 points)
• The greedy approach works for fractional knapsack.
Problem 1.b. (2 points)
• The greedy approach may fail for 0-1 knapsack.
Problem 2.
Consider the following symbols with their corresponding frequencies:
A : 1, B : 1, C : 2, D : 3, E : 5, F : 8, G : 13, H : 21
Problem 2.a. (3 points)
• Construct the Huffman coding of these symbols along with its optimal coding tree.
Problem 2.b. (3 points)
• Use your coding tree to decode 0001001000010000000001001
Problem 3.
Consider the problem of making change for n cents using the fewest number of coins. Assume that
each coin’s value is an integer.
Problem 3.a. (4 points)
• Suppose that the available coins are in the denominations that are powers of c, i.e., the
denominations are c
0
, c1
, · · · , ck
for some integers c > 1 and k ≥ 1. Show that the greedy
algorithm of picking the largest denomation first always yields an optimal solution. You are
expected to reason about why this approach gives an optimal solution. (Hint: Show that for
each denomination c
i
, the optimal solution must have less than c coins.)
Problem 3.b. (4 points)
• Design an O(nk) time algorithm that makes change for any set of k different coin denominations, assuming that one of the coins is a penny.
1
Problem 4. (7 points)
Implementation: Implement the make change algorithm you designed in the previous problem.
Your program should read a text file “data.txt” where each line in “data.txt” contains three values
c, k and n. Please make sure you take your input in the specified order c, k and n. For example, a
line in “data.txt” may look like the following:
3 4 38
where c = 3, k = 4, n = 38. That is, the set of denominations is {3
0
, 3
1
, 3
2
, 3
3
, 3
4} = {1, 3, 9, 27, 81},
and we would like to make change for n = 38. The file “data.txt” may include multiple lines like
above.
The output will be written to a file called “change.txt”, where the output corresponding to each
input line contains a few lines. Each line has two numbers, where the first number denotes a denomination and the second number represents the cardinality of that denomination in the solution.
For example, for the above input line ‘3 4 38’, the optimal solution is the multiset {27, 9, 1, 1}, and
the output in the file “change.txt” is as follows:
27 1
9 1
1 2
which means the solution contains 1 coin of denomination 27, one coin of 9 and two coins of
denomination 1. You can use a delimiter line to separate the outputs generated for different input
lines.
Problem 5. (3 points)
Extra credit: Can you generalize the results you found in the construction of Problem 2 (i.e., the
Huffman coding tree )? Write a statement/theorem that captures your generalization.
Submit a copy of all your code files and a README file that explains how
to compile and run your code in a ZIP file to TEACH. We will only test
execution with an input file named data.txt.
