Description
Problem 1. (4 points)
What does dynamic programming have in common with divide-and-conquer? What is a principal
difference between them?
Problem 2. (6 points)
Shortest path counting: A chess rook can move horizontally or vertically to any square in the
same row or the same column of a chessboard. Find the number of shortest paths by which a
rook can move from one corner of a chessboard to the diagonally opposite corner. The length of
a path is measured by the number of squares it passes through, including the first and the last
squares. Solve the problem:
a) by a dynamic programming algorithm.
b) by using elementary combinatorics.
Problem 3. (6 points)
Maximum square submatrix: Given an m×n Boolean matrix B, find its largest square submatrix
whose elements are all zeros. Design a dynamic programming algorithm and indicate its time
efficiency. (The algorithm may be useful for, say, finding the largest free square area on a
computer screen or for selecting a construction site.)
Problem 4. (14 points)
Consider the following instance of the knapsack problem with capacity W = 6
Item Weight Value
1 3 $25
2 2 $20
3 1 $15
4 4 $40
5 5 $50
a) Apply the bottom-up dynamic programming algorithm to that instance.
b) How many different optimal subsets does the instance of part (a) have?
c) In general, how can we use the table generated by the dynamic programming algorithm to tell
whether there is more than one optimal subset for the knapsack problem’s instance?
d) Implement the bottom-up dynamic programming algorithm for the knapsack problem. The
program should read inputs from a file called “data.txt”, and the output will be written to screen,
indicating the optimal subset(s).
e) For the bottom-up dynamic programming algorithm, prove that its time efficiency is in
Θ(nW), its space efficiency is in Θ(nW) and the time needed to find the composition of an
optimal subset from a filled dynamic programming table is in O(n).
EXTRA CREDIT (4 points)
Implement an algorithm that finds the composition of an optimal subset from the table generated
by the bottom-up dynamic programming algorithm for the knapsack problem.
Programs can be written in C, C++ or Python, but all code must run on the
OSU engr servers. Submit to TEACH a copy of all your code files and a
README file that explains how to compile and run your code in a ZIP file.
We will only test execution with an input file named data.txt.