## Description

Comp 251: Assignment 3

Programming component

• You are provided some starter code that you should fill in as requested. Add your code

only where you are instructed to do so. You can add some helper methods. Do not

modify the code in any other way and in particular, do not change the methods or

constructors that are already given to you, do not import extra code and do not touch

the method headers. The format that you see on the provided code is the only format

accepted for programming questions. Any failure to comply with these rules will

result in an automatic 0.

COMP 251 – HW3 Page 2 of 6 Fall 2020

• Public tests cases are available on codePost. You can run them on your code at any

time. If your code fails those tests, it means that there is a mistake somewhere. Even

if your code passes those tests, it may still contain some errors. We will grade your

code with a more challenging, private set of test cases. We therefore highly encourage

you to modify that tester class, expand it and share it with other students on the

discussion board. Do not include it in your submission.

• Your code should be properly commented and indented.

• Do not change or alter the name of the files you must submit, or the method

headers in these files. Files with the wrong name will not be graded. Make sure

you are not changing file names by duplicating them. For example, main (2).java will

not be graded. Make sure to double-check your zip file.

• You can submit either a zip file or individual files on codePost. If you get

more than 0 on the public tests, it means codePost accepted your files.

• You will automatically get 0 if the files you submitted on codePost do not

compile, since you can ensure yourself that they do. Note that public test cases

do not cover every situation and your code may crash when tested on a method that

is not checked by the public tests. This is why you need to add your own test cases

and compile and run your code from command line on linux.

COMP 251 – HW3 Page 3 of 6 Fall 2020

1. (50 points) Ford-Fulkerson

We will implement the Ford-Fulkerson algorithm to calculate the Maximum Flow of a directed weighted graph. Here, you will use the files WGraph.java and FordFulkerson.java,

which are available on the course website. Your role will be to complete two methods in

the template FordFulkerson.java.

The file WGraph.java is similar to the file that you used in your previous assignment to

build graphs. The only differences are the addition of setter and getter methods for the

Edges and the addition of the parameters “source” and “destination”. There is also an

additional constructor that will allow the creation of a graph cloning a WGraph object.

Graphs are encoded using a similar format than the one used in the previous assignment.

The only difference is that now the first line corresponds to two integers, separated by

one space, that represent the “source” and the “destination” nodes. An example of such

file can be found on the course website in the file ff2.txt. These files will be used

as an input in the program FordFulkerson.java to initialize the graphs. This graph

corresponds to the same graph depicted in [CLRS2009] page 727.

Your task will be to complete the two static methods fordfulkerson WGraph graph)

and pathDFS(Integer source, Integer destination, WGraph graph). The second

method pathDFS finds a path via Depth First Search (DFS) between the nodes “source”

and “destination” in the “graph”. You must return an ArrayList of Integers with the

list of unique nodes belonging to the path found by the DFS. The first element in the

list must correspond to the “source” node, the second element in the list must be the

second node in the path, and so on until the last element (i.e., the “destination” node) is

stored. If the path does not exist, return an empty path. The method fordfulkerson

must compute an integer corresponding to the max flow of the “graph”, as well as the

graph encoding the assignment associated with this max flow.

Once completed, compile all the java files and run the command line java FordFulkerson

ff2.txt. Your program will output a String containing the relevant information. An

example of the expected output is available in the file ff2testout.txt. This output

keeps the same format than the file used to build the graph; the only difference is that

the first line now represents the maximum flow (instead of the “source” and “destination” nodes). The other lines represent the same graph with the weights updated to the

values that allow the maximum flow. The file ff226testout.txt represents the answer

to the example showed in [CLRS2009] Page 727. You are invited to run other examples

of your own to verify that your program is correct.

2. (50 points) Bellman-Ford

We want to implement the Bellman-Ford algorithm for finding the shortest path in

a graph where edges can have negative weights. Once again, you will use the object

WGraph. Your task is to complete the method BellmanFord(WGraph g, int source)

and shortestPath(int destination) in the file BellmanFord.java.

The method BellmanFord takes an object WGraph named g as an input and an integer

that indicates the source of the paths. If the input graph g contains a negative cycle,

then the method should throw an exception (see template). Otherwise, it will return

COMP 251 – HW3 Page 4 of 6 Fall 2020

an object BellmanFord that contains the shortest path estimates (the private array of

integers distances), and for each node, its predecessor in the shortest path from the

source (the private array of integers predecessors).

The method shortestPath will return the list of nodes as an array of integers along

the shortest path from the source to the destination. If this path does not exists, the

method should throw an exception (see template).

An example input is available in bf1.txt.

3. (0 points) Knapsack Problem

We have seen in class the Knapsack problem and a dynamic programming algorithm.

One could define the Knapsack problem as following:

Definition. Let n 0 be the number of distinct items and W 0 be the knapsack

capacity. For each item i, wi 0 denotes the item weight and vi 0 denotes its value.

The goal is to maximize the total value

Xn

i=1

vixi

while

Xn

i=1

wixi ≤ W

where xi ∈ {0, 1} for i ∈ {1, . . . , n}.

Algorithm. We recall the recursive form of the dynamic programming algorithm.

Let OP T(i, w) be the maximum profit subset of items 1, . . . , i with weight limit w. If

OP T does not select the item i, then OP T selects among items {1, . . . , i−1} with weight

limit w. Otherwise, OP T selects among items {1, . . . , i − 1} with weight limit w − wi

.

We could formalize as

OP T(i, w) =

0 if i = 0

OP T(i − 1, w) if wi w

max{OP T(i − 1, w), vi + OP T(i − 1, w − wi)} otherwise

COMP 251 – HW3 Page 5 of 6 Fall 2020

Correctness of dynamic programming algorithm (0)

Usually, a dynamic programming algorithm can be seen as a recursion and proof by

induction is one of the easiest way to show its correctness. The structure of a proof by

strong induction for one variable, say n, contains three parts. First, we define the

Proposition P(n) that we want to prove for the variable n. Next, we show that the

proposition holds for Base case(s), such as n = 0, 1, . . . etc. Finally, in the Inductive

step, we assume that P(n) holds for any value of n strictly smaller than n

0

, then we

prove that P(n

0

) also holds.

Use the proof by strong induction properly to show that the algorithm of the Knapsack

problem above is correct.

Bounded Knapsack Problem (0)

Let us consider a similar problem, in which each item i has ci 0 copies (ci

is an

integer). Thus, xi

is no longer a binary value, but a non-negative integer at most equal

to ci

, 0 ≤ xi ≤ ci

. Modify the dynamic programming algorithm seen at class for this

problem.

Note: One could consider a new set, in which item i has ci occurrences. Then, the

algorithm seen as class can be applied. However, this could be costly since ci might be

large. Therefore, the algorithm you propose should be different than this one.

Unbounded Knapsack Problem (0-bonus)

In this question, we consider the case where the quantity of each item is unlimited. Thus,

xi could be any non-negative integer. Provide a dynamic programming algorithm for

this problem.