# Assignment 3 Solution

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1. (40 points) 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 FordFulker-son.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 the similar to the file that you used in your previous assignment to build graphs. The only differences are the addition of setters and getters methods for the Edges and the addition of the parameters “source” and “destination”. There is also an addi-tional constructor that will allow the creation of a graph cloning a WGraph object. Graphs are also 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 with 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(Integer source, Integer destination, WGraph graph, String filePath) and pathDFS( Integer source, Integer destination, WGraph graph). The second method pathDFS finds a path through a Depth First Search (DFS) between the nodes “source” and “destination” in the “graph” through non-zero weight edges. You must return an ArrayList of Integers with the list of unique nodes belonging to the path found by the DFS. If no path is found, return an empty ArrayList. 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. The method fordfulkerson must compute

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an integer corresponding to the max flow of the “graph” and the graph itself. The method fordfulkerson has a variable called myMcGillID, which must be initialized with your McGill ID number.

Once completed, compile all the java files and run the command line java FordFulkerson ff2.txt. Your program must use the function writeAnswer to save your output in a file. An example of the expected output file is available in the file ff226000000.txt. This output keeps the same format than the file used to build the graph; the only difference is that the a line has been added; the first line now represents the maximum flow (instead of the “source” and “destination” nodes). If the fordfulkerson method was unable to compute the maximum flow, it should output a result of -1 (and not throw an exception). The other lines represent the same graph with the weights updated with the values that represent the maximum flow. The file ff226000000.txt represent the answer of the example showed in [CLRS2009] page

727. You are invited to run other examples of your own to verify that your program is correct.

1. (40 points) We want to implement the Bellman-Ford algorithm for finding the shortest path in a graph where edge can have negative weights. This question extends the previous question on the implementation of the Dijkstra’s algorithm done in the assignment 2. You will need to ex-ecute this program to use the same auxiliary class Wgraph used in question 1. Your task is to fill the method BellmanFord(WGraph g, int source) and shortestPath(int destination) in the file BellmanFord.java.

The method BellmanFord takes a object WGraph named g as an input (See Assignment 2) and an integer that indicates the source of the paths. If the input graph g contains a neg-ative cycle, then the method should throw an exception. Otherwise, it will return an ob-ject 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 pri-vate 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 node destination. If this path does not exists, the method should throw an exception.

Please take a look at the code, we defined some exceptions that you should use when appro-priate if one of your methods fails to terminate.

Input graphs are available on the course webpage to test your program. Nonetheless, we invite you to also make your own graphs to test your program.

1. (20 points) You will complete this section through MyCourses. Note that you MUST use your own results to answer those questions. Answers to this quiz that would not conceptually match the output of your program will be considered plagiarism (refer to course outline).

You will submit FordFulkerson.java and BellmanFord.java in a single zip file.

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