Homework 08: GraphAlgorithms Solution

Description

Collaboration Policy

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When working on homework assignments, you may not directly copy code from any source (other than your own past submissions). You are welcome to collaborate with peers and consult external re-sources, but you must personally write all of the code you submit. You must list, at the top of each le in your submission, every student with whom you collaborated and every resource you consulted while completing the assignment.

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Style and Formatting

It is important that your code is not only functional, but written clearly and with good programming style. Your code will be checked against a style checker. The style checker is provided to you, and is

Homework 08: GraphAlgorithms Due: See Canvas

located on Canvas. It can be found under Files, along with instructions on how to use it. A point is deducted for every style error that occurs. If there is a discrepancy between what you wrote in accordance with good style and the style checker, then address your concerns with the Head TA.

Javadocs

Write javadoc comments for any helper methods you create in a style similar to the existing javadocs. Any javadocs you write must be useful and describe the contract, parameters, and return value of the method. Random or useless javadocs added only to appease checkstyle will lose points.

Vulgar/Obscene Language

Any submission that contains profanity, vulgar, or obscene language will receive an automatic zero on the assignment. This policy applies not only to comments/javadocs, but also things like variable names.

Exceptions

When throwing exceptions, you must include a message by passing in a String as a parameter. The message must be useful and tell the user what went wrong. \Error”, \BAD THING HAP-PENED”, and \fail” are not good messages. The name of the exception itself is not a good message. For example:

Bad: throw new IndexOutOfBoundsException(‘‘Index is out of bounds.’’);

Good: throw new IllegalArgumentException(‘‘Cannot insert null data into data structure.’’); In addition, you may not use try catch blocks to catch an exception unless you catching an exception you have explicitly thrown yourself with the throw new ExceptionName(‘‘Exception Message’’); syn-

tax (replacing ExceptionName and Exception Message with the actual exception name and message respectively).

Generics

If available, use the generic type of the class; do not use the raw type of the class. For example, use new LinkedList<Integer>() instead of new LinkedList(). Using the raw type of the class will result in a penalty.

Forbidden Statements

You may not use these in your code at any time in CS 1332.

• package

• System.arraycopy()

• clone()

• assert()

• Arrays class

• Array class

• Thread class

• Collections class

• Collection.toArray()

• Re ection APIs

Homework 08: GraphAlgorithms Due: See Canvas

• Inner or nested classes

• Lambda Expressions

• Method References (using the :: operator to obtain a reference to a method)

If you’re not sure on whether you can use something, and it’s not mentioned here or anywhere else in the homework les, just ask.

Debug print statements are ne, but nothing should be printed when we run your code. We expect clean runs – printing to the console when we’re grading will result in a penalty. If you submit these, we will take o points.

JUnits

We have provided a very basic set of tests for your code. These tests do not guarantee the correctness of your code (by any measure), nor do they guarantee you any grade. You may additionally post your own set of tests for others to use on the Georgia Tech GitHub as a gist. Do NOT post your tests on the public GitHub. There will be a link to the Georgia Tech GitHub as well as a list of JUnits other students have posted on the class Piazza.

If you need help on running JUnits, there is a guide, available on Canvas under Files, to help you run JUnits on the command line or in IntelliJ.

Homework 08: GraphAlgorithms Due: See Canvas

GraphAlgorithms

For this assignment, you will code 3 di erent graph algorithms. This homework has many les included, so be sure to read ALL of the documentation given before asking questions.

Graph Data Structure

You are provided a Graph class. The important methods to note from this class are:

• getVertices returns a Set of Vertex objects (another class provided to you) associated with a graph.

• getEdges returns a Set of Edge objects (another class provided to you) associated with a graph.

• getAdjList returns a Map that maps Vertex objects to Lists of VertexDistance objects. This Map is especially important for traversing the graph, as it will e ciently provide you the edges adjacent to any vertex (the outgoing edges of any vertex). For example, consider an adjacency list where vertex A is associated with a list that includes a VertexDistance object with vertex B and distance 2 and another VertexDistance object with vertex C and distance 3. This implies that in this graph, there is an edge from vertex A to vertex B of weight 2 and another edge from vertex A to vertex C of weight 3.

Vertex Distance Data Structure

In the Graph class and Dijkstra’s algorithm, you will be using the VertexDistance class implementation that we have provided. In the Graph class, this data structure is used by the adjacency list to represent which vertices a vertex is connected to. In Dijkstra’s algorithm, you should use this data structure along with a PriorityQueue. When utilizing VertexDistance in this algorithm, the vertex attribute should represent the destination vertex and the distance attribute should represent the minimum cumulative path cost from the source vertex to the destination vertex.

DFS

Depth-First Search is a search algorithm that visits vertices in a depth based order. Similar to pre/post/in-order traversal in BSTs, it depends on a Stack-like behavior to work. In your implementation, the Stack will be the recursive stack, meaning you should not create a Stack data structure. It searches along one path of vertices from the start vertex and backtracks once it hits a dead end or a visited vertex until it nds another path to continue along. Your implementation of DFS must be recursive to receive credit.

Single-Source Shortest Path (Dijkstra’s Algorithm)

The next algorithm is Dijkstra’s Algorithm. This algorithm nds the shortest path from one vertex to all of the other vertices in the graph. This algorithm only works for non-negative edge weights, so you may assume all edge weights for this algorithm will be non-negative. In order to keep track of the cumulative distance from the source vertex to the vertices you visit in this algorithm, you will need to use the VertexDistance data structure we are providing you. At any stage throughout the algorithm, the PriorityQueue of VertexDistance objects will tell you which vertex currently has the minimum cumulative distance from the source vertex.

There are two commonly implemented terminating condition variants for Dijkstra’s Algorithm. The rst variant is where you depend purely on the PriorityQueue to determine when to terminate. You only terminate once the PriorityQueue is empty. The other variant, the classic variant, is the version where you maintain both a PriorityQueue and a visited set. To terminate, still check if the PriorityQueue

Homework 08: GraphAlgorithms Due: See Canvas

is empty, but you can also terminate early once all the vertices are in the visited set. You should implement the classic variant for this assignment. The classic variant, while using more memory, is usually more time e cient since there is an extra condition that could allow it to terminate early.

Self-Loops and Parallel Edges

In this framework, self-loops and parallel edges work as you would expect. If you recall, self-loops are edges from a vertex to itself. Parallel edges are multiple edges with the same orientation between two vertices. In other words, parallel edges are edges that are incident on precisely the same vertices. These cases are valid test cases, and you should expect them to be tested. However, most implementations of these algorithms handle these cases automatically, so you shouldn’t have to worry too much about them when implementing the algorithms.

Prim’s Algorithm

A tree is a graph that is acyclic and connected. A spanning tree is a subgraph that contains all the vertices of the original graph and is a tree. An MST has two main qualities: being minimum and a spanning tree. Being minimum dictates that the spanning tree’s sum of edge weights must be minimized.

By the properties of a spanning tree, any valid MST must have jV j 1 edges in it. However, since all undirected edges are speci ed as two directional edges, a valid MST for your implementation will have 2(jV j 1) edges in it.

Prim’s algorithm builds the MST outward from a single component, starting with a starting vertex. At each step, the algorithm adds the cheapest edge connected to the incomplete MST that does not cause a cycle. Cycle detection can be handled with a visited set like in Dijkstra’s..

Visualizations of Graphs

The directed graph used in the student tests is:

1 4 6

2 3 5 7

The undirected graph used in the student tests is:

7

A B

5

8

• 3

C			F
2	D	E	6
		1

Homework 08: GraphAlgorithms Due: See Canvas

Grading

Here is the grading breakdown for the assignment. There are various deductions not listed that are incurred when breaking the rules listed in this PDF and in other various circumstances.

Methods:
DFS	20pts
Dijkstra’s	30pts
Prim’s	25pts
Other:
Checkstyle	10pts
E ciency	15pts
Total:	100pts

Provided

The following le(s) have been provided to you. There are several, but we’ve noted the ones to edit.

1. GraphAlgorithms.java

This is the class in which you will implement the di erent graph algorithms. Feel free to add private static helper methods but do not add any new public methods, new classes, in-stance variables, or static variables.

2. Graph.java

This class represents a graph. Do not modify this le.

3. Vertex.java

This class represents a vertex in the graph. Do not modify this le.

4. Edge.java

This class represents an edge in the graph. It contains the vertices connected to this edge and its weight. Do not modify this le.

5. VertexDistance.java

This class holds a vertex and a distance together as a pair. It is meant to be used with Dijk-stra’s algorithm. Do not modify this le.

6. GraphAlgorithmsStudentTests.java

This is the test class that contains a set of tests covering the basic algorithms in the GraphAlgorithms class. It is not intended to be exhaustive and does not guarantee any type of grade. Write your own tests to ensure you cover all edge cases. The graphs used for these tests are shown above in the pdf.

Deliverables

You must submit all of the following le(s). Make sure all le(s) listed below are in each submission, as only the last submission will be graded. Make sure the lename(s) matches the lename(s) below, and that only the following le(s) are present. Do NOT submit Graph.java, Vertex.java, Edge.java, or

Homework 08: GraphAlgorithms Due: See Canvas

VertexDistance.java, or else your submission will not compile on Gradescope.

Once submitted, double check that it has uploaded properly on Gradescope. To do this, download your uploaded le(s) to a new folder, copy over the support le(s), recompile, and run. It is your sole responsibility to re-test your submission and discover editing oddities, upload issues, etc.

1. GraphAlgorithms.java

7