Comp 251: Assignment 2

• Your solution must be submitted electronically on MyCourses.

• 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 give you an automatic 0.

• The starter code includes a tester class. 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 set of examples. We therefore highly encourage you to modify

that tester class, expand it and share it with other students on the myCourses 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 one of the files you must submit. 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.

• Do not submit individual files. Include all your files into a .zip file and, when appropriate, answer

the complementary quiz online on MyCourses.

• You will automatically get 0 if the files you submitted on MyCourses do not compile.

• You should compile all your files as a single package from command line, by using the command

javac -d . *.java

Exercise 1 (Disjoint sets (20 points)) We want to implement a disjoint set data structure with union

and find operations. The template for this program is available on the course website and named

DisjointSets.java.

In this question, we model a partition of n elements with distinct integers ranging from 0 to n − 1 (i.e.

each element is represented by an integer in [0, · · · , n − 1], and each integer in [0, · · · , n − 1] represent

one element). We choose to represent the disjoint sets with trees, and to implement the forest of trees

with an array named par. More precisely, the value stored in par[i] is parent of the element i, and

par[i]==i when i is the root of the tree and thus the representative of the disjoint set.

You will and implement union by rank and the path compression technique seen in class. The rank is an

integer associated with each node. Initially (i.e. when the set contains one single object) its value is 0.

Union operations link the root of the tree with smaller rank to the root of the tree with larger rank. In

case of the rank of both trees is the same, the rank of the new root increases by 1. You can implement

the rank with an specific array (called rank) that has been added to the template) or use the array par

(This is tricky). Note that path compression does not change the rank of a node.

Download the file DisjointSets.java, and complete the methods find(int i) as well as

union(int i, int j). The constructor takes one argument n (a strictly positive integer) that indicates the number of elements in the partition, and initialize it by assigning a separate set to each element.

The method find(int i) will return the representative of the disjoint set that contains i (do not

forget to implement path compression here!). The method union(int i, int j) will merge the

set with smaller rank (for instance i) in the disjoint set with larger rank (in that case j). In that case,

the root of the tree containing i will become a child of the root of the tree containing j), and return the

representative (as an integer) of the new merged set. Do not forget to update the ranks. In case of the

ranks are identical, you will merge i into j.

Once completed, compile and run the file DisjointSets.java. It should produce the output available in the file unionfind.txt available on the course website.

Note: You will need to complete this question to implement Question 2.

Exercise 2 (Minimum Spanning trees (40 points)) We will implement the Kruskal algorithm to calculate the minimum spanning tree (MST) of a undirected weighted graph. Here, you will use the

file DisjointSets.java completed in the previous question, and two other files WGraph.java,

Kruskal.java available on the course website. You will need the classes DisjointSets and

WGraph to execute Kruskal.java. Your role will be to complete two methods in the template

Kruskal.java.

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The file WGraph.java implements two classes WGraph and Edge. An object of Edge stores all informations about edges (i.e. the two vertices and the weight of the edge), which are used to build graphs.

The class WGraph has two constructors WGraph() and WGraph(String file). The first one

creates an empty graph and the second uses a file to initialize a graph. Graphs are encoded using the

following format. The first line of this file is a single integer n that indicates the number of nodes in

the graph. Each vertex is labelled with an number in [0, · · · , n − 1], and each integer in [0, · · · , n − 1]

represents one and only one vertex. The following lines respect the syntax “n1 n2 w”, where n1 and n2

are integers representing the nodes connected by an edge, and w the weight of this edge. n1, n2, and w

must be separated by space(s). An example of such file can be found on the course website with the file

g1.txt. These files will be used as an input in the program Kruskal.java to initialize the graphs.

Thus, an example of a command line is java Kruskal g1.txt.

The class WGraph also provide a method addEdge(Edge e) that adds an edge to a graph (i.e. an

object of this class). Another method listOfEdgesSorted() allows you to access the list of edges

of a graph in increasing order of their weight.

You task will be to complete the two static methods isSafe(DisjointSets p, Edge e) and

kruskal(WGraph g) in Kruskal.java. The method isSafe considers a partition of the nodes

p and an edge e, and should return True if it is safe to add the edge e to the MST, and False otherwise.

The method kruskal will take a graph object of the class WGraph as an input, and return another

WGraph object which will be the MST of the input graph.

Once completed, compile all the java files and run the command line java A2.Kruskal g1.txt.

An example of the expected output is available in the file mst1.txt. You are invited to run other

examples of your own to verify that your program is correct.

Exercise 3 (Greedy algorithms (30 points)) In this exercise, you will plan out your homework with

a greedy algorithm. You are given as input a list of homeworks defined by two arrays, an array of

weights (the relative importance of the homework towards your final grade), and an array of deadlines.

Those arrays are not sorted. Each index on those arrays, which are of the same size, represents a single

homework to submit. Weights and deadlines are both integers between 1 and 100. Each homework

takes exactly one hour to complete. Your task is to output a homeworkPlan, which will take the form

of an array of length equal to the weights and deadlines. Each index in this array represents the same

homework as each index in the input arrays. For each homework, indicate the time at which you plan on

completing the homework. This time should be defined as an integer between 1 and the latest deadline of

your homeworks, divided into slots of one hour. If you do not plan on completing a specific homework

at all, indicate 0 in the corresponding slot of the homeworkPlan . The homework is considered due

at the end of the time slot. In other words, if the homework is due at t=14, then you can complete it

before or during the slot t=14. For example, Homework 2 is defined as the homework with deadline

deadlines[2] and weight weights[2] If your solution plans on doing Homework 2 first, then

homeworkPlan[2]=1 should appear in your output. You can only complete a single homework in a

1 hour slot. Note that sometimes you will be given too much homework to complete in time, and that is

okay.

Your homework plan should maximize the sum of the weights of completed assignments.

To organize your schedule, we give you a class HW_sched.java, which defines an Assignment

object, with a number (it’s index in the input array), a weight and a deadline.In addition, we provide

you the file GreedyTester.java that demonstrates how to use this class to initialize an ArrayList of

homework objects of the appropriate size.

In order to organize your schedule efficiently, you will have to sort the homeworks. For your convenience, we provide a compare method, which takes as input two assignments, compares them, and

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outputs the order they should appear it. You have to determine what comparison criterion you want to

use to compare assignments in this problem. Given two assignments A1 and A2, the method should

output:

• 0, if the two items are equivalent

• 1, if a1 should appear after a2 in the sorted list

• -1, if a2 should appear after a1 in the sorted list

Your compare method should be the only tool you use to re-organize lists and arrays in this problem.

You will submit all the Java files you modified in this assignment, in a single zip file.

Exercise 4 (10 points) You will answer to this section through MyCourses. Note that you MUST use

your own results to answer those questions. Answers to this quiz that would not match the output of your

program will be considered as plagiarism (refer to course outline).

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