## Description

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 indi-cates 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 avail-able 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 cal-culate 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 in-formations 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 “n_{1} n_{2} w”, where n_{1} and n_{2} are integers representing the nodes connected by an edge, and w the weight of this edge. n_{1}, n_{2}, 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 con-venience, 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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