Description

General instructions (Read carefully!)
• Your solution must be submitted electronically on codePost. Here is a short tutorial to help you understand how the platform works. You should already be on the platform since this is the second assignment.
Technical considerations
• 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.
• Note that your code will be evaluated with Java 8 on a linux environment (similar to ubuntu). It is your responsibility to make sure your code compiles and runs correctly when executed from command line in this type of environment.
• 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. Do not add packages or change the import statements.
• Submit your files individually on codePost (no zips)
• You will automatically get 0 if the files you submitted on codePost do not compile, since you can ensure yourself that they do.
Exercise 1 (Disjoint sets (24 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 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 the case where the rank of both trees is the same, the rank of the new root increases by 1. You can implement the rank with a 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 initializes 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 the case where 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 Exercise 2.
Exercise 2 (Minimum Spanning trees (24 points)) We will implement the Kruskal algorithm to calculate the minimum spanning tree (MST) of an undirected weighted graph. Here you will use the file
DisjointSets.java completed in the previous question and two other files: WGraph.java and 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.
The file WGraph.java implements two classes WGraph and Edge. An Edge object 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 the 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 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. Note that you need to compile it with the two other files for it to work.
You should also update grade in HW_Sched, which is your total grade based on the sum of the weights of the homework you completed. Completing all but one homework with a weight of 10 and no late penalties should yield a grade of 90%.
The input arrays are unsorted, as part of the greedy algorithm, the template we provide sorts the homeworks using Java’s Collections.sort(). This sort function uses Java’s compare() method, which takes two objects as input, compares them, and outputs the order they should appear in. The template will ask you to override this compare() method, which will alter the way in which the assignments will be ordered.
Given two assignments A1 and A2, the method compare should output:
0, if the two items are equivalent
1, if A1 should appear after A1 in the sorted list
-1, if A2 should appear after A1 in the sorted list
The compare method (called by Collections.sort()) should be the only tool you use to reorganize lists and arrays in this problem. You will then implement the rest of the SelectAssignments() method.
Submit your three Java files on codePost.