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Exercise 1 (Disjoint sets (15 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 Question 2.

Exercise 2 (Minimum Spanning trees (25 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.

Exercise 3 (Greedy algorithms (20 points))
In this exercise, you will plan your homework with a greedy
algorithm. The input is a list of homeworks defined by two arrays: deadlines and weights (the relative importance of the homework towards your final grade). These arrays have the same size and they
contain integers between 1 and 100. The index of each entry in the arrays represents a single homework, for example, Homework 2 is defined as a homework with deadline deadlines[2] and weight
weights[2]. Each homework takes exactly one hour to complete.
Your task is to output a homeworkPlan: an array of length equal to the last deadline. Each entry in
the array represents a one-hour timeslot, starting at 0 and ending at ’last deadline – 1’. For each time slot,
homeworkPlan indicates the homework which you plan to do during that slot. You can only complete
a single homework in one 1-hour slot. The homeworks are due at the beginning of a time slot, in other
words if an assignment’s deadline is x, then the last time slot when you can do it is x – 1. For example,
if the homework is due at t=14, then you can complete it before or during the slot t=13. If your solution
plans to do Homework 2 first, then you should have homeworkPlan[0]=2 in the output. 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 (its 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.
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
Assignments will be ordered. You have to determine what comparison criterion you want to use. 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
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.