In this assignment we will implement three variations of the disjoint union-find algorithm
alongwith its data structure as discussed in class. 
We are to implement this using object oriented approaches and C#/C++/Java. 
The data structure for the disjoint union-find is such that 
every set is kept in a tree with the nodes pointing to their parents. 
Further, there would be a pointer from each item to its corresponding node 
in the tree of the set to which it currently belongs. 
Each set's internal name is the smallest item in the set. 
Each node in the tree would keep a parent pointer, 
NumberOfItemsInSubtree field indicating number of items in the subtree rooted at the node, 
InternalNameOfSet (the root of a tree would have the correct internal set name; all others would be obsolete), 
and any other necessary information.

Three variations of the disjoint union-find algorithm are 
(Soln1) weighting rule is applied during a union operation and path-compression is performed durng find, 
(Soln2) use the weighting rule but no path-compression, 
and (Soln3) no weighting rule and no path-compression during union’s and find’s, respectively.

Given a find operation, such as f(123), 
your find algorithm should output the internal set to which item 123 currently belongs (and for Soln1 performs a path compression). 
Given a union operation, such as u(12,45), first find the sets, say x and y, 
to which items 12 and 45 belong and then union the sets x and y (for Soln1 and Soln2 use weighting rule).

Given a sequence of finds and unions, 
record the time taken by this sequence and the heights of non-singleton sets 
(i.e., sets having more than one items) for each of the three variations Soln1, Soln2 and Soln3. 
Analyze these values for different sequences and size of universes. 
Print the values in a nice format along with non-singleton sets. 
Your program should be well documented, structured and follow object oriented approaches. 
Your output should be neat, clean and readable. Highlight important portions of your output.

The data file is hw4.dat and its format is as follows. (For each of the ni value run the k sequences as given below.)

n1, n2, n3, ..., nl - sizes of total number of items for each of the l test runs

m1 - size of the first sequence of finds and unions
sequence of length m1 with find being f(x) and union being u(x,y), such as f(x), f(c), u(x,y), u(y,z),f(d)
m2 - size of the second sequence of finds and unions
sequence of length m2

mk - size of the k-th sequence of finds and unions
sequence of length mk