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## A. Union-Find

Consider a partition of a contiguous set of integers 1 to V.
Initially, the partition consists of a set of V sets, each containing
a single integer. We want to support the operations of
finding which partition any given
integer belongs to and merging two partitions into one ("union"). We identify
partitions by choosing one representative member for each.
Fill in the file `UnionFind.java`

to implement the `find`

and `union`

operations.
Do ** not** use any data structures from the Java library (the utilities
in the

`Arrays`

class are OK), and do **define any classes other than**

*not*`UnionFind`

. Just use arrays and primitive values.## B. Minimal Spanning Trees

Consider an undirected graph whose vertices are numbered 1 to V and
whose edges are labeled with non-negative integer weights. Given such a graph,
we would like to find a minimal spanning tree in the form of a list of edges
from the graph that are included in the tree. To give you some exercise in
true data-structure hacking, for this problem we'll represent the graph using
only arrays and ints. Fill in the skeleton file `MST.java`

to use Kruskal's
Algorithm to create this tree.
Assume that we will be giving you *large* graphs to operate on.
As for part A, do ** not** use any data structures from the
Java library.

The class `MSTTest`

runs some JUnit tests on your MST algorithm. If a test case
fails, you can use the main program of `Utils.java`

to see exactly what graph was
fed to it, if desired. The test cases will print out parameters to
the main program that generate the graph in the test case. Executing

` java Utils V MINEDGES MAXWEIGHT RANDOM-SEED true`

prints the corresponding test case and the MST your program produces from it,.
(The arguments V through RANDOM-SEED tell `Utils.randomConnectedGraph`

what graph to generate). This is less helpful on really large test cases,
of course.