$1$ Instructions

Implement two algorithms

$1.$ UNION$-$FIND: You must implement this data structure where

$$ UNION operation is done by rank

$2.$ UNION$-$FIND$-$WITH$-$PATH$-$COMPRESSION: You must implement this data structure where

$$ UNION operation is done by rank and

$$ FIND$-$SET is implemented with path compression

$2\mathrm{.}1$ Test Cases

Each test case will be input in the form of a text file

Input format

$$ First line $-$ n $($Number of elements$)$

$$ Number of UNION Operations will be n $1$

$$ Second line $-$ m $($Number of FIND$-$SET Operations$)$

$$ Next m $+($n $1)$ lines will either be a

$1.$ Union operation Uxy, where $1<=$ x$,$ y $<=$ n

$2.$ Find operation Fx$,$ where $1<=$ x $<=$ n

Output format

$$ m lines for each FIND$-$SET operation, where each line outputs the root$/$representative

of the set to which x belongs

Sample Input

$4$

$6$

F $2$

U $12$

F $2$

U $23$

F $3$

F $2$

F $4$

U $14$

F $4$

Sample Output

$2$

$1$

$1$

$1$

$4$

$1$

Details

$$ Uxy should merge the two sets containing x and y

$$ Definition of Rank

$$ For union by rank, a node stores its rank which is an upper bound for its height.

$$ When a node is initialized, its rank is set to zero.

$$ To merge trees with roots x and y$,$ first compare their ranks.

$$ If the ranks are different, then the larger rank tree becomes the parent, and the

ranks of x and y do not change.

$$ Tie Break

$$ Let x in T$1,$ y in T$2$

$$ If rank$($T$1)=$ rank$($T$2),$ root of T$1$ must be made the root of the resultant

merged tree

$$ Fx should output the root of the tree to which x belongs in the output file

$2\mathrm{.}2$ Test Case Execution

$$ The input file name will be given as a command line argument with the option $-$i or

$--$input $[$See below for example$]$

$$ Assume the file exists in the current directory where the program is located

$$ Output file must be named output.txt $[$See below for example$]$

$$ Output file must be created in the current directory where the program is located

See the following commands for python $($but could be extrapolated for any other language$)$

$ python$3$ union$$f i n d $.$ py $$i t e s t $1.$ t x t

$ python$3$ union$$f i n d $.$ py $$i n p u t t e s t $1.$ t x t

In this case, the output file must be created in the current directory with the name

test $1$ output.txt