Disjoint Data Structures

Let's try another approach.

Idea

Arr[ A ] is a parent of A.

Consider the root element of each subset, which is only a special element in that subset having itself as the parent. Assume that R is the root element, then arr[ R ] = R.

For more clarity, consider the subset S = {0, 1, 2, 3, 4, 5}

Initially each element is the root of itself in all the subsets because arr[ i ] = i, where i is the element in the set. Therefore root(i) = i.

Performing union(1, 0) will connect 1 to 0 and will set root (0) as the parent of root (1). As root(1) = 1 and root(0) = 0, the value of arr[ 1 ] will change from 1 to 0. Therefore, 0 will be the root of the subset that contains the elements {0, 1}.

Performing union (0, 2), will indirectly connect 0 to 2 by setting root(2) as the parent of root(0). As root(0) is 0 and root(2) is 2, it will change value of arr[ 0 ] from 0 to 2. Therefore, 2 will be the root of the subset that contains the elements {2, 0, 1}.

Performing union (3, 4) will indirectly connect 3 to 4 by setting root(4) as the parent of root(3). As root(3) is 3 and root(4) is 4, it will change the value of arr[ 3 ] from 3 to 4. Therefore, 4 will be the root of the subset that contains the elements {3, 4}.

Performing union (1, 4) will indirectly connect 1 to 4 by setting root(4) as the parent of root(1). As root(4) is 4 and root(1) is 2, it will change value of arr[ 2 ] from 2 to 4. Therefore, 4 will be the root of the set containing elements {0, 1, 2, 3, 4}.

After each step, you will see the change in the array arr also.

After performing the required union(A, B) operations, you can perform the find(A, B) operation easily to check whether A and B are connected. This can be done by calculating the roots of both A and B. If the roots of A and B are the same, then it means that both A and B are in the same subset and are connected.

Calculating the root of an element

Arr[ i ] is the parent of i (where i is the element of the set). The root of i is Arr[ Arr[ Arr[ ...Arr[ i ]...... ] ] ] until arr[ i ] is not equal to i. You can run a loop until you get an element that is a parent of itself.

Note This can only be done when there is no cycle in the elements of the subset, else the loop will run infinitely.

1. Find(1, 4): 1 and 4 have the same root i.e. 4. Therefore, it means that they are connected and this operation will give the result True.

2. Find(3, 5): 3 and 5 do not have same root because root(3) is 4 and root(5) is 5. This means that they are not connected and this operation will give the result False.

Implementation

Initially each element is a parent of itself, which can be done by using the initialize function as discussed above.

``  //finding root of an element    int root(int Arr[ ],int i)    {        while(Arr[ i ] != i)           //chase parent of current element until it reaches root        {         i = Arr[ i ];        }        return i;    }    /*modified union function where we connect the elements by changing the root of one of the elements*/    int union(int Arr[ ] ,int A ,int B)    {        int root_A = root(Arr, A);               int root_B = root(Arr, B);          Arr[ root_A ] = root_B ;       //setting parent of root(A) as root(B)    }    bool find(int A,int B)    {        if( root(A)==root(B) )       //if A and B have the same root, it means that they are connected.        return true;        else        return false;    }``
In the worst case, this idea will also take linear time in connecting 2 elements and determining (finding) whether two elements are connected. Another disadvantage is that while connecting two elements, which subset has more elements is not checked. This may sometimes create a big problem because you will have to perform approximately linear time operations.
Implementation of above approach:
#include<iostream>
using namespace std;
void initialize( int Arr[ ], int N)
{
for(int i = 0;i<N;i++)
Arr[ i ] = i ;
}
//returns true if A and B are connected, else returns false
//finding root of an element
int root(int Arr[ ],int i)
{
while(Arr[ i ] != i) //chase parent of current element until it reaches root
{
i = Arr[ i ];
}
return i;
}

/*modified union function where we connect the elements by changing the root of one of the elements*/

void unionr(int Arr[ ] ,int A ,int B)
{
int root_A = root(Arr, A);
int root_B = root(Arr, B);
Arr[ root_A ] = root_B ; //setting parent of root(A) as root(B)
}
bool find(int * a,int A,int B)
{
if( root(a,A)==root(a,B) ) //if A and B have the same root, it means that they are connected.
return true;
else
return false;
}
int main(){
int a[11],n=10;
initialize(a,n);
cout<<"Printing array entries:\n";
for(int i=0;i<n;i++){
cout<<a[i]<<" ";
}
cout<<endl;
cout<<"Checking whether 2 and 3 belongs to same array:\n";
if(!find(a,2,3)){
cout<<"No";
}
else{
cout<<"Yes";
}
cout<<endl;
cout<<"Making 2 and 3 to be in same group\n";
unionr(a,2,3);
cout<<"Checking whether 2 and 3 belongs to same array:\n";
if(!find(a,2,3)){
cout<<"No";
}
else{
cout<<"Yes";
}
cout<<endl;

}
Output:
`Printing array entries:0 1 2 3 4 5 6 7 8 9Checking whether 2 and 3 belongs to same array:NoMaking 2 and 3 to be in same groupChecking whether 2 and 3 belongs to same array:Yes`

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