The Plot Thickens: Unveiling the Mystery of Union Operation

Let's continue our detective adventure to find out if our city's road network forms a loop. After investigating the first road connecting intersections 0 and 1, we've found they belong to different neighborhoods. Time to merge them!

Step 1: Merging Neighborhoods with Union(0, 1)

We will combine the neighborhoods of 0 and 1 into one. To do this, we'll update our parent array, which is our tracking mechanism.

Visual Update of the Investigation Board

Step 2: Understanding the Change in Parent Array

In the parent array, changing the value at index 0 to 1 is akin to saying that neighborhood 1 has now absorbed neighborhood 0. In family terms, it's like saying 0 is now under the 'parental guidance' of 1.

What Does This Update Mean?

This change signifies that intersection 1 is now the representative of the new, larger neighborhood {0, 1}. This merged neighborhood is like a bigger family now, and 1 is the head of the family.

Step 3: Examining the Next Clue - Edge (0,2)

Now, let's investigate another road—this one connecting intersections 0 and 2.

What will happen? Is 2 also part of this new, larger neighborhood, or is it a separate entity? Is there a loop forming, or are we still in the clear?

1Find(0) = 1
2Find(2) = 2

Here's some code to illustrate how this would work:

1// Define the Union function to merge neighborhoods
2function union(vertex1, vertex2) {
3    const root1 = find(vertex1);
4    const root2 = find(vertex2);
5    parent[root1] = root2;
6}
7
8// Let's merge the neighborhoods of 0 and 1
9union(0, 1);
10
11// Now, investigate another road connecting intersections 0 and 2
12const rootOfVertex0 = find(0);  // Find(0) should now return 1, as 0 and 1 are in the same neighborhood
13const rootOfVertex2 = find(2);  // Find(2) should return 2
14
15console.log(`Find(0) = ${rootOfVertex0}`);  // Outputs: Find(0) = 1
16console.log(`Find(2) = ${rootOfVertex2}`);  // Outputs: Find(2) = 2