The Climax: Unveiling the Existence of a Cycle

The suspense peaks as we look at the last remaining road, connecting intersections 3 and 4. These intersections are in separate neighborhoods, so once again, it's time for a merge.

Step 1: The Final Merge with Union(3, 4)

We're merging the neighborhoods of 3 and 4. Given that 3 is already the head of a large neighborhood {0, 1, 2, 3}, this merge will add 4 to this expanding community.

The Last Update on the Investigation Board

Step 2: Interpreting the Final State of the Parent Array

By updating the value under 3 to 4, we signify that 4 is now the head of this mega-neighborhood, representing the union set {0, 1, 2, 3, 4}. It's as if 4 has assumed the role of the ultimate community leader, consolidating all the smaller communities into one.

Step 3: The Moment of Truth

Now, let's pay attention to the last edge that connects 3 and 4. According to our parent array, both intersections are part of the same large neighborhood. Using our find operation, we discover:

1Find(3) = 4
2Find(4) = 4

The results are clear as day. Both 3 and 4 are part of the same neighborhood, which means we've found a cycle!

The Final Verdict: A Cycle Detected!

That's our "Aha!" moment! Finding that both elements belong to the same group confirms the presence of a cycle in this graph. Our detective journey using the Union-Find algorithm has reached its pinnacle, successfully revealing a loop in our network of roads.

xxxxxxxxxx

}

// Initialize the parent array to represent isolated neighborhoods.

const parent = [0, 1, 2, 3, 4];

​

// Define the find function to identify the 'root' or 'parent' of each neighborhood.

function find(vertex) {

    if (parent[vertex] === vertex) {

        return vertex;

    }

    return find(parent[vertex]);

}

​

// Define the Union function to merge neighborhoods.

function union(vertex1, vertex2) {

    const root1 = find(vertex1);

    const root2 = find(vertex2);

    parent[root1] = root2;

}

​

// Merging previous neighborhoods.

union(0, 1);

union(1, 2);

union(2, 3);

​

// Investigate the last remaining road connecting intersections 3 and 4.

union(3, 4);

​

// Display the updated parent array.

console.log(`Updated Parent Array: ${parent}`);

​