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) = 4The 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}`);