Question

Illustrate Kruskal’s algorithm on the graph Γ. In addition

show how the Union-Find data structure changes throughout the
algorithm.

Answer 1

Kruskal's algorithm is used to find a minimum spanning tree in a graph. The Union-Find data structure is used to track connected components and avoid creating cycles. An example graph Γ is illustrated, and the algorithm is applied step by step with changes in the Union-Find data structure.

Step-by-step explanation:

Kruskal's algorithm is a widely used algorithm in graph theory to find a minimum spanning tree of a connected undirected graph. It starts by sorting all the edges in increasing order of their weights. Then, it repeatedly adds the smallest edge to the spanning tree as long as it doesn't create a cycle. The Union-Find data structure is used to keep track of the connected components and detect cycles.

For example, let's consider the graph Γ with the following edges and weights:

• AB with weight 2
• BC with weight 3
• CD with weight 1
• DE with weight 4
• EA with weight 5

Applying Kruskal's algorithm, we start by sorting the edges in increasing order of their weights:

• CD with weight 1
• AB with weight 2
• BC with weight 3
• DE with weight 4
• EA with weight 5

Then, we initialize the Union-Find data structure with each vertex in its own component. As we process the edges one by one, we check if the two vertices of the edge belong to the same component. If they don't, we merge the components and add the edge to the minimum spanning tree. Otherwise, we ignore the edge to avoid creating a cycle.

In this case, we can follow these steps:

1. Create separate components for each vertex: A, B, C, D, E
2. Choose the edge CD with weight 1 and merge components C and D
3. The Union-Find data structure now has two components: {A, B} and {C, D}
4. Choose the edge AB with weight 2 and merge components A and B
5. The Union-Find data structure now has two components: {A, B, C, D} and {E}
6. Choose the edge BC with weight 3. Since B and C are already in the same component, we ignore this edge
7. Choose the edge DE with weight 4 and merge components D and E
8. The Union-Find data structure now has one component: {A, B, C, D, E}
9. Choose the edge EA with weight 5 and ignore it to avoid creating a cycle

Finally, the resulting minimum spanning tree is:

• AB with weight 2
• CD with weight 1
• DE with weight 4

The Union-Find data structure keeps track of the connected components and changes throughout the algorithm as the components are merged.