How to use Boruvka's algorithm for minimum spanning tree?

How to use Boruvka's algorithm for minimum spanning tree?

Boruvka's algorithm is a greedy algorithm used to find the minimum spanning tree of a graph. It works by repeatedly adding the cheapest edge that connects two components until all vertices are connected.

To use Boruvka's algorithm, we start by initializing each vertex as a separate component. Then, we repeat the following steps until all vertices are connected:

1. For each component, find the cheapest edge that connects it to another component.
2. Add these edges to the minimum spanning tree.
3. Merge the components that are connected by these edges.

We continue this process until all vertices are connected, and we have a minimum spanning tree.

Let's illustrate this algorithm with an example. Consider the following graph:

```
2
A-----B
|\ /|
| \ / |
3 | X | 1
| / \ |
|/ \|
C-----D
4
```

We start by initializing each vertex as a separate component:

```
A B C D
```

Then, we repeat the following steps:

1. For each component, find the cheapest edge that connects it to another component:
- A is connected to B with weight 2.
- B is connected to A with weight 2.
- C is connected to D with weight 4.
- D is connected to C with weight 4.
2. Add these edges to the minimum spanning tree:
- AB with weight 2.
- CD with weight 4.
3. Merge the components that are connected by these edges:
- A and B are merged into one component.
- C and D are merged into one component.

Now, we have the following components:

```
AB C D
```

We repeat the same steps:

1. For each component, find the cheapest edge that connects it to another component:
- AB is connected to CD with weight 1.
- CD is connected to AB with weight 1.
2. Add these edges to the minimum spanning tree:
- AB-CD with weight 1.
3. Merge the components that are connected by these edges:
- AB and CD are merged into one component.

Now, all vertices are connected, and we have the minimum spanning tree:

```
2
A-----B
\