How to find maximum spanning tree?

For the problem of finding the maximum spanning tree

In graph theory, a spanning tree is a connected acyclic subgraph that includes all vertices of the graph. The maximum spanning tree refers to the spanning tree with the maximum sum of edge weights. The problem of finding the maximum spanning tree frequently arises in fields such as network design and circuit design. The commonly used algorithms for solving this problem are Prim's Algorithm and Kruskal's Algorithm. These algorithms are typically used for finding the minimum spanning tree, but by appropriately processing the edge weights, they can also be used to find the maximum spanning tree.

Prim's Algorithm

The basic idea of Prim's Algorithm is to start from a single vertex in the graph and incrementally construct a spanning tree that includes all vertices. In each iteration, the edge with the maximum weight connecting the current spanning tree to the remaining vertices is added.

1. Select any vertex in the graph as the starting point.
2. Find the edge with the maximum weight connecting the current spanning tree to the remaining vertices.
3. Add this edge and its corresponding vertex to the current spanning tree.
4. Repeat steps 2 and 3 until all vertices are included in the spanning tree.

Kruskal's Algorithm

The basic idea of Kruskal's Algorithm is to sort all edges of the graph in descending order of weight and then select edges in sequence to construct the maximum spanning tree.

1. Sort all edges of the graph in descending order of weight.
2. Initialize a forest containing all vertices but no edges (each vertex is its own connected component).
3. Consider each edge in sequence; if the two vertices connected by the edge belong to different connected components, add the edge and merge the corresponding components.
4. Repeat step 3 until all vertices are in the same connected component, forming a spanning tree.

Example

Suppose we have a graph with 4 vertices and 5 edges, with the following edge weights:

• A-B: 7
• A-D: 6
• B-C: 9
• B-D: 8
• C-D: 5

The steps for finding the maximum spanning tree using Kruskal's Algorithm are as follows:

1. Sort the edges: B-C(9), B-D(8), A-B(7), A-D(6), C-D(5).
2. Start by adding the edge with the largest weight: first add B-C.
3. Next, add B-D; at this point, the spanning tree includes vertices B, C, D.
4. Then add A-B; at this point, all vertices are included in the spanning tree.
5. At this point, the maximum spanning tree consists of edges B-C, B-D, and A-B, with a total weight of 24.

Prim's Algorithm can also yield the same maximum spanning tree, though the iterative process differs.

For both algorithms, whether finding the maximum or minimum spanning tree, the key lies in how edge weights are defined and compared. By negating the edge weights, we can utilize these algorithms to find the maximum spanning tree.