3.10.3 Minimum Spanning Trees

In a situation that can be modelled by a graph, Kruskal’s algorithm is a mathematical tool that can be used to reduce costs, materials or time.

Why do we use Kruskal’s Algorithm?

• Kruskal’s algorithm is a series of steps that when followed will produce the minimum spanning tree for a connected graph
• Finding the minimum spanning tree is useful in a lot of practical applications to connect all of the vertices in the most efficient way possible
• The number of edges in a minimum spanning tree will always be one less than the number of vertices in the graph
• A cycle is a walk that starts at a given vertex and ends at the same vertex.
• A minimum spanning tree cannot contain any cycles.

What is Kruskal’s Algorithm?

• STEP 1
  Sort the edges in terms of increasing weight
• STEP 2
  Select the edge of least weight (if there is more than one edge of the same weight, either may be used)
• STEP 3
  Select the next edge of least weight that has not already been chosen and add it to your tree provided that it does not make a cycle with any of the previously selected edges
• STEP 4
  Repeat STEP 3 until all of the vertices in the graph are connected

Prim’s algorithm is a second method of finding the minimum spanning tree of a graph.

What is Prim’s Algorithm?

• Prim’s algorithm involves adding edges from vertices that are already connected to the tree.
• Cycles are avoided by only adding edges that are not already connected at one end.
• STEP 1
  Start at any vertex and choose the edge of least weight that is connected to it
• STEP 2
  Choose the edge of least weight that is incident (connected) to any of the vertices already connected and does not connect to another vertex that is already in the tree
• STEP 3
  Repeat STEP 2 until all of the vertices are added to the tree

Information may be given to you either in the form of a graph or as a weighted adjacency table. Prim’s algorithm can be adapted to be used from the adjacency matrix.

How do you apply Prim’s algorithm to a matrix?

• A minimum spanning tree is built up from the least weight edges that are incident to vertices already in the tree by looking at the relevant rows in the adjacency table
• STEP 1
  Select any vertex to start from, cross out the values in the column associated with that vertex and label the row associated with the vertex 1
• STEP 2
  Circle the lowest value in any cell along that row and add the edge to your tree, cross out the remaining values in the column of the cell that you have circled
• STEP 3
  Label the row associated with the same vertex as the column in the previous STEP with the next
• STEP 4
  Circle the lowest value in any cell along any of the rows that have been labelled and add the edge to your tree, cross out the remaining values in the column of the cell that you have circled
• STEP 5
  Repeat STEPS 3 and 4 until all rows have been labelled and all vertices have been added to the tree

Which should I use Prim’s or Kruskal’s Algorithm?

• Kruskal’s algorithm can be used when the information is in graph form whereas Prim’s algorithm can be used in either graph or matrix form.
• Prim’s algorithm is sometimes considered to be more efficient that Kruskal’s algorithm as
  • the edges do not need to be ordered at the start and
  • it does not rely on checking for cycles at each step

• An exam question will usually specify which method should be used, otherwise you have the choice
• If you are asked to find the minimum spanning tree and the information given in the question is in the form of a table, you should use Prim’s algorithm