Kruskal's Algorithm (Edexcel IAL Maths)

Spec point — 2.1

Kruskal's Algorithm

What is Kruskal's algorithm?

• Kruskal's algorithm is a mathematical tool that can be used to connect all of the vertices in a network in a way that reduces costs, materials or time

  • It achieves this by finding a minimum spanning tree (MST).

Why do we use Kruskal’s Algorithm?

• Kruskal’s algorithm is a series of steps that when followed will produce a minimum spanning tree for a network

• Finding a minimum spanning tree is useful in a lot of practical applications

  • It connects 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 minimum spanning tree cannot contain any cycles

What are the steps of 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

  • Reject it if it forms a **cycle **with the other selected edges
  • Otherwise, add it to your tree

• STEP 4 Repeat STEP 3 until all of the vertices in the graph are** connected**

  • If there are n nodes then there will be n - 1 edges in the minimum spanning tree

• STEP 5 List the edges in the minimum spanning tree in the order they were added

Exam Hint Draw the edges included in the minimum spanning tree as you go along; this will make it easier to spot potential cycles.

Remember to state the order in which the edges are selected or rejected to get full marks for working!

Worked Example Consider the network below.

a) Use Kruskal’s algorithm to find the minimum spanning tree. Show each step of the algorithm clearly.

Answer:

• *STEP 1 *** List the edges in order of increasing weight

EH (1) DH (2) AB (4) EF (5) FG (5) GH (6) BC (6) BF (7) CD (8) AD (9) AE (11) CG (15)

• STEP 2* * Add the edge of least weight (EH)

• STEP 3 Add the next edge of least weight (DH)

• *STEP 4 *** Repeat Step 3 Repeat adding the next edge of least weight each time provided it does not create a cycle until all vertices are connected

  • Add AB
  • Add EF
  • Add FG
  • Reject GH as it forms a cycle
  • Add BC
  • Add BF

Final answer: Edges are added in the order: EH (1), DH (2), AB (4), EF (5), FG (5), reject GH, BC (6), BF (7)**

b) State the total weight of the minimum spanning tree.

Answer:

Add up the weights of the edges in the minimum spanning tree

1 + 2 + 4 + 5 + 5 + 6 + 7 = 30

Final answer: Total weight = 30