minimum spanning tree
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- A minimum spanning tree (MST) connects all nodes in a graph with the minimum total edge weight while avoiding cycles. - There are different algorithms that can find an MST, such as Kruskal's algorithm and Prim's algorithm, which were introduced in the document. - Kruskal's algorithm works by sorting the edges by weight and building the MST by adding the shortest edges that do not create cycles. Prim's algorithm grows the MST from an initial node by always adding the shortest edge connected to the current MST.
minimum spanning tree
- 1. Minimum spanning treesMinimum spanning trees 1
- 2. MST A minimum spanning tree connects all nodes in a given graph A MST must be a connected and undirected graph A MST can have weighted edges Multiple MSTs can exist within a given undirected graph Multiple MSTs can be generated depending on which algorithm is used If you wish to have an MST start at a specific node However, if there are weighted edges and all weighted edges are unique, only one MST will exist 2
- 3. Borůvka’s Algorithm • The first MST Algorithm was created by Otakar Borůvka in 1926 • The algorithm was used to create efficient connections between the electricity network in the Czech Republic • No longer used since Prim’s and Kruskal’s algorithms were discovered 3
- 4. Prim’s Algorithm • Initially discovered in 1930 by Vojtěch Jarník, then rediscovered in 1957 by Robert C. Prim • Similar to Dijkstra’s Algorithm regarding a connected graph • Starts off by picking any node within the graph and growing from there 4
- 5. Kruskal’s Algorithm • Created in 1957 by Joseph Kruskal • Finds the MST by taking the smallest weight in the graph and connecting the two nodes and repeating until all nodes are connected to just one tree • This is done by creating a priority queue using the weights as keys • Each node starts off as it’s own tree • While the queue is not empty, if the edge retrieved connects two trees, connect them, if not, discard it • Once the queue is empty, you are left with the minimum spanning tree 5
- 6. Minimum Connector Algorithms Kruskal’s algorithm 1. Select the shortest edge in a network 2. Select the next shortest edge which does not create a cycle 3. Repeat step 2 until all vertices have been connected Prim’s algorithm 1. Select any vertex 2. Select the shortest edge connected to that vertex 3. Select the shortest edge connected to any vertex already connected 4. Repeat step 3 until all vertices have been connected Prim’s algorithm 1. Select any vertex 2. Select the shortest edge connected to that vertex 3. Select the shortest edge connected to any vertex already connected 4. Repeat step 3 until all vertices have been connected 6
- 7. A cable company want to connect five villages to their network which currently extends to the market town of Addis Ababa. What is the minimum length of cable needed? Addis Hawassa Adama Awash Dilla Wolita 2 7 4 5 8 6 4 5 3 8 Example 7
- 8. We model the situation as a network, then the problem is to find the minimum connector for the network A F B C D E 2 7 4 5 8 6 4 5 3 8 8
- 9. A F B C D E 2 7 4 5 8 6 4 5 3 8 List the edges in order of size: ED 2 AB 3 AE 4 CD 4 BC 5 EF 5 CF 6 AF 7 BF 8 CF 8 Kruskal’s Algorithm 9
- 10. Select the shortest edge in the network ED 2 Kruskal’s Algorithm A F B C D E 2 7 4 5 8 6 4 5 3 8 10
- 11. Select the next shortest edge which does not create a cycle ED 2 AB 3 Kruskal’s Algorithm A F B C D E 2 7 4 5 8 6 4 5 3 8 11
- 12. Select the next shortest edge which does not create a cycle ED 2 AB 3 CD 4 (or AE 4) Kruskal’s Algorithm A F B C D E 2 7 4 5 8 6 4 5 3 8 12
- 13. Select the next shortest edge which does not create a cycle ED 2 AB 3 CD 4 AE 4 Kruskal’s Algorithm A F B C D E 2 7 4 5 8 6 4 5 3 8 13
- 14. Select the next shortest edge which does not create a cycle ED 2 AB 3 CD 4 AE 4 BC 5 – forms a cycle EF 5 Kruskal’s Algorithm A F B C D E 2 7 4 5 8 6 4 5 3 8 14
- 15. All vertices have been connected. The solution is ED 2 AB 3 CD 4 AE 4 EF 5 Total weight of tree: 18 Kruskal’s Algorithm A F B C D E 2 7 4 5 8 6 4 5 3 8 15
- 16. A F B C D E 2 7 4 5 8 6 4 5 3 8 Select any vertex A Select the shortest edge connected to that vertex AB 3 Prim’s Algorithm 16
- 17. A F B C D E 2 7 4 5 8 6 4 5 3 8 Select the shortest edge connected to any vertex already connected. AE 4 Prim’s Algorithm 17
- 18. Select the shortest edge connected to any vertex already connected. ED 2 Prim’s Algorithm A F B C D E 2 7 4 5 8 6 4 5 3 8 18
- 19. Select the shortest edge connected to any vertex already connected. DC 4 Prim’s Algorithm A F B C D E 2 7 4 5 8 6 4 5 3 8 19
- 20. Select the shortest edge connected to any vertex already connected. EF 5 Prim’s Algorithm A F B C D E 2 7 4 5 8 6 4 5 3 8 20
- 21. Prim’s Algorithm A F B C D E 2 7 4 5 8 6 4 5 3 8 All vertices have been connected. The solution is AB 3 AE 4 ED 2 DC 4 EF 5 Total weight of tree: 18 21
- 22. •Both algorithms will always give solutions with the same length. •They will usually select edges in a different order •Occasionally they will use different edges – this may happen when you have to choose between edges with the same length. In this case there is more than one minimum connector for the network. Some points to note 22
- 23. Questions? 23