1. Minimum Spanning Trees

2. Example Application for MST
Minimum Spanning Tree for
Electrical wiring

3. Spanning Trees Connected and having no cycles.
A graph having more than V – 1 edges contains at least one cycle.

4. Minimum Spanning Tree (MST)
What is minimum spanning tree?
A tree that covers (spans) all the vertices of a connected graph which has the minimum total cost of edges in the tree.
A MST exists for a graph iff the graph is connected.
The same problem makes sense for directed graphs too, but the solution is more difficult.

5. Minimum Spanning Tree (MST)
Our problem is finding a minimum spanning tree in an undirected and connected graph.

6. Example: Minimum Spanning Treev1 2 v2 4 1 3 10 2 v3 v4 7 v5 8 4 5 6 v6 1 v7
Example undirected and connected graph v1 v2 v5 v3 v6 v7 2 1 4 6 v4
The corresponding minimum spanning tree

7. Properties: Minimum Spanning Tree
If the number of vertices of a connected undirected graph is |V|, then its minimum spanning tree will have
|V| vertices
|V| - 1 edges.
A MST does not contain any cycle, since it is a tree.
If we add an extra edge to a MST, then it will have a cycle.

8. Example Application for MST
power outlet
or light
Electrical wiring of a house using minimum amount of wires (cables)

9. Example Application for MST
Minimum Spanning Tree for
Electrical wiring

10. Prim’s Algorithm for MST
MST is constructed in successive stages.
At each stage:
A new vertex is added to the tree by choosing the edge (u,v) such that the cost of (u,v), w(u,v), is the smallest among all edges where u is in the tree and v is not.

11. 1 2 3 4 5 6 7

14. 6 8 5 3 7 9 15

15. Prim’s Algorithm: Example
Start with vertex v1. It is the initial current tree which
we will grow to an MST
V1 0
V4 1
V2 2
V3 4
V6  v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4
Content of the priority queue
A connected, undirected graph G is given above.

16. Prim’s Algorithm: Example
Select an edge from graph: that is not in the current tree,
that has the minimum cost, and that can be connected to the current tree.
Step 1 v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4
V4 1
V2 2
V3 4
Content of the priority queue

17. Prim’s Algorithm: Example
The edges that can be connected are:
(v1,v2): cost 2
(v1,v4): cost 1
(v1,v3): cost 2
Step 1 v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4
V4 1
V2 2
V3 4
Content of the priority queue

18. Prim’s Algorithm: Example
We include the vertex v4, that is connected to the selected edge, to the current tree. In this way we grow the tree.
Step 2 v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4
V2 2
V3 2
V7 4
V5 7
V6 8
Content of the priority queue

19. Content of the priority queue
Prim’s Algorithm
Repeat steps: 1, and 2
Add either edge (v4, v3) v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4
V3 2
V7 4
V5 7
V6 8
Content of the priority queue

20. Content of the priority queue
Prim’s Algorithm
Grow the tree! v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4
V7 4
V6 5
V5 7
Content of the priority queue

21. Content of the priority queue
Prim’s Algorithm
Add edge (v4, v7) v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4
V7 4
V6 5
V5 7
Content of the priority queue

22. Content of the priority queue
Prim’s Algorithm
Grow the tree! v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4
V6 1
V5 6
Content of the priority queue

23. Content of the priority queue
Prim’s Algorithm
Add edge (v7, v6) v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4
V6 1
V5 6
Content of the priority queue

24. Content of the priority queue
Prim’s Algorithm
Grow the tree! v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4
V5 6
Content of the priority queue

25. Content of the priority queue
Prim’s Algorithm
Add edge (v7, v5) v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4
V5 6
Content of the priority queue

26. Prim’s Algorithm: Example
Grow the tree! v1 v2 v5 v3 v6 v7 4 2 1 5 8 7 3 10 6 v4
Content of the priority queue

27. Prim’s Algorithm: Examplev1 v2 v5 v3 v6 v7 2 1 4 6 v4
Finished! The resulting MST is shown below!

32. PRIM’s algorithm is similar to the Dijkstra’s shortest path algorithm
PRIM’s algorithm is similar to the Dijkstra’s shortest path algorithm. Complexities are the same.