1. Minimum spanning trees
Mohamad Ayache

2. What is MST? A graph that connects ALL nodes together
The SHORTEST PATH between all the places you need to go
NO CYCLES ALLOWED

3. Mainly used in designing networks
Why we use mst?
Mainly used in designing networks
To reduce costs
EX:
electric power-water-sewage lines-telephone lines

4. Problem: Laying Telephone Wire4
Problem: Laying Telephone Wire
Central office

5. Wiring: Naïve Approach5
Wiring: Naïve Approach
Central office
Expensive!

6. Wiring: Better Approach6
Wiring: Better Approach
Central office
Minimize the total length of wire connecting the customers

7. Finding spanning trees
Kruskal’s algorithm
Prim’s algorithm

8. Kruskal’s algorithm Pick the smallest edge
Repeat until all nodes connected with no cycles

9. Kruskal Complete Graph 4 B C 4 2 1 A 4 E F 1 2 D 3 10 5 G 5 6 3 4 I HJ

10. Kruskal 4 1 A B A D 4 4 B C B D 4 10 2 B C B J C E 4 2 1 1 5 C F D H A6 2 2 D J E G D 3 10 5 G 3 5 F G F I 5 6 3 4 3 4 I G I G J H 3 2 J 2 3 H J I J

11. Kruskal
Sort Edges
(in reality they are placed in a priority queue - not sorted - but sorting them makes the algorithm easier to visualize) 1 1 A D C F 2 2 C E E G 4 2 3 B C H J F G 4 2 1 3 3 G I I J A 4 E F 1 4 4 2 A B B D D 3 10 5 G 4 4 B C G J 5 6 3 4 5 5 I F I D H H 3 2 J 6 10 D J B J

12. Kruskal Add Edge 1 1 A D C F 2 2 C E E G 4 2 3 B C H J F G 4 2 1 3 3 GI I J A 4 E F 1 4 4 2 A B B D D 3 10 5 G 4 4 B C G J 5 6 3 4 5 5 I F I D H H 3 2 J 6 10 D J B J

13. Kruskal Add Edge 1 1 A D C F 2 2 C E E G 4 2 3 B C H J F G 4 2 1 3 3 GI I J A 4 E F 1 4 4 2 A B B D D 3 10 5 G 4 4 B C G J 5 6 3 4 5 5 I F I D H H 3 2 J 6 10 D J B J

14. Kruskal Add Edge 1 1 A D C F 2 2 C E E G 4 2 3 B C H J F G 4 2 1 3 3 GI I J A 4 E F 1 4 4 2 A B B D D 3 10 5 G 4 4 B C G J 5 6 3 4 5 5 I F I D H H 3 2 J 6 10 D J B J

15. Kruskal Add Edge 1 1 A D C F 2 2 C E E G 4 2 3 B C H J F G 4 2 1 3 3 GI I J A 4 E F 1 4 4 2 A B B D D 3 10 5 G 4 4 B C G J 5 6 3 4 5 5 I F I D H H 3 2 J 6 10 D J B J

16. Kruskal Add Edge 1 1 A D C F 2 2 C E E G 4 2 3 B C H J F G 4 2 1 3 3 GI I J A 4 E F 1 4 4 2 A B B D D 3 10 5 G 4 4 B C G J 5 6 3 4 5 5 I F I D H H 3 2 J 6 10 D J B J

17. Kruskal Cycle Don’t Add Edge 1 1 A D C F 2 2 C E E G 4 2 3 B C H J F GI I J A 4 E F 1 4 4 2 A B B D D 3 10 5 G 4 4 B C G J 5 6 3 4 5 5 I F I D H H 3 2 J 6 10 D J B J

18. Kruskal Add Edge 1 1 A D C F 2 2 C E E G 4 2 3 B C H J F G 4 2 1 3 3 GI I J A 4 E F 1 4 4 2 A B B D D 3 10 5 G 4 4 B C G J 5 6 3 4 5 5 I F I D H H 3 2 J 6 10 D J B J

19. Kruskal Add Edge 1 1 A D C F 2 2 C E E G 4 2 3 B C H J F G 4 2 1 3 3 GI I J A 4 E F 1 4 4 2 A B B D D 3 10 5 G 4 4 B C G J 5 6 3 4 5 5 I F I D H H 3 2 J 6 10 D J B J

20. Kruskal Add Edge 1 1 A D C F 2 2 C E E G 4 2 3 B C H J F G 4 2 1 3 3 GI I J A 4 E F 1 4 4 2 A B B D D 3 10 5 G 4 4 B C G J 5 6 3 4 5 5 I F I D H H 3 2 J 6 10 D J B J

21. Kruskal Cycle Don’t Add Edge 1 1 A D C F 2 2 C E E G 4 2 3 B C H J F GI I J A 4 E F 1 4 4 2 A B B D D 3 10 5 G 4 4 B C G J 5 6 3 4 5 5 I F I D H H 3 2 J 6 10 D J B J

22. Kruskal Add Edge 1 1 A D C F 2 2 C E E G 4 2 3 B C H J F G 4 2 1 3 3 GI I J A 4 E F 1 4 4 2 A B B D D 3 10 5 G 4 4 B C G J 5 6 3 4 5 5 I F I D H H 3 2 J 6 10 D J B J

23. Kruskal Minimum Spanning Tree Complete Graph 4 B C 4 B C 4 4 2 1 2 1 AF E 1 F 1 2 D 2 D 3 10 G 5 G 3 5 6 3 4 I I H H 3 2 3 J 2 J

24. Cost of tree 1+1+2+2+2+3+3+4+4 = 22 Kruskal Minimum Spanning Tree 4 BD G 3 I H 3 2 J

25. prim’s algorithm Pick any node
Connect to the smallest edge to your tree
Repeat until all nodes connected with no cycles

26. PRIM Complete Graph 4 B C 4 2 1 A 4 E F 1 2 D 3 10 5 G 5 6 3 4 I H 3 2J

27. PRIM Old Graph New Graph 4 B C 4 4 B C 2 1 4 2 1 A 4 E F 1 A 4 E F 1 23 2 10 D 3 5 G 10 5 G 5 6 3 5 6 3 4 I 4 H I H 3 2 J 3 2 J

28. PRIM Old Graph New Graph 4 B C 4 4 B C 2 1 4 2 1 A 4 E F 1 A 4 E F 1 23 2 10 D 3 5 G 10 5 G 5 6 3 5 6 3 4 I 4 H I H 3 2 J 3 2 J

29. PRIM Old Graph New Graph 4 B C 4 4 B C 2 1 4 2 1 A 4 E F 1 A 4 E F 1 23 2 10 D 3 5 G 10 5 G 5 6 3 5 6 3 4 I 4 H I H 3 2 J 3 2 J

30. PRIM Old Graph New Graph 4 B C 4 4 B C 2 1 4 2 1 A 4 E F 1 A 4 E F 1 23 2 10 D 3 5 G 10 5 G 5 6 3 5 6 3 4 I 4 H I H 3 2 J 3 2 J

31. PRIM Old Graph New Graph 4 B C 4 4 B C 2 1 4 2 1 A 4 E F 1 A 4 E F 1 23 2 10 D 3 5 G 10 5 G 5 6 3 5 6 3 4 I 4 H I H 3 2 J 3 2 J

32. PRIM Old Graph New Graph 4 B C 4 4 B C 2 1 4 2 1 A 4 E F 1 A 4 E F 1 23 2 10 D 3 5 G 10 5 G 5 6 3 5 6 3 4 I 4 H I H 3 2 J 3 2 J

33. PRIM Old Graph New Graph 4 B C 4 4 B C 2 1 4 2 1 A 4 E F 1 A 4 E F 1 23 2 10 D 3 5 G 10 5 G 5 6 3 5 6 3 4 I 4 H I H 3 2 J 3 2 J

34. PRIM Old Graph New Graph 4 B C 4 4 B C 2 1 4 2 1 A 4 E F 1 A 4 E F 1 23 2 10 D 3 5 G 10 5 G 5 6 3 5 6 3 4 I 4 H I H 3 2 J 3 2 J

35. PRIM Old Graph New Graph 4 B C 4 4 B C 2 1 4 2 1 A 4 E F 1 A 4 E F 1 23 2 10 D 3 5 G 10 5 G 5 6 3 5 6 3 4 I 4 H I H 3 2 J 3 2 J

36. PRIM Old Graph New Graph 4 B C 4 4 B C 2 1 4 2 1 A 4 E F 1 A 4 E F 1 23 2 10 D 3 5 G 10 5 G 5 6 3 5 6 3 4 I 4 H I H 3 2 J 3 2 J

37. PRIM Complete Graph Minimum Spanning Tree 4 B C 4 4 B C 2 1 4 2 1 A 4F 1 A E F 1 2 D 3 2 10 D 5 G G 5 6 3 3 4 I H I H 3 2 J 3 2 J

38. Cost of tree 1+1+2+2+2+3+3+4+4 = 22 PRIM Minimum Spanning Tree 4 1 2 3B C D E F G H I J
Minimum Spanning Tree
Cost of tree
= 22

39. Analysis Kruskal’s algorithm Prim’s algorithm
consider the spanning tree to consist of edges only
works best if the number of edges is kept to a minimum
consider the spanning tree to consist of both nodes and edges
works best if the number of edges and nodes is kept to a minimum
For small graphs, the edges matter more, while for large graphs the number of nodes matters more.

40. Thank you for listening