Prim Algorithm and kruskal algorithm
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The document discusses minimum spanning trees and two algorithms for finding them: Prim's algorithm and Kruskal's algorithm. Prim's algorithm works by growing a spanning tree from an initial node, always adding the edge with the lowest weight that connects to a node not yet in the tree. Kruskal's algorithm sorts the edges by weight and builds up a spanning tree by adding edges in order as long as they do not form cycles. Both algorithms run on undirected, weighted graphs and produce optimal minimum spanning trees.
Prim Algorithm and kruskal algorithm
- 1. Minimum Spanning Trees Text • Read Weiss, §9.5 Prim’s Algorithm • Weiss §9.5.1 • Similar to Dijkstra’s Algorithm Kruskal’s Algorithm • Weiss §9.5.2 • Focuses on edges, rather than nodes
- 2. Definition • A Minimum Spanning Tree (MST) is a subgraph of an undirected graph such that the subgraph spans (includes) all nodes, is connected, is acyclic, and has minimum total edge weight
- 3. Algorithm Characteristics • Both Prim’s and Kruskal’s Algorithms work with undirected graphs • Both work with weighted and unweighted graphs but are more interesting when edges are weighted • Both are greedy algorithms that produce optimal solutions
- 4. Prim’s Algorithm • Similar to Dijkstra’s Algorithm except that dv records edge weights, not path lengths
- 5. Walk-Through Initialize array K dv pv A F ∞ − B F ∞ − C F ∞ − D F ∞ − E F ∞ − F F ∞ − G F ∞ − H F ∞ − 4 25 A H B F E D C G 7 2 10 18 3 4 3 7 8 9 3 10 2
- 6. 4 25 A H B F E D C G 7 2 10 18 3 4 3 7 8 9 3 10 Start with any node, say D K dv pv A B C D T 0 − E F G H 2
- 7. 4 25 A H B F E D C G 7 2 10 18 3 4 3 7 8 9 3 10 Update distances of adjacent, unselected nodes K dv pv A B C 3 D D T 0 − E 25 D F 18 D G 2 D H 2
- 8. 4 25 A H B F E D C G 7 2 10 18 3 4 3 7 8 9 3 10 Select node with minimum distance K dv pv A B C 3 D D T 0 − E 25 D F 18 D G T 2 D H 2
- 9. 4 25 A H B F E D C G 7 2 10 18 3 4 3 7 8 9 3 10 Update distances of adjacent, unselected nodes K dv pv A B C 3 D D T 0 − E 7 G F 18 D G T 2 D H 3 G 2
- 10. 4 25 A H B F E D C G 7 2 10 18 3 4 3 7 8 9 3 10 Select node with minimum distance K dv pv A B C T 3 D D T 0 − E 7 G F 18 D G T 2 D H 3 G 2
- 11. 4 25 A H B F E D C G 7 2 10 18 3 4 3 7 8 9 3 10 Update distances of adjacent, unselected nodes K dv pv A B 4 C C T 3 D D T 0 − E 7 G F 3 C G T 2 D H 3 G 2
- 12. 4 25 A H B F E D C G 7 2 10 18 3 4 3 7 8 9 3 10 Select node with minimum distance K dv pv A B 4 C C T 3 D D T 0 − E 7 G F T 3 C G T 2 D H 3 G 2
- 13. 4 25 A H B F E D C G 7 2 10 18 3 4 3 7 8 9 3 10 Update distances of adjacent, unselected nodes K dv pv A 10 F B 4 C C T 3 D D T 0 − E 2 F F T 3 C G T 2 D H 3 G 2
- 14. 4 25 A H B F E D C G 7 2 10 18 3 4 3 7 8 9 3 10 Select node with minimum distance K dv pv A 10 F B 4 C C T 3 D D T 0 − E T 2 F F T 3 C G T 2 D H 3 G 2
- 15. 4 25 A H B F E D C G 7 2 10 18 3 4 3 7 8 9 3 10 Update distances of adjacent, unselected nodes K dv pv A 10 F B 4 C C T 3 D D T 0 − E T 2 F F T 3 C G T 2 D H 3 G 2 Table entries unchanged
- 16. 4 25 A H B F E D C G 7 2 10 18 3 4 3 7 8 9 3 10 Select node with minimum distance K dv pv A 10 F B 4 C C T 3 D D T 0 − E T 2 F F T 3 C G T 2 D H T 3 G 2
- 17. 4 25 A H B F E D C G 7 2 10 18 3 4 3 7 8 9 3 10 Update distances of adjacent, unselected nodes K dv pv A 4 H B 4 C C T 3 D D T 0 − E T 2 F F T 3 C G T 2 D H T 3 G 2
- 18. 4 25 A H B F E D C G 7 2 10 18 3 4 3 7 8 9 3 10 Select node with minimum distance K dv pv A T 4 H B 4 C C T 3 D D T 0 − E T 2 F F T 3 C G T 2 D H T 3 G 2
- 19. 4 25 A H B F E D C G 7 2 10 18 3 4 3 7 8 9 3 10 Update distances of adjacent, unselected nodes K dv pv A T 4 H B 4 C C T 3 D D T 0 − E T 2 F F T 3 C G T 2 D H T 3 G 2 Table entries unchanged
- 20. 4 25 A H B F E D C G 7 2 10 18 3 4 3 7 8 9 3 10 Select node with minimum distance K dv pv A T 4 H B T 4 C C T 3 D D T 0 − E T 2 F F T 3 C G T 2 D H T 3 G 2
- 21. 4 A H B F E D C G 2 3 4 3 3 Cost of Minimum Spanning Tree = Σ dv = 21 K dv pv A T 4 H B T 4 C C T 3 D D T 0 − E T 2 F F T 3 C G T 2 D H T 3 G 2 Done
- 22. Kruskal’s Algorithm • Work with edges, rather than nodes • Two steps: – Sort edges by increasing edge weight – Select the first |V| – 1 edges that do not generate a cycle
- 23. Walk-Through Consider an undirected, weight graph 5 1 A H B F E D C G 3 2 4 6 3 4 3 4 8 4 3 10
- 24. Sort the edges by increasing edge weight edge dv (D,E) 1 (D,G) 2 (E,G) 3 (C,D) 3 (G,H) 3 (C,F) 3 (B,C) 4 5 1 A H B F E D C G 3 2 4 6 3 4 3 4 8 4 3 10 edge dv (B,E) 4 (B,F) 4 (B,H) 4 (A,H) 5 (D,F) 6 (A,B) 8 (A,F) 10
- 25. Select first |V|–1 edges which do not generate a cycle edge dv (D,E) 1 √ (D,G) 2 (E,G) 3 (C,D) 3 (G,H) 3 (C,F) 3 (B,C) 4 5 1 A H B F E D C G 3 2 4 6 3 4 3 4 8 4 3 10 edge dv (B,E) 4 (B,F) 4 (B,H) 4 (A,H) 5 (D,F) 6 (A,B) 8 (A,F) 10
- 26. Select first |V|–1 edges which do not generate a cycle edge dv (D,E) 1 √ (D,G) 2 √ (E,G) 3 (C,D) 3 (G,H) 3 (C,F) 3 (B,C) 4 5 1 A H B F E D C G 3 2 4 6 3 4 3 4 8 4 3 10 edge dv (B,E) 4 (B,F) 4 (B,H) 4 (A,H) 5 (D,F) 6 (A,B) 8 (A,F) 10
- 27. Select first |V|–1 edges which do not generate a cycle edge dv (D,E) 1 √ (D,G) 2 √ (E,G) 3 χ (C,D) 3 (G,H) 3 (C,F) 3 (B,C) 4 5 1 A H B F E D C G 3 2 4 6 3 4 3 4 8 4 3 10 edge dv (B,E) 4 (B,F) 4 (B,H) 4 (A,H) 5 (D,F) 6 (A,B) 8 (A,F) 10 Accepting edge (E,G) would create a cycle
- 28. Select first |V|–1 edges which do not generate a cycle edge dv (D,E) 1 √ (D,G) 2 √ (E,G) 3 χ (C,D) 3 √ (G,H) 3 (C,F) 3 (B,C) 4 5 1 A H B F E D C G 3 2 4 6 3 4 3 4 8 4 3 10 edge dv (B,E) 4 (B,F) 4 (B,H) 4 (A,H) 5 (D,F) 6 (A,B) 8 (A,F) 10
- 29. Select first |V|–1 edges which do not generate a cycle edge dv (D,E) 1 √ (D,G) 2 √ (E,G) 3 χ (C,D) 3 √ (G,H) 3 √ (C,F) 3 (B,C) 4 5 1 A H B F E D C G 3 2 4 6 3 4 3 4 8 4 3 10 edge dv (B,E) 4 (B,F) 4 (B,H) 4 (A,H) 5 (D,F) 6 (A,B) 8 (A,F) 10
- 30. Select first |V|–1 edges which do not generate a cycle edge dv (D,E) 1 √ (D,G) 2 √ (E,G) 3 χ (C,D) 3 √ (G,H) 3 √ (C,F) 3 √ (B,C) 4 5 1 A H B F E D C G 3 2 4 6 3 4 3 4 8 4 3 10 edge dv (B,E) 4 (B,F) 4 (B,H) 4 (A,H) 5 (D,F) 6 (A,B) 8 (A,F) 10
- 31. Select first |V|–1 edges which do not generate a cycle edge dv (D,E) 1 √ (D,G) 2 √ (E,G) 3 χ (C,D) 3 √ (G,H) 3 √ (C,F) 3 √ (B,C) 4 √ 5 1 A H B F E D C G 3 2 4 6 3 4 3 4 8 4 3 10 edge dv (B,E) 4 (B,F) 4 (B,H) 4 (A,H) 5 (D,F) 6 (A,B) 8 (A,F) 10
- 32. Select first |V|–1 edges which do not generate a cycle edge dv (D,E) 1 √ (D,G) 2 √ (E,G) 3 χ (C,D) 3 √ (G,H) 3 √ (C,F) 3 √ (B,C) 4 √ 5 1 A H B F E D C G 3 2 4 6 3 4 3 4 8 4 3 10 edge dv (B,E) 4 χ (B,F) 4 (B,H) 4 (A,H) 5 (D,F) 6 (A,B) 8 (A,F) 10
- 33. Select first |V|–1 edges which do not generate a cycle edge dv (D,E) 1 √ (D,G) 2 √ (E,G) 3 χ (C,D) 3 √ (G,H) 3 √ (C,F) 3 √ (B,C) 4 √ 5 1 A H B F E D C G 3 2 4 6 3 4 3 4 8 4 3 10 edge dv (B,E) 4 χ (B,F) 4 χ (B,H) 4 (A,H) 5 (D,F) 6 (A,B) 8 (A,F) 10
- 34. Select first |V|–1 edges which do not generate a cycle edge dv (D,E) 1 √ (D,G) 2 √ (E,G) 3 χ (C,D) 3 √ (G,H) 3 √ (C,F) 3 √ (B,C) 4 √ 5 1 A H B F E D C G 3 2 4 6 3 4 3 4 8 4 3 10 edge dv (B,E) 4 χ (B,F) 4 χ (B,H) 4 χ (A,H) 5 (D,F) 6 (A,B) 8 (A,F) 10
- 35. Select first |V|–1 edges which do not generate a cycle edge dv (D,E) 1 √ (D,G) 2 √ (E,G) 3 χ (C,D) 3 √ (G,H) 3 √ (C,F) 3 √ (B,C) 4 √ 5 1 A H B F E D C G 3 2 4 6 3 4 3 4 8 4 3 10 edge dv (B,E) 4 χ (B,F) 4 χ (B,H) 4 χ (A,H) 5 √ (D,F) 6 (A,B) 8 (A,F) 10
- 36. Select first |V|–1 edges which do not generate a cycle edge dv (D,E) 1 √ (D,G) 2 √ (E,G) 3 χ (C,D) 3 √ (G,H) 3 √ (C,F) 3 √ (B,C) 4 √ 5 1 A H B F E D C G 2 3 3 3 edge dv (B,E) 4 χ (B,F) 4 χ (B,H) 4 χ (A,H) 5 √ (D,F) 6 (A,B) 8 (A,F) 10 Done Total Cost = Σ dv = 21 4 }not considered