Data Structures and Algorithm - Week 8 - Minimum Spanning Trees
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The document discusses minimum spanning trees and provides examples of Prim's and Kruskal's algorithms. It includes: - A definition of minimum spanning tree as a subgraph that spans all nodes with minimum total edge weight. - Characteristics of Prim's and Kruskal's algorithms such as working with undirected, weighted/unweighted graphs and producing optimal solutions greedily. - A walk-through example of Prim's algorithm on a graph and calculating the minimum spanning tree cost.
Data Structures and Algorithm - Week 8 - Minimum Spanning Trees
- 1. Data Structures and Algorithms Week 8: Minimum Spanning Trees Ferdin Joe John Joseph, PhD Faculty of Information Technology Thai-Nichi Institute of Technology, Bangkok
- 2. Week 8 • Minimum Spanning Trees • Implementation in Java Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 2
- 3. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 3
- 4. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 4
- 5. Prim’s Algorithm Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 5
- 6. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 6
- 7. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 7
- 8. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 8
- 9. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 9
- 10. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 10
- 11. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 11
- 12. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 12
- 13. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 13
- 14. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 14
- 15. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 15
- 16. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 16
- 17. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 17
- 18. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 18
- 19. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 19
- 20. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 20
- 21. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 21
- 22. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 22
- 23. Java Implementation https://www.geeksforgeeks.org/prims-minimum- spanning-tree-mst-greedy-algo-5/ Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 23
- 24. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 24
- 25. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 25
- 26. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 26
- 27. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 27
- 28. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 28
- 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 Accepting edge (E,G) would create a cycle Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 29
- 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 30
- 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 31
- 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 (B,F) 4 (B,H) 4 (A,H) 5 (D,F) 6 (A,B) 8 (A,F) 10 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 32
- 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 (B,F) 4 (B,H) 4 (A,H) 5 (D,F) 6 (A,B) 8 (A,F) 10 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 33
- 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 (B,H) 4 (A,H) 5 (D,F) 6 (A,B) 8 (A,F) 10 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 34
- 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 (A,H) 5 (D,F) 6 (A,B) 8 (A,F) 10 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 35
- 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 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 36
- 37. 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 Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 37
- 38. 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 considere d Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 38
- 39. Java Implementation https://www.geeksforgeeks.org/kruskals-minimum- spanning-tree-algorithm-greedy-algo-2/ Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 39
- 40. Next Week Search Algorithms Lecture series for Data Structures and Algorithms, Data Science and Analytics, Thai-Nichi Institute of Technology 40