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Chapter 7: Greedy Algorithms

Chapter 7: Greedy Algorithms. Kruskal’s, Prim’s, and Dijkstra’s Algorithms. Kruskal’s Algorithm. Solves the Minimum Spanning Tree Problem Input: List of edges in a graph n – the number of vertices Output: Prints the list of edges in the Minimum Spanning Tree. 4. 5. A. B. C. 3. 6. 4.

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Chapter 7: Greedy Algorithms

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  1. Chapter 7: Greedy Algorithms Kruskal’s, Prim’s, and Dijkstra’s Algorithms

  2. Kruskal’s Algorithm • Solves the Minimum Spanning Tree Problem • Input: • List of edges in a graph • n – the number of vertices • Output: • Prints the list of edges in the Minimum Spanning Tree

  3. 4 5 A B C 3 6 4 5 6 7 3 4 G D E F 7 6 4 5 3 5 5 J H I

  4. 4 5 A B C Kruskal’s 3 6 4 5 6 7 3 4 G D E F kruskal(e, n) { sort(e); 7 6 4 5 3 5 5 J H I

  5. 4 5 5 A B C B B C C 3 6 4 5 6 6 6 7 3 4 7 G D E F G D E E E F 7 6 4 5 7 6 3 5 5 A J H I H I 3 D 3 C E F D 3 4 4 4 G F A B H F G F A 5 H I 4 5 5 5 J J I J E Kruskal’s kruskal(e, n) { sort(e);

  6. 4 5 5 A B C B B C C 3 6 4 5 6 6 6 7 3 4 7 G D E F G D E E E F 7 6 4 5 7 6 3 5 5 A J H I H I 3 D 3 C E F D 3 4 4 4 G F A B H F G F A 5 H I 4 5 5 5 J J I J E A B C D E F G H I J kruskal(e, n) { sort(e); for (i = A to J) makeset(i)

  7. 4 5 5 A B C B B C C 3 6 4 5 6 6 6 7 3 4 7 G D E F G D E E E F 7 6 4 5 7 6 3 5 5 A J H I H I 3 D 3 C E F D 3 4 4 4 G F A B H F G F A 5 H I 4 5 5 5 J J I J E A B C D E F G H I J i 1 Count 0 kruskal(e, n) { ... count = 0; i = 1

  8. 5 B B C C 6 6 7 G D E E E F 7 6 A H I 3 D 3 C E F D 3 4 4 4 G F A B H F G F A 5 H I 4 5 5 5 J J I J E A B C D E F G H I J n 10 count 0 i 1 kruskal(e, n) { while (count < n-1) { if (findset(e[i].v) != findset(e[i].w)) { print(e[i].v + “ ”+ e[i].w); count++; union(e[i].v, e[i].w); } i++; }

  9. 5 B B C C 6 6 7 G D E E E F 7 6 A H I 3 D 3 C E F D 3 4 4 4 G F A B H F G F A 5 H I 4 5 5 5 J J I J E A B C DH E F G I J n 10 Count 1 i 2 kruskal(e, n) { while (count < n-1) { if (findset(e[i].v) != findset(e[i].w)) { print(e[i].v + “ ”+ e[i].w); count++; union(e[i].v, e[i].w); } i++; }

  10. 5 B B C C 6 6 7 G D E E E F 7 6 A H I 3 D 3 C E F D 3 4 4 4 G F A B H F G F A 5 H I 4 5 5 5 J J I J E A B C DH EF G I J n 10 Count 2 i 3 kruskal(e, n) { while (count < n-1) { if (findset(e[i].v) != findset(e[i].w)) { print(e[i].v + “ ”+ e[i].w); count++; union(e[i].v, e[i].w); } i++; }

  11. 5 B B C C 6 6 7 G D E E E F 7 6 A H I 3 D 3 C E F D 3 4 4 4 G F A B H F G F A 5 H I 4 5 5 5 J J I J E ADH B C EF G I J n 10 Count 3 i 4 kruskal(e, n) { while (count < n-1) { if (findset(e[i].v) != findset(e[i].w)) { print(e[i].v + “ ”+ e[i].w); count++; union(e[i].v, e[i].w); } i++; }

  12. 5 B B C C 6 6 7 G D E E E F 7 6 A H I 3 D 3 C E F D 3 4 4 4 G F A B H F G F A 5 H I 4 5 5 5 J J I J E ADH B C EFG I J n 10 Count 4 i 5 kruskal(e, n) { while (count < n-1) { if (findset(e[i].v) != findset(e[i].w)) { print(e[i].v + “ ”+ e[i].w); count++; union(e[i].v, e[i].w); } i++; }

  13. 5 B B C C 6 6 7 G D E E E F 7 6 A H I 3 D 3 C E F D 3 4 4 4 G F A B H F G F A 5 H I 4 5 5 5 J J I J E ADHB C EFG I J n 10 Count 5 i 6 kruskal(e, n) { while (count < n-1) { if (findset(e[i].v) != findset(e[i].w)) { print(e[i].v + “ ”+ e[i].w); count++; union(e[i].v, e[i].w); } i++; }

  14. 5 B B C C 6 6 7 G D E E E F 7 6 A H I 3 D 3 C E F D 3 4 4 4 G F A B H F G F A 5 H I 4 5 5 5 J J I J E ADHB CEFG I J n 10 Count 6 i 7 kruskal(e, n) { while (count < n-1) { if (findset(e[i].v) != findset(e[i].w)) { print(e[i].v + “ ”+ e[i].w); count++; union(e[i].v, e[i].w); } i++; }

  15. 5 B B C C 6 6 7 G D E E E F 7 6 A H I 3 D 3 C E F D 3 4 4 4 G F A B H F G F A 5 H I 4 5 5 5 J J I J E ADHB CEFGJ I n 10 Count 7 i 8 kruskal(e, n) { while (count < n-1) { if (findset(e[i].v) != findset(e[i].w)) { print(e[i].v + “ ”+ e[i].w); count++; union(e[i].v, e[i].w); } i++; }

  16. 5 B B C C 6 6 7 G D E E E F 7 6 A H I 3 D 3 C E F D 3 4 4 4 G F A B H F G F A 5 H I 4 5 5 5 J J I J E ADHBCEFGJ I n 10 Count 8 i 9 kruskal(e, n) { while (count < n-1) { if (findset(e[i].v) != findset(e[i].w)) { print(e[i].v + “ ”+ e[i].w); count++; union(e[i].v, e[i].w); } i++; }

  17. 5 B B C C 6 6 7 G D E E E F 7 6 A H I 3 D 3 C E F D 3 4 4 4 G F A B H F G F A 5 H I 4 5 5 5 J J I J E ADHBCEFGJ I n 10 Count 8 i 10 kruskal(e, n) { while (count < n-1) { if (findset(e[i].v) != findset(e[i].w)) { print(e[i].v + “ ”+ e[i].w); count++; union(e[i].v, e[i].w); } i++; }

  18. 5 B B C C 6 6 7 G D E E E F 7 6 A H I 3 D 3 C E F D 3 4 4 4 G F A B H F G F A 5 H I 4 5 5 5 J J I J E ADHBCEFGJI n 10 Count 9 i 11 kruskal(e, n) { while (count < n-1) { if (findset(e[i].v) != findset(e[i].w)) { print(e[i].v + “ ”+ e[i].w); count++; union(e[i].v, e[i].w); } i++; }

  19. 4 5 5 B B C C 6 6 7 G D E E E F 7 6 A H I 3 D 3 C E F D 3 4 4 4 G F A B H F G F A 5 H I 4 5 5 5 J J I J E A B C 3 6 4 5 6 7 3 4 G D E F 7 6 4 5 3 5 5 J H I

  20. 4 5 5 B B C C 6 6 7 G D E E E F 7 6 A H I 3 D 3 C E F D 3 4 4 4 G F A B H F G F A 5 H I 4 5 5 5 J J I J E A B C 3 6 4 5 6 7 3 4 G D E F 7 6 4 5 3 5 5 J H I

  21. 4 5 A B C A 3 6 4 5 6 7 3 4 G D E F D E B 7 6 4 5 3 5 5 J H I H F G C J I

  22. Theorem 7.2.5 pp. 280 • Let G be a connected, weighted graph, and let G’ be a sub-graph of a minimal spanning tree of G. Let C be a component of G’, and let S be the set of all Edges with one vertex in C and the other not in C. If we add a minimum weight edge in S to G’, the resulting graph is also contained in a minimal spanning tree of G

  23. 4 5 A B C G Minimal Spanning Tree of G Theorem 7.2.5 pp. 280 3 6 4 5 6 A 7 3 4 G D E F 7 6 4 5 D E B 3 • Let G be a connected, weighted graph, and let G’ be a sub-graph of a minimal spanning tree of G. Let C be a component of G’, and let S be the set of all Edges with one vertex in C and the other not in C. If we add a minimum weight edge in S to G’, the resulting graph is also contained in a minimal spanning tree of G 5 5 J H I H F G C J I

  24. 4 5 4 A A B 5 7 D D E E 3 H G’ Subset of Minimal Spanning Tree of G A B C G Theorem 7.2.5 pp. 280 3 6 4 5 6 7 3 4 G D E F A C 7 6 4 5 3 • G’ be a sub-graph of a minimal spanning tree of G. Let C be a component of G’, and let S be the set of all Edges with one vertex in C and the other not in C. 5 5 J H I D E S

  25. 4 5 4 A A B 5 7 D D E E 3 H G’ Subset of Minimal Spanning Tree of G A B C G Theorem 7.2.5 pp. 280 3 6 4 5 6 7 3 4 G D E F A C 7 6 4 5 3 • If we add a minimum weight edge in S to G’, the resulting graph is also contained in a minimal spanning tree of G 5 5 J H I D E S

  26. Theorem 7.2.6: Kruskal’s Algorithm finds minimum spanning tree Proof by induction • G’ is a sub-graph constructed by Kruskal’s Algorithm • G’ is initially empty but each step of the Algorithm increases the size of G’ • Inductive Assumption: G’ is contained in the MST.

  27. Theorem 7.2.6: Kruskal’s Algorithm finds minimum spanning tree Proof by induction • Let (v,w) be the next edge selected by Kruskal’s Algorithm • Kruskal’s algorithm finds the minimum weight edge (v,w) such that v and w are not already in G’ • C can be any subset of the MST, so you can always construct a C such that v is in C and w is not. • Therefore, by Theorem 7.2.5, when (v,w) is added to G’, the resulting graph is also contained in the MST.

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