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More Chapter 7: Greedy Algorithms

More Chapter 7: Greedy Algorithms. Kruskal’s Minimum Spanning Tree Algorithm. Minimum Spanning Tree (MST) Problem. Given a weighted graph, i.e. a graph with edge weights… try to find a sub-graph that (i) connects all the nodes and (ii) the sum of the edge weights is minimal.

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More Chapter 7: Greedy Algorithms

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  1. More Chapter 7: Greedy Algorithms Kruskal’s Minimum Spanning Tree Algorithm.

  2. Minimum Spanning Tree (MST) Problem • Given a weighted graph, i.e. a graph with edge weights… • try to find a sub-graphthat(i) connects all the nodes and (ii) the sum of the edge weights is minimal. • This sub-graph will always be a tree. 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

  3. MST Example • Given the followinggraph, find the MST • First, we want allthe nodes connected • Second, we want topick the lowest weight edges. 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. MST Example • Greedy step 1: • Start with Aand select theminimum edge • This would connect D. 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 Total Weight: 3

  5. MST Example • Greedy step 2: • From Dselect theminimum edge • This would connect H. 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 Total Weight: 6

  6. MST Example • Greedy step 3: • From Hselect theminimum edge • This would connect I. 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 Total Weight: 11

  7. MST Example • Greedy step 4: • From Iselect theminimum edge • This would connect J. 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 Total Weight: 16

  8. MST Example • Greedy step 5: • From Jselect theminimum edge • This would connect G. 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 Total Weight: 20

  9. MST Example • Greedy step 6: • From Gselect theminimum edge • This would connect F. 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 Total Weight: 24

  10. MST Example • Greedy step 6: • From Gselect theminimum edge • This would connect F. 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 Total Weight: 24

  11. MST Example • What is the running timeof the algorithm? 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 Total Weight: 27

  12. MST Example • N steps • Each step mustfind the minimumedge from a node • Worst case:N-1 + N-1 + N-1 + … + N-1 = O(N2) 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

  13. MST Example • You might think the worst case is:N-1 + N-2 + N-3 + … + 3 + 2 + 1 = O(N2)because at every step you connect a node and don’t have to consider it’s edges. • However, think about how the algorithm would actually be implemented, and how you would keep track of this info?

  14. MST Example • You might think the worst case is:N-1 + N-2 + N-3 + … + 3 + 2 + 1 = O(N2) pick node v from the node_list. while node_listis not empty { mark v as visited. min_hop = infinity; foreach of v’s edges (v,w) if (w is not visited) if (edge (v,w) < min_hop) { min_hop = edge(v,w)min_edge = w; } remove v from node from node_list v = w; } The first while loop will always take N iterations The foreach loop could take N-1 iteration in a complete graph

  15. MST Example • Greedy step 8: • From Eselect theminimum edge • This would connect B. 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 Total Weight: 33

  16. MST Example • Greedy step 9: • From Bselect theminimum edge • This would connect C. 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 Total Weight: 38

  17. MST Example • This greedyalgorithm failed • Why? 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 Total Weight: 38

  18. MST Example • It makes a localdecision. • From E, itchooses to go toB • We have to consider other options. 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 Total Weight: 38

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

  20. 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

  21. 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

  22. 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);

  23. 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)

  24. 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

  25. 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++; }

  26. 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++; }

  27. 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++; }

  28. 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++; }

  29. 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++; }

  30. 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++; }

  31. 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++; }

  32. 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++; }

  33. 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++; }

  34. 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++; }

  35. 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++; }

  36. 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

  37. 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

  38. 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

  39. 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

  40. 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

  41. 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

  42. 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 from S to G’, the resulting graph is also contained in a minimal spanning tree of G 5 5 J H I D E S

  43. 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.

  44. 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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