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Minimum Spanning Trees

Learn about Minimum Spanning Trees through Prim's Algorithm for designing networks, circuits, and airline routes efficiently. Discover key applications and the concept of light edges in MST.

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Minimum Spanning Trees

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  1. Minimum Spanning Trees

  2. Minimum Spanning Tree • Problem: given a connected, undirected, weighted graph 6 4 5 9 14 2 10 15 3 8

  3. Minimum Spanning Tree • Problem: given a connected, undirected, weighted graph, find a spanning tree using edges that minimize the total weight 6 4 5 9 14 2 10 15 3 8

  4. Minimum Spanning Tree Number of edges = |V| -1 Acyclic Minimum total weight

  5. Minimum Spanning Tree • Which edges form the minimum spanning tree (MST) of this graph? A 6 4 5 9 H B C 14 2 10 15 G E D 3 8 F

  6. Minimum Spanning Tree • Answer: A 6 4 5 9 H B C 14 2 10 15 G E D 3 8 F

  7. Minimum Spanning Tree • Applications • Design of networks • Design of electrical circuits • Airline routes

  8. Minimum Spanning Tree • Thm: (Light Edges are in MST) • Let T be MST of G, and let A  T be subtree of T • Let (u,v) be min-weight edge connecting A to V-A • Then (u,v)  T

  9. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);

  10. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 6 4 9 5 14 2 10 15 3 8 Run on example graph

  11. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);  6 4 9 5    14 2 10 15    3 8  Run on example graph

  12. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);  6 4 9 5    14 2 10 15 r 0   3 8  Pick a start vertex r

  13. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);  6 4 9 5    14 2 10 15 u 0   3 8  Grey vertices have been removed from Q

  14. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);  6 4 9 5    14 2 10 15 u 0   3 8 3 Red arrows indicate parent pointers

  15. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);  6 4 9 5 14   14 2 10 15 u 0   3 8 3

  16. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);  6 4 9 5 14   14 2 10 15 0   3 8 3 u

  17. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);  6 4 9 5 14   14 2 10 15 0 8  3 8 3 u

  18. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);  6 4 9 5 10   14 2 10 15 0 8  3 8 3 u

  19. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);  6 4 9 5 10   14 2 10 15 0 8  3 8 3 u

  20. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);  6 4 9 5 10 2  14 2 10 15 0 8  3 8 3 u

  21. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);  6 4 9 5 10 2  14 2 10 15 0 8 15 3 8 3 u

  22. Prim’s Algorithm u MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);  6 4 9 5 10 2  14 2 10 15 0 8 15 3 8 3

  23. Prim’s Algorithm u MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v);  6 4 9 5 10 2 9 14 2 10 15 0 8 15 3 8 3

  24. Prim’s Algorithm u MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 4 6 4 9 5 10 2 9 14 2 10 15 0 8 15 3 8 3

  25. Prim’s Algorithm u MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 4 6 4 9 5 5 2 9 14 2 10 15 0 8 15 3 8 3

  26. Prim’s Algorithm u MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 4 6 4 9 5 5 2 9 14 2 10 15 0 8 15 3 8 3

  27. Prim’s Algorithm u MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 4 6 4 9 5 5 2 9 14 2 10 15 0 8 15 3 8 3

  28. Prim’s Algorithm u MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 4 6 4 9 5 5 2 9 14 2 10 15 0 8 15 3 8 3

  29. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); 4 6 4 9 5 5 2 9 u 14 2 10 15 0 8 15 3 8 3

  30. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); What is the hidden cost in this code?

  31. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; DecreaseKey(v, w(u,v));

  32. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; DecreaseKey(v, w(u,v)); How often is ExtractMin() called? How often is DecreaseKey() called?

  33. Prim’s Algorithm MST-Prim(G, w, r) Q = VG; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u; key[v] = w(u,v); What will be the running time? Using heap: O(E lg V)

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