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Minimum Spanning Trees

Minimum Spanning Trees. c. b. a. d. e. g. f. c. b. a. d. e. g. f. c. b. a. d. e. g. f. c. 5. b. -1. 2. 7. 5. a. d. 0. e. 8. 0. 5. -3. 0. -1. g. f. c. 5. b. -1. 2. 7. 5. a. d. 0. e. 8. 0. 5. -3. 0. -1. g. f.

harper-richardson
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Minimum Spanning Trees

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  1. Minimum Spanning Trees

  2. c b a d e g f

  3. c b a d e g f

  4. c b a d e g f

  5. c 5 b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  6. c 5 b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f Minimum spanning tree weight = 0

  7. Systematic Approach: -Grow a spanning tree A -Always add a safe edgeto A that we know belongs to some minimal spanning tree

  8. c 5 R b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  9. c 5 R b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f There must go exactly one edge from b to the minimum spanning tree.Which edge should we choose?Are all three a possibility (maybe suboptimal)?Could the 7 edge be optimal to choose?What about the two 5 edges?

  10. c 5 R b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f There must go exactly one edge from b to the spanning tree.Which edge should we choose?Are all three a possibility? Yes, every vertex will connect to the tree Could the 7 edge be optimal to choose? No, we must connect with a light edgeWhat about the two 5 edges? The red is possible, but they both are !!

  11. c 5 R’ b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  12. c 5 R’ b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  13. c 5 R’ b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  14. c 5 R’ b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  15. c 5 R’ b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  16. c 5 R’ b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  17. c 5 R’ b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  18. c 5 R’ b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  19. c 5 R’’ b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  20. c 5 R’’ b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  21. c 5 R’’’ b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  22. Kruskal’s Algorithm

  23. Idea: Rethink the previous algorithm from vertices to edges-Initially each vertex forms a singleton tree -In each iteration add the lightest edge connecting two trees in the forest.

  24. c 5 b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  25. c 5 b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  26. c 5 b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  27. c 5 b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  28. c 5 b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  29. c 5 b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  30. c 5 b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  31. c 5 b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  32. c 5 R’’’ b -1 2 7 5 a d 0 e 8 0 5 -3 0 -1 g f

  33. MSTkruskal(G,w) • 1 AØ • for each vertex vV[G] • do MakeSet(v) • sort the edges of E into nondecreasing order by weight w • for each edge (u,v) E, taken in nondecreasing order by weight • doif FindSet(u) FindSet(v) • thenAA {(u,v)} • Union(u,v) • returnA

  34. MSTkruskal(G,w) • 1 AØ • for each vertex vV[G] • do MakeSet(v) • sort the edges of E into nondecreasing order by weight w • for each edge (u,v) E, taken in nondecreasing order by weight • doif FindSet(u) FindSet(v) • thenAA {(u,v)} • Union(u,v) • returnA Time?

  35. MSTkruskal(G,w) • 1 AØ • for each vertex vV[G] • doMakeSet(v) • sort the edges of E into nondecreasing order by weight w • for each edge (u,v) E, taken in nondecreasing order by weight • doifFindSet(u) FindSet(v) • thenAA {(u,v)} • Union(u,v) • returnA O(E*lgE) Time?

  36. MSTkruskal(G,w) • 1 AØ • for each vertex vV[G] • doMakeSet(v) • sort the edges of E into nondecreasing order by weight w • for each edge (u,v) E, taken in nondecreasing order by weight • doifFindSet(u) FindSet(v) • thenAA {(u,v)} • Union(u,v) • returnA O(E*lgE) Theorem 21.13: m FindSet, Union, and MakeSet operations of which n are MakeSet operations takes O(m*α(n)) Time?

  37. MSTkruskal(G,w) • 1 AØ • for each vertex vV[G] • doMakeSet(v) • sort the edges of E into nondecreasing order by weight w • for each edge (u,v) E, taken in nondecreasing order by weight • doifFindSet(u) FindSet(v) • thenAA {(u,v)} • Union(u,v) • returnA O(E*lg(E)) O(V+E*α(V)) =O(V+E*lg(V)) = O(V+E*lg(E)) =O(E*lg(E)) Time?

  38. MSTkruskal(G,w) • 1 AØ • for each vertex vV[G] • doMakeSet(v) • sort the edges of E into nondecreasing order by weight w • for each edge (u,v) E, taken in nondecreasing order by weight • doifFindSet(u) FindSet(v) • thenAA {(u,v)} • Union(u,v) • returnA O(E*lg(E)) O(V+E*α(V)) =O(V+E*lg(V)) = O(V+E*lg(E)) =O(E*lg(E)) O(E*lg(E))

  39. c 5 R b -1 2 7 5 a d 1 e 8 0 5 -3 0 -1 g f Obviously tree. Which edge should we choose?

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