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- 1. Minimum Cut 郭至軒（KuoE0） KuoE0.tw@gmail.com KuoE0.ch
- 2. Cut
- 3. cut (undirected)
- 4. A partition of the vertices of a graph into twodisjoint subsets 3 1 7 4 6 9 2 8 5 undirected graph
- 5. A partition of the vertices of a graph into twodisjoint subsets 3 1 7 4 6 9 2 8 5 undirected graph
- 6. A partition of the vertices of a graph into twodisjoint subsets 3 1 7 4 6 9 2 8 5 undirected graph
- 7. A partition of the vertices of a graph into twodisjoint subsets 1 4 2 6 3 7 5 8 9 undirected graph
- 8. Cut-set of the cut is the set of edges whoseend points are in diﬀerent subsets. 1 4 2 6 3 7 5 8 9 undirected graph
- 9. Cut-set of the cut is the set of edges whoseend points are in diﬀerent subsets. 1 4 2 6 3 7 5 8 9 Cut-set undirected graph
- 10. weight = number of edges or sum of weighton edges 1 4 2 6 3 7 5 8 9 weight is 7 undirected graph
- 11. cut (directed)
- 12. A partition of the vertices of a graph into twodisjoint subsets 3 1 7 4 6 9 2 8 5 directed graph
- 13. A partition of the vertices of a graph into twodisjoint subsets 3 1 7 4 6 9 2 8 5 directed graph
- 14. A partition of the vertices of a graph into twodisjoint subsets 3 1 7 4 6 9 2 8 5 directed graph
- 15. A partition of the vertices of a graph into twodisjoint subsets 1 4 2 6 3 7 5 8 9 directed graph
- 16. Cut-set of the cut is the set of edges whoseend points are in diﬀerent subsets. 1 4 2 6 3 7 5 8 9 directed graph
- 17. Cut-set of the cut is the set of edges whoseend points are in diﬀerent subsets. 1 4 2 6 3 7 5 8 9 directed graph
- 18. Cut-set of the cut is the set of edges whoseend points are in diﬀerent subsets. 1 4 2 6 3 7 5 8 9 Cut-set directed graph
- 19. weight = number of edges or sum of weighton edges 1 4 2 6 3 7 5 8 9 weight is 5⇢ or 2⇠ directed graph
- 20. s-t cut1. one side is source2. another side is sink3. cut-set only consists of edges goingfrom source’s side to sink’s side
- 21. Source Sink Other 31 7 4 6 92 8 5 ﬂow network
- 22. Source Sink 31 7 4 6 92 8 5 ﬂow network
- 23. Source Sink 31 7 4 6 92 8 5 ﬂow network
- 24. cut-set only consists of edges goingfrom source’s side to sink’s side 1 4 2 6 3 7 5 8 9 ﬂow network
- 25. cut-set only consists of edges goingfrom source’s side to sink’s side 1 4 2 6 3 7 5 8 9 weight is 6 ﬂow network
- 26. Max-Flow Min-Cut Theorem
- 27. Observation 1The network ﬂow sent across any cut is equal to theamount reaching sink. 4/4 2 4 3/3 4/4 1 1/4 6 3/8 2/3 3 5 2/2 total ﬂow = 6, ﬂow on cut = 3 + 3 = 6
- 28. Observation 1The network ﬂow sent across any cut is equal to theamount reaching sink. 4/4 2 4 3/3 4/4 1 1/4 6 3/8 2/3 3 5 2/2 total ﬂow = 6, ﬂow on cut = 3 + 3 = 6
- 29. Observation 1The network ﬂow sent across any cut is equal to theamount reaching sink. 4/4 2 4 3/3 4/4 1 1/4 6 3/8 2/3 3 5 2/2 total ﬂow = 6, ﬂow on cut = 3 + 4 - 1 = 6
- 30. Observation 1The network ﬂow sent across any cut is equal to theamount reaching sink. 4/4 2 4 3/3 4/4 1 1/4 6 3/8 2/3 3 5 2/2 total ﬂow = 6, ﬂow on cut = 3 + 4 - 1 = 6
- 31. Observation 1The network ﬂow sent across any cut is equal to theamount reaching sink. 4/4 2 4 3/3 4/4 1 1/4 6 3/8 2/3 3 5 2/2 total ﬂow = 6, ﬂow on cut = 4 + 2= 6
- 32. Observation 1The network ﬂow sent across any cut is equal to theamount reaching sink. 4/4 2 4 3/3 4/4 1 1/4 6 3/8 2/3 3 5 2/2 total ﬂow = 6, ﬂow on cut = 4 + 2= 6
- 33. Observation 1The network ﬂow sent across any cut is equal to theamount reaching sink. 4/4 2 4 3/3 4/4 1 1/4 6 3/8 2/3 3 5 2/2 total ﬂow = 6, ﬂow on cut = 4 + 2= 6
- 34. Observation 1The network ﬂow sent across any cut is equal to theamount reaching sink. 4/4 2 4 3/3 4/4 1 1/4 6 3/8 2/3 3 5 2/2 total ﬂow = 6, ﬂow on cut = 4 + 2= 6
- 35. Observation 2Then the value of the ﬂow is at most the capacity ofany cut. 4 2 4 3 4 1 4 6 8 3 3 5 2 It’s trivial!
- 36. Observation 2Then the value of the ﬂow is at most the capacity ofany cut. 4 2 4 3 4 1 4 6 8 3 3 5 2 It’s trivial!
- 37. Observation 3Let f be a ﬂow, and let (S,T) be an s-t cut whosecapacity equals the value of f. 4/4 2 4 3/3 4/4 1 1/4 6 3/8 2/3 3 5 2/2 f is the maximum ﬂow (S,T) is the minimum cut
- 38. Observation 3Let f be a ﬂow, and let (S,T) be an s-t cut whosecapacity equals the value of f. 4/4 2 4 3/3 4/4 1 1/4 6 3/8 2/3 3 5 2/2 f is the maximum ﬂow (S,T) is the minimum cut
- 39. Max-FlowEQUALMin-Cut
- 40. Example
- 41. 4 2 4 3 41 4 6 8 3 3 5 2
- 42. Maximum Flow = 6 4/4 2 4 3/3 4/41 1/4 6 3/8 2/3 3 5 2/2
- 43. Residual Network 0/4 2 4 0/3 0/41 3/4 6 5/8 1/3 3 5 0/2
- 44. Minimum Cut = 6 0/4 2 4 0/3 0/41 3/4 6 5/8 1/3 3 5 0/2
- 45. Minimum Cut = 6 0/4 2 4 0/3 0/41 3/4 6 5/8 1/3 3 5 0/2
- 46. Minimum Cut = 6 0/4 2 4 0/3 0/41 3/4 6 5/8 1/3 3 5 0/2
- 47. Minimum Cut = 6 0/4 2 4 0/3 0/41 3/4 6 5/8 1/3 3 5 0/2
- 48. The minimum capacity limit the maximum ﬂow!
- 49. ﬁnd a s-t cut
- 50. Maximum Flow = 6 4/4 2 4 3/3 4/41 1/4 6 3/8 2/3 3 5 2/2
- 51. Travel on Residual Network 0/4 2 4 0/3 0/41 3/4 6 5/8 1/3 3 5 0/2
- 52. start from source 0/4 2 4 0/3 0/41 3/4 6 5/8 1/3 3 5 0/2
- 53. don’t travel through full edge 0/4 2 4 0/3 0/41 3/4 6 5/8 1/3 3 5 0/2
- 54. don’t travel through full edge 0/4 2 4 0/3 0/41 3/4 6 5/8 1/3 3 5 0/2
- 55. no residual edge 0/4 2 4 0/3 0/41 3/4 6 5/8 1/3 3 5 0/2
- 56. no residual edge 0/4 2 4 0/3 0/41 3/4 6 5/8 1/3 3 5 0/2
- 57. s-t cut 0/4 2 4 0/3 0/41 3/4 6 5/8 1/3 3 5 0/2
- 58. s-t cut 0/4 2 4 0/3 0/41 3/4 6 5/8 1/3 3 5 0/2
- 59. result of starting from sink 0/4 2 4 0/3 0/41 3/4 6 5/8 1/3 3 5 0/2
- 60. result of starting from sink 0/4 2 4 0/3 0/41 3/4 6 5/8 1/3 3 5 0/2
- 61. Minimum cut is non-unique!
- 62. time complexity:based on max-ﬂow algorithm Ford-Fulkerson algorithm O(EF) Edmonds-Karp algorithm O(VE2) Dinic algorithm O(V2E)
- 63. Stoer Wagner only for undirected graphtime complexity: O(N3) or O(N2log2N)
- 64. Practice Now UVa 10480 - Sabotage
- 65. Problem List UVa 10480 UVa 10989 POJ 1815 POJ 2914 POJ 3084 POJ 3308 POJ 3469
- 66. Reference• http://www.ﬂickr.com/photos/dgjones/335788038/• http://www.ﬂickr.com/photos/njsouthall/3181945005/• http://www.csie.ntnu.edu.tw/~u91029/Cut.html• http://en.wikipedia.org/wiki/Cut_(graph_theory)• http://en.wikipedia.org/wiki/Max-ﬂow_min-cut_theorem• http://www.cs.princeton.edu/courses/archive/spr04/cos226/lectures/ maxﬂow.4up.pdf• http://www.cnblogs.com/scau20110726/archive/ 2012/11/27/2791523.html
- 67. Thank You for Your Listening.

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