[ACM-ICPC] Minimum Cut

Transcript

1. Minimum Cut ֲࢸݢʢKuoE0ʣ KuoE0.tw@gmail.com KuoE0.ch
2. Cut
3. cut (undirected)
4. 1 3 5 6 7 8 9 2 4 undirected graph A partition of the vertices of a graph into two disjoint subsets
5. 1 3 5 6 7 8 9 2 4 undirected graph A partition of the vertices of a graph into two disjoint subsets
6. 1 3 5 6 7 8 9 2 4 undirected graph A partition of the vertices of a graph into two disjoint subsets
7. 1 2 8 5 4 7 9 3 6 A partition of the vertices of a graph into two disjoint subsets undirected graph
8. 1 2 8 5 4 7 9 3 6 Cut-set of the cut is the set of edges whose end points are in different subsets. undirected graph
9. 1 2 8 5 4 7 9 3 6 Cut-set of the cut is the set of edges whose end points are in different subsets. Cut-set undirected graph
10. 1 2 8 5 4 7 9 3 6 weight = number of edges or sum of weight on edges weight is 7 undirected graph
11. cut (directed)
12. 1 3 5 6 7 8 9 2 4 directed graph A partition of the vertices of a graph into two disjoint subsets
13. 1 3 5 6 7 8 9 2 4 directed graph A partition of the vertices of a graph into two disjoint subsets
14. 1 3 5 6 7 8 9 2 4 directed graph A partition of the vertices of a graph into two disjoint subsets
15. 1 2 8 5 4 7 9 3 6 directed graph A partition of the vertices of a graph into two disjoint subsets
16. 1 2 8 5 4 7 9 3 6 directed graph Cut-set of the cut is the set of edges whose end points are in different subsets.
17. 1 2 8 5 4 7 9 3 6 directed graph Cut-set of the cut is the set of edges whose end points are in different subsets.
18. 1 2 8 5 4 7 9 3 6 Cut-set directed graph Cut-set of the cut is the set of edges whose end points are in different subsets.
19. 1 2 8 5 4 7 9 3 6 weight is 5⇢ or 2⇠ directed graph weight = number of edges or sum of weight on edges
20. s-t cut 1. one side is source 2. another side is sink 3. cut-set only consists of edges going from source’s side to sink’s side
21. 1 3 5 6 7 8 9 2 4 flow network Source Sink Other
22. 1 3 5 6 7 8 9 2 4 flow network Source Sink
23. 1 3 5 6 7 8 9 2 4 flow network Source Sink
24. 1 2 8 5 4 7 9 3 6 flow network cut-set only consists of edges going from source’s side to sink’s side
25. 1 2 8 5 4 7 9 3 6 weight is 6 flow network cut-set only consists of edges going from source’s side to sink’s side
26. Max-Flow Min-Cut Theorem
27. Observation 1 The network flow sent across any cut is equal to the amount reaching sink. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 total flow = 6, flow on cut = 3 + 3 = 6
28. Observation 1 The network flow sent across any cut is equal to the amount reaching sink. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 total flow = 6, flow on cut = 3 + 3 = 6
29. Observation 1 The network flow sent across any cut is equal to the amount reaching sink. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 total flow = 6, flow on cut = 3 + 4 - 1 = 6
30. Observation 1 The network flow sent across any cut is equal to the amount reaching sink. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 total flow = 6, flow on cut = 3 + 4 - 1 = 6
31. Observation 1 The network flow sent across any cut is equal to the amount reaching sink. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 total flow = 6, flow on cut = 4 + 2= 6
32. Observation 1 The network flow sent across any cut is equal to the amount reaching sink. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 total flow = 6, flow on cut = 4 + 2= 6
33. Observation 1 The network flow sent across any cut is equal to the amount reaching sink. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 total flow = 6, flow on cut = 4 + 2= 6
34. Observation 1 The network flow sent across any cut is equal to the amount reaching sink. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 total flow = 6, flow on cut = 4 + 2= 6
35. Observation 2 Then the value of the flow is at most the capacity of any cut. 1 3 2 4 5 6 3 8 4 2 4 4 3 It’s trivial!
36. Observation 2 Then the value of the flow is at most the capacity of any cut. 1 3 2 4 5 6 3 8 4 2 4 4 3 It’s trivial!
37. Observation 3 Let f be a flow, and let (S,T) be an s-t cut whose capacity equals the value of f. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 f is the maximum flow (S,T) is the minimum cut
38. Observation 3 Let f be a flow, and let (S,T) be an s-t cut whose capacity equals the value of f. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 f is the maximum flow (S,T) is the minimum cut
39. Max-Flow EQUAL Min-Cut
40. Example
41. 1 3 2 4 5 6 3 8 4 2 4 4 3
42. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 Maximum Flow = 6
43. 1 3 2 4 5 6 0/3 5/8 0/4 0/2 3/4 0/4 1/3 Residual Network
44. 1 3 2 4 5 6 0/3 5/8 0/4 0/2 3/4 0/4 1/3 Minimum Cut = 6
45. 1 3 2 4 5 6 0/3 5/8 0/4 0/2 3/4 0/4 1/3 Minimum Cut = 6
46. 1 3 2 4 5 6 0/3 5/8 0/4 0/2 3/4 0/4 1/3 Minimum Cut = 6
47. 1 3 2 4 5 6 0/3 5/8 0/4 0/2 3/4 0/4 1/3 Minimum Cut = 6
48. The minimum capacity limit the maximum flow!
49. find a s-t cut
50. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 Maximum Flow = 6
51. 1 3 2 4 5 6 0/3 5/8 0/4 0/2 3/4 0/4 1/3 Travel on Residual Network
52. 1 3 2 4 5 6 0/3 5/8 0/4 0/2 3/4 0/4 1/3 start from source
53. 1 3 2 4 5 6 0/3 5/8 0/4 0/2 3/4 0/4 1/3 don’t travel through full edge
54. 1 3 2 4 5 6 0/3 5/8 0/4 0/2 3/4 0/4 1/3 don’t travel through full edge
55. 1 3 2 4 5 6 0/3 5/8 0/4 0/2 3/4 0/4 1/3 no residual edge
56. 1 3 2 4 5 6 0/3 5/8 0/4 0/2 3/4 0/4 1/3 no residual edge
57. 1 3 2 4 5 6 0/3 5/8 0/4 0/2 3/4 0/4 1/3 s-t cut
58. 1 3 2 4 5 6 0/3 5/8 0/4 0/2 3/4 0/4 1/3 s-t cut
59. result of starting from sink 1 3 2 4 5 6 0/3 5/8 0/4 0/2 3/4 0/4 1/3
60. result of starting from sink 1 3 2 4 5 6 0/3 5/8 0/4 0/2 3/4 0/4 1/3
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 graph time complexity: O(N3) or O(N2log2N)
64. UVa 10480 - Sabotage Practice Now
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.