[ACM-ICPC] Bipartite Matching

Transcript

1. Bipartite Matching ֲࢸݢʢKuoE0ʣ KuoE0.tw@gmail.com KuoE0.ch
2. Bipartite Graph two disjoint sets every edge connects between them
3. Matching A subset of edges, no two of which share an endpoint.
4. Valid Matching bachelor not matching edge matching edge
5. Invalid Matching shared not matching edge matching edge
6. Bipartite Matching objective: more matching
7. 1 2 8 4 5 7 9 3 6 not matching edge matching edge
8. Alternating Path A path in which the edges belong alternatively to the matching and not to the matching.
9. 1 2 8 4 5 7 9 3 6 not matching edge [1 - 5 - 2 - 9] is an alternating path. matching edge
10. 1 2 8 4 5 7 9 3 6 not matching edge [1 - 5 - 2 - 9] is an alternating path. matching edge
11. 1 2 8 4 5 7 9 3 6 not matching edge matching edge [1 - 5 - 2 - 6 - 3] is not an alternating path.
12. 1 2 8 4 5 7 9 3 6 not matching edge matching edge [1 - 5 - 2 - 6 - 3] is not an alternating path.
13. Alternating Cycle A cycle in which the edges belong alternatively to the matching and not to the matching.
14. 1 2 8 4 5 7 9 3 6 not matching edge [1 - 5 - 2 - 9 - 1] is an alternating cycle. matching edge
15. 1 2 8 4 5 7 9 3 6 not matching edge [1 - 5 - 2 - 9 - 1] is an alternating cycle. matching edge
16. 1 2 8 4 5 7 9 3 6 not matching edge matching edge Reverse the alternating cycle, cardinality won’t change.
17. Augmenting Path Algorithm Maximum Matching
18. Augmenting Path An alternating path that starts from and ends on unmatched vertices.
19. 1 2 8 4 5 7 9 3 6 not matching edge matching edge [2 - 5 - 1 - 9] is an augmenting path.
20. 1 2 8 4 5 7 9 3 6 not matching edge matching edge [2 - 5 - 1 - 9] is an augmenting path.
21. 1 2 8 4 5 7 9 3 6 not matching edge matching edge Reverse the augmenting path, cardinality will increase.
22. Augmenting Path • The length of an augmenting path always is odd. • Reversing an augmenting path will increase the cardinality. • If there is no augmenting path, the cardinality is maximum.
23. 1.Try to build augmenting paths from all vertices in one side. 2.Travel on graph. 3.If an augmenting path exists, reverse the augmenting path to increase cardinality. 4.If no augmenting path exists, ignore this vertex. 5.Repeat above step until there no augmenting path exists. Algorithm
24. 1 2 8 4 5 7 9 3 6 not matching edge matching edge [1 - 5] is an augmenting path. current cardinality: 0
25. 1 2 8 4 5 7 9 3 6 not matching edge matching edge [1 - 5] is an augmenting path. current cardinality: 0
26. 1 2 8 4 5 7 9 3 6 Reverse it! not matching edge matching edge current cardinality: 1
27. 1 2 8 4 5 7 9 3 6 [2 - 5 - 1- 9] is an augmenting path. not matching edge matching edge current cardinality: 1
28. 1 2 8 4 5 7 9 3 6 [2 - 5 - 1- 9] is an augmenting path. not matching edge matching edge current cardinality: 1
29. 1 2 8 4 5 7 9 3 6 not matching edge matching edge current cardinality: 2 Reverse it!
30. 1 2 8 4 5 7 9 3 6 not matching edge matching edge current cardinality: 2 [3 - 6] is an augmenting path.
31. 1 2 8 4 5 7 9 3 6 not matching edge matching edge current cardinality: 2 [3 - 6] is an augmenting path.
32. 1 2 8 4 5 7 9 3 6 not matching edge matching edge current cardinality: 3 Reverse it!
33. 1 2 8 4 5 7 9 3 6 not matching edge matching edge current cardinality: 3 [4 - 5 - 2 - 6 - 3 - 7] is an augmenting path.
34. 1 2 8 4 5 7 9 3 6 not matching edge matching edge current cardinality: 3 [4 - 5 - 2 - 6 - 3 - 7] is an augmenting path.
35. 1 2 8 4 5 7 9 3 6 not matching edge matching edge current cardinality: 4 Reverse it!
36. 1 2 8 4 5 7 9 3 6 not matching edge matching edge Max cardinality is 4.
37. Time Complexity O(V × E) V: number of vertices E: number of edges
38. // find an augmenting path bool find_aug_path( int x ) { for ( int i = 0; i < ( int )vertex[ x ].size(); ++i ) { int next = vertex[ x ][ i ]; // not in augmenting path if ( !visit[ next ] ) { // setup this vertex in augmenting path visit[ next ] = 1; /* * If this vertex is a unmatched vertex, reverse augmenting path and return. * If this vector is a matched vertex, try to reverse augmenting path and continue find an unmatched vertex. */ if ( lnk2[ next ] == -1 || find_aug_path( lnk2[ next ] ) ) { lnk1[ x ] = next, lnk2[ next ] = x; return 1; } } } return 0; } Source Code
39. UVa 670 - The dog task Practice Now
40. Hungarian Algorithm (Kuhn-Munkres Algorithm) Maximum Weight Matching
41. 1 4 2 3 5 6 5 3 2 1 -2 5
42. weight adjustment If add (subtract) some value on all edges connected with vertex X, the maximum matching won’t be effected. 1 4 2 3 5 6 a b c not matching edge matching edge 1 4 2 3 5 6 a+d b+d c+d
43. vertex labeling For convenient, add a variable on vertices to denote the value add (subtract) on edges connected with them. 1 4 2 3 5 6 a b c not matching edge matching edge 1 4 2 3 5 6 a+d b+d c+d d ≡
44. vertex labeling l(x)+l(y)≥w(x,y), for each edge(x,y) 1 4 2 3 5 6 3 5 3 2 1 -2 5 5 5 0 0 0
45. 1 4 2 3 5 6 0 5 3 2 1 -2 5 2 4 1 3 0 minimize vertex labeling Admissible Edge: l(x)+l(y)=w(x,y)
46. convert maximum weight problem to minimum vertex labeling problem
47. Augmenting Path with Admissible Edge [1 - 4] is an augment path with admissible edges. 1 4 2 3 5 6 3 5 3 2 1 -2 5 5 5 0 0 0
48. Augmenting Path with Admissible Edge [1 - 4] is an augment path with admissible edges. 1 4 2 3 5 6 3 5 3 2 1 -2 5 5 5 0 0 0
49. Augmenting Path with Admissible Edge Reverse it! 1 4 2 3 5 6 3 5 3 2 1 -2 5 5 5 0 0 0 Current Weight = 5
50. Augmenting Path with Admissible Edge No augmenting path found start from vertex 2. 1 4 2 3 5 6 3 5 3 2 1 -2 5 5 5 0 0 0
51. Augmenting Path with Admissible Edge No augmenting path found start from vertex 2. 1 4 2 3 5 6 3 5 3 2 1 -2 5 5 5 0 0 0
52. Adjust Vertex Labeling 1 4 2 3 5 6 3 5 3 2 1 -2 5 5 5 0 0 0 relax value d = min(l(x)+l(y)-w(x,y)) x: vertex in alternating path y: vertex y not in alternating path
53. Adjust Vertex Labeling 1 4 2 3 5 6 3 5 3 2 1 -2 5 5 5 0 0 0 R(1,6)=3 R(2,5)=2 R(2,6)=5 relax d = 2
54. Adjust Vertex Labeling l(x) subtract value d. l(y) add value d. 1 4 2 3 5 6 1 5 3 2 1 -2 5 3 5 0 2 0
55. Continue to Find Augmenting Path 1 4 2 3 5 6 1 5 3 2 1 -2 5 3 5 0 2 0 [2 - 5] is an augment path with admissible edges.
56. Continue to Find Augmenting Path 1 4 2 3 5 6 1 5 3 2 1 -2 5 3 5 0 2 0 [2 - 5] is an augment path with admissible edges.
57. Continue to Find Augmenting Path 1 4 2 3 5 6 1 5 3 2 1 -2 5 3 5 0 2 0 Reverse it! Current Weight = 6
58. Continue to Find Augmenting Path 1 4 2 3 5 6 1 5 3 2 1 -2 5 3 5 0 2 0 No augmenting path found start from vertex 3.
59. Continue to Find Augmenting Path 1 4 2 3 5 6 1 5 3 2 1 -2 5 3 5 0 2 0 No augmenting path found start from vertex 3.
60. Adjust Vertex Labeling 1 4 2 3 5 6 1 5 3 2 1 -2 5 3 5 0 2 0 R(1,6)=1 R(2,6)=3 relax d = 1
61. Adjust Vertex Labeling 1 4 2 3 5 6 0 5 3 2 1 -2 5 2 4 1 3 0 l(x) subtract value d. l(y) add value d.
62. Continue to Find Augmenting Path 1 4 2 3 5 6 0 5 3 2 1 -2 5 2 4 1 3 0 [3 - 5 - 2 - 4 - 1 - 6] is an augment path with admissible edges.
63. Continue to Find Augmenting Path 1 4 2 3 5 6 0 5 3 2 1 -2 5 2 4 1 3 0 [3 - 5 - 2 - 4 - 1 - 6] is an augment path with admissible edges.
64. Continue to Find Augmenting Path 1 4 2 3 5 6 0 5 3 2 1 -2 5 2 4 1 3 0 Reverse it! Current Weight = 10
65. Maximum Weight Matching = 10 1 4 2 3 5 6 0 5 3 2 1 -2 5 2 4 1 3 0
66. Algorithm 1.Initial vertex labeling to fit l(x)+l(y)≥w(x,y) 2.Find augmenting paths composed with admissible edges from all vertices in one side. 3.If no augmenting path exists, adjust vertex labeling until the augmenting path found. 4.If augmenting path exists, continue to find next augmenting path. 5.Repeat above step until there no augmenting path exists. 6.Calculate the sum of weight on matching edges.
67. POJ 2195 - Going Home Practice Now
68. Problem List UVa 670 UVa 753 UVa 10080 UVa 10092 UVa 10243 UVa 10418 UVa 10984 POJ 3565
69. Reference • http://en.wikipedia.org/wiki/Bipartite_graph • http://en.wikipedia.org/wiki/Matching_(graph_theory) • http://www.flickr.com/photos/marceau_r/5244129689 • http://www.sanfilippo-chianti.it/offerta-svalentino.html • http://www.csie.ntnu.edu.tw/~u91029/Matching.html
70. Thank You for Your Listening.