[ACM-ICPC] Tree Isomorphism

Transcript

1. Tree Isomorphism ֲࢸݢʢKuoE0ʣ KuoE0.tw@gmail.com KuoE0.ch
2. Latest update: May 1, 2013 Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) http://creativecommons.org/licenses/by-sa/3.0/
3. Isomorphism The structures of two trees are equal.
4. so abstract...
5. 7 6 1 2 4 5 3 7 1 6 2 4 5 3 4 3 1 5 7 2 6 isomorphism
6. How to judge two rooted tree are isomorphic?
8. If the trees are isomorphic, all their sub-trees are also isomorphic.
9. If the trees are isomorphic, all their sub-trees are also isomorphic.
10. If the trees are isomorphic, all their sub-trees are also isomorphic.
11. If the trees are isomorphic, all their sub-trees are also isomorphic.
12. Hash the tree
13. Hash all sub-tree
14. Hash all sub-tree recursively
15. initial value = 191 single vertex
16. initial value = 191 single vertex
17. initial value = 191 single vertex 191 191 191 191
18. initial value = 191 single vertex 191 191 191 191 define by yourself
19. 191 191 191 191 two-level sub-tree child = (191)
20. 191 191 191 191 two-level sub-tree child = (191)
21. 191 191 191 191 two-level sub-tree child = (191) (191 × 701 xor 191) mod 34943 = 29247 29247 29247
22. 191 191 191 191 two-level sub-tree child = (191) (191 × 701 xor 191) mod 34943 = 29247 29247 29247 define by yourself
23. 191 191 191 191 two-level sub-tree child = (191) (191 × 701 xor 191) mod 34943 = 29247 29247 29247 define by yourself
24. 29247 29247 191 191 191 191 three-level sub-tree child = (191,29247,191)
25. 29247 29247 191 191 191 191 three-level sub-tree child = (191,29247,191)
26. 29247 29247 191 191 191 191 three-level sub-tree child = (191,29247,191) child = (191,191,29247) sort
27. 29247 29247 191 191 191 191 three-level sub-tree child = (191,29247,191) child = (191,191,29247) sort (((((191 × 701 xor 191) mod 34943) × 701 xor 191) mod 34943) × 701 xor 29247) mod 34943 = 33360 33360
28. 29247 29247 191 191 191 191 three-level sub-tree child = (191,29247,191) child = (191,191,29247) sort (((((191 × 701 xor 191) mod 34943) × 701 xor 191) mod 34943) × 701 xor 29247) mod 34943 = 33360 33360
29. 33360 29247 29247 191 191 191 191 the total tree child = (33360,29247)
30. 33360 29247 29247 191 191 191 191 the total tree child = (33360,29247)
31. 33360 29247 29247 191 191 191 191 the total tree child = (33360,29247) child = (29247,33360) sort
32. 33360 29247 29247 191 191 191 191 the total tree child = (33360,29247) child = (29247,33360) sort (((191 × 701 xor 29247) mod 34943) × 701 xor 33360) mod 34943 = 4687 4687
33. 33360 29247 29247 191 191 191 191 the total tree child = (33360,29247) child = (29247,33360) sort (((191 × 701 xor 29247) mod 34943) × 701 xor 33360) mod 34943 = 4687 4687
34. hash value of the tree is 4687
35. initial value = 191 single vertex
36. initial value = 191 single vertex
37. initial value = 191 single vertex 191 191 191 191
38. 191 191 191 191 two-level sub-tree child = (191)
39. 191 191 191 191 two-level sub-tree child = (191)
40. 191 191 191 191 two-level sub-tree child = (191) (191 × 701 xor 191) mod 34943 = 29247 29247 29247
41. 191 29247 29247 191 191 191 three-level sub-tree child = (191,191,29247)
42. 191 29247 29247 191 191 191 three-level sub-tree child = (191,191,29247)
43. 191 29247 29247 191 191 191 three-level sub-tree child = (191,191,29247) child = (191,191,29247) sort
44. 191 29247 29247 191 191 191 three-level sub-tree child = (191,191,29247) child = (191,191,29247) (((((191 × 701 xor 191) mod 34943) × 701 xor 191) mod 34943) × 701 xor 29247) mod 34943 = 33360 sort 33360
45. 191 33360 29247 29247 191 191 191 the total tree child = (33360,29247)
46. 191 33360 29247 29247 191 191 191 the total tree child = (33360,29247)
47. 191 33360 29247 29247 191 191 191 the total tree child = (33360,29247) child = (29247,33360) sort
48. 191 33360 29247 29247 191 191 191 the total tree child = (33360,29247) child = (29247,33360) sort (((191 × 701 xor 29247) mod 34943) × 701 xor 33360) mod 34943 = 4687 4687
49. hash value of the tree is 4687
51. ≡
52. Algorithm HASH_TREE(T): 1. hash all sub-trees 2. sort hash value of sub-trees (unique) 3. calculate hash value (any hash function)
53. Time Complexity O(Nlog2N) N is number of vertices height of tree
54. Source Code int hash( TREE &now, int root ) { int value = INIT; vector< int > sub; //get all hash value of subtree for ( int i = 0; i < now[ root ].size(); ++i ) sub.push_back( hash( now, now[ root ] [ i ] ) ); //sort them to keep unique order sort( sub.begin(), sub.end() ); //hash this this tree for ( int i = 0; i < sub.size(); ++i ) value = ( ( value * P1 ) ^ sub[ i ] ) % P2; return value % P2; }
55. Representation of Tree Let the height of left child less than the right one.
56. Representation of Tree Let the height of left child less than the right one.
57. 4 3 2 1 2 1 1 1
58. 3 2 1 1 1 4 3 2 1 2 1 1 1 4 2 1
59. 2 1 3 2 1 1 1 4 3 2 1 2 1 1 1 4 2 1 3 1 1 4 2 1
60. Algorithm SORT_CHILD(T): 1. sort all sub-trees 2. compare the height 3. if height is equal, compare child recursively 4. put the lower at left and the higher at right
61. How about unrooted tree?
62. Find a Root
63. eliminate leaves
64. eliminate leaves
65. eliminate leaves
66. eliminate leaves
67. eliminate leaves
68. eliminate leaves
69. Enumerate Possible Root(s)
70. Rebuild The Tree
71. Rebuild The Tree
72. Rebuild The Tree
73. Rebuild The Tree
74. Rebuild The Tree
75. Rebuild The Tree
76. Apply the Isomorphism Detection
77. [POJ] 1635 - Subway tree systems Practice Now
78. Thank You for Your Listening.