Tree isomorphism. Alexander Smal. Joint Advanced Student School St.Petersburg State University of Information Technologies, Mechanics and Optics
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1 Tree isomorphism Alexander Smal St.Petersburg State University of Information Technologies, Mechanics and Optics Joint Advanced Student School / 22
2 Motivation In some applications the chemical structures are often trees with millions of vertices: gene splicing, protein analysis, molecular biology. Difference between O(n), O(n log n), and O(n 2 ) isomorphism algorithms is not just theoretical importance. 2 / 22
3 Graph isomorphism Definition Isomorphism of graphs G 1 (V 1, E 1 ) and G 2 (V 2, E 2 ) is a bijection between the vertex sets φ : V 1 V 2 such that u, v V 1 (u, v) E 1 (φ(u), φ(v)) E 2. 3 / 22
4 Graph isomorphism Definition Isomorphism of graphs G 1 (V 1, E 1 ) and G 2 (V 2, E 2 ) is a bijection between the vertex sets φ : V 1 V 2 such that u, v V 1 (u, v) E 1 (φ(u), φ(v)) E 2. Facts No algorithm, other than brute force, is known for testing whether two arbitrary graphs are isomorphic. 3 / 22
5 Graph isomorphism Definition Isomorphism of graphs G 1 (V 1, E 1 ) and G 2 (V 2, E 2 ) is a bijection between the vertex sets φ : V 1 V 2 such that u, v V 1 (u, v) E 1 (φ(u), φ(v)) E 2. Facts No algorithm, other than brute force, is known for testing whether two arbitrary graphs are isomorphic. It is still an open question (!) whether graph isomorphism is NP complete. 3 / 22
6 Graph isomorphism Definition Isomorphism of graphs G 1 (V 1, E 1 ) and G 2 (V 2, E 2 ) is a bijection between the vertex sets φ : V 1 V 2 such that u, v V 1 (u, v) E 1 (φ(u), φ(v)) E 2. Facts No algorithm, other than brute force, is known for testing whether two arbitrary graphs are isomorphic. It is still an open question (!) whether graph isomorphism is NP complete. Polynomial time isomorphism algorithms for various graph subclasses such as trees are known. 3 / 22
7 Rooted trees Definition Rooted tree (V, E, r) is a tree (V, E) with selected root r V. 4 / 22
8 Rooted trees Definition Rooted tree (V, E, r) is a tree (V, E) with selected root r V. Definition Isomorphism of rooted trees T 1 (V 1, E 1, r 1 ) and T 2 (V 2, E 2, r 2 ) is a bijection between the vertex sets φ : V 1 V 2 such that and φ(r 1 ) = r 2. u, v V 1 (u, v) E 1 (φ(u), φ(v)) E 2 4 / 22
9 Rooted trees Definition Rooted tree (V, E, r) is a tree (V, E) with selected root r V. Definition Isomorphism of rooted trees T 1 (V 1, E 1, r 1 ) and T 2 (V 2, E 2, r 2 ) is a bijection between the vertex sets φ : V 1 V 2 such that and φ(r 1 ) = r 2. u, v V 1 (u, v) E 1 (φ(u), φ(v)) E 2 4 / 22
10 Rooted trees Definition Rooted tree (V, E, r) is a tree (V, E) with selected root r V. Definition Isomorphism of rooted trees T 1 (V 1, E 1, r 1 ) and T 2 (V 2, E 2, r 2 ) is a bijection between the vertex sets φ : V 1 V 2 such that and φ(r 1 ) = r 2. u, v V 1 (u, v) E 1 (φ(u), φ(v)) E 2 Example T 1 and T 2 are isomorphic as graphs... a B T 1 b c A C T 2 4 / 22
11 Rooted trees Definition Rooted tree (V, E, r) is a tree (V, E) with selected root r V. Definition Isomorphism of rooted trees T 1 (V 1, E 1, r 1 ) and T 2 (V 2, E 2, r 2 ) is a bijection between the vertex sets φ : V 1 V 2 such that and φ(r 1 ) = r 2. u, v V 1 (u, v) E 1 (φ(u), φ(v)) E 2 Example T 1 and T 2 are isomorphic as graphs but not as rooted trees! a B T 1 b c A C T 2 4 / 22
12 Rooted trees (part 2) Lemma If there is O(n) algorithm for rooted trees isomorphism, then there is O(n) algorithm for ordinary trees isomorphism. 5 / 22
13 Rooted trees (part 2) Lemma If there is O(n) algorithm for rooted trees isomorphism, then there is O(n) algorithm for ordinary trees isomorphism. Proof. 1 Let A to be O(n) algorithm for rooted trees. 5 / 22
14 Rooted trees (part 2) Lemma If there is O(n) algorithm for rooted trees isomorphism, then there is O(n) algorithm for ordinary trees isomorphism. Proof. 1 Let A to be O(n) algorithm for rooted trees. 2 Let T 1 and T 2 to be ordinary trees. 5 / 22
15 Rooted trees (part 2) Lemma If there is O(n) algorithm for rooted trees isomorphism, then there is O(n) algorithm for ordinary trees isomorphism. Proof. 1 Let A to be O(n) algorithm for rooted trees. 2 Let T 1 and T 2 to be ordinary trees. 3 Lets find centers of this trees. There are three cases: 5 / 22
16 Rooted trees (part 2) Lemma If there is O(n) algorithm for rooted trees isomorphism, then there is O(n) algorithm for ordinary trees isomorphism. Proof. 1 Let A to be O(n) algorithm for rooted trees. 2 Let T 1 and T 2 to be ordinary trees. 3 Lets find centers of this trees. There are three cases: 1 each tree has only one center (c 1 and c 2 respectively) return A(T 1, c 1, T 2, c 2 ) 5 / 22
17 Rooted trees (part 2) Lemma If there is O(n) algorithm for rooted trees isomorphism, then there is O(n) algorithm for ordinary trees isomorphism. Proof. 1 Let A to be O(n) algorithm for rooted trees. 2 Let T 1 and T 2 to be ordinary trees. 3 Lets find centers of this trees. There are three cases: 1 each tree has only one center (c 1 and c 2 respectively) return A(T 1, c 1, T 2, c 2 ) 2 each tree has exactly two centers (c 1, c 1 and c 2, c 2 respectively) return A(T 1, c 1, T 2, c 2 ) or A(T 1, c 1, T 2, c 2 ) 5 / 22
18 Rooted trees (part 2) Lemma If there is O(n) algorithm for rooted trees isomorphism, then there is O(n) algorithm for ordinary trees isomorphism. Proof. 1 Let A to be O(n) algorithm for rooted trees. 2 Let T 1 and T 2 to be ordinary trees. 3 Lets find centers of this trees. There are three cases: 1 each tree has only one center (c 1 and c 2 respectively) return A(T 1, c 1, T 2, c 2 ) 2 each tree has exactly two centers (c 1, c 1 and c 2, c 2 respectively) return A(T 1, c 1, T 2, c 2 ) or A(T 1, c 1, T 2, c 2 ) 3 trees has different number of centers return False 5 / 22
19 Diameter and center Definition The diameter of tree is the length of the longest path. 6 / 22
20 Diameter and center Definition The diameter of tree is the length of the longest path. Definition A center is a vertex v such that the longest path from v to a leaf is minimal over all vertices in the tree. 6 / 22
21 Diameter and center Definition The diameter of tree is the length of the longest path. Definition A center is a vertex v such that the longest path from v to a leaf is minimal over all vertices in the tree. Algorithm 1: Choose a random root r. 2: Find a vertex v 1 the farthest form r. 3: Find a vertex v 2 the farthest form v 1. 4: Diameter is a length of path from v 1 to v 2. 5: Centers are median elements of path from v 1 to v 2. 6 / 22
22 Diameter and center Definition The diameter of tree is the length of the longest path. Definition A center is a vertex v such that the longest path from v to a leaf is minimal over all vertices in the tree. Algorithm 1: Choose a random root r. 2: Find a vertex v 1 the farthest form r. 3: Find a vertex v 2 the farthest form v 1. 4: Diameter is a length of path from v 1 to v 2. 5: Centers are median elements of path from v 1 to v 2. It is O(n) algorithm. 6 / 22
23 The idea Let s try to find complete invariant of rooted trees isomorphism. 7 / 22
24 The idea Let s try to find complete invariant of rooted trees isomorphism. Definition Isomorphism invariant is a function f (T ) such that f (T 1 ) = f (T 2 ) for all pairs of isomorphic trees T 1 and T 2. 7 / 22
25 The idea Let s try to find complete invariant of rooted trees isomorphism. Definition Isomorphism invariant is a function f (T ) such that f (T 1 ) = f (T 2 ) for all pairs of isomorphic trees T 1 and T 2. Definition Complete isomorphism invariant is a function f (T ) such that two trees T 1 and T 2 are isomorphic if and only if f (T 1 ) = f (T 2 ). 7 / 22
26 The idea Let s try to find complete invariant of rooted trees isomorphism. Definition Isomorphism invariant is a function f (T ) such that f (T 1 ) = f (T 2 ) for all pairs of isomorphic trees T 1 and T 2. Definition Complete isomorphism invariant is a function f (T ) such that two trees T 1 and T 2 are isomorphic if and only if f (T 1 ) = f (T 2 ). So if we find complete isomorphism invariant we can obtain algorithm from it. 7 / 22
27 The idea Let s try to find complete invariant of rooted trees isomorphism. Definition Isomorphism invariant is a function f (T ) such that f (T 1 ) = f (T 2 ) for all pairs of isomorphic trees T 1 and T 2. Definition Complete isomorphism invariant is a function f (T ) such that two trees T 1 and T 2 are isomorphic if and only if f (T 1 ) = f (T 2 ). So if we find complete isomorphism invariant we can obtain algorithm from it. Note Starting from the next slide tree always means rooted tree! 7 / 22
28 Candidate 1 Observation The level number of a vertex is a tree isomorphism invariant. 8 / 22
29 Candidate 1 Observation The level number of a vertex is a tree isomorphism invariant. Conjecture Two trees are isomorphic if and only if they have the same number of levels and the same number of vertices on each level. 8 / 22
30 Candidate 1 Observation The level number of a vertex is a tree isomorphism invariant. Conjecture Two trees are isomorphic if and only if they have the same number of levels and the same number of vertices on each level. Observation The number of the leaves is a tree isomorphism invariant. 8 / 22
31 Candidate 1 Observation The level number of a vertex is a tree isomorphism invariant. Conjecture Two trees are isomorphic if and only if they have the same number of levels and the same number of vertices on each level. Observation The number of the leaves is a tree isomorphism invariant. Contrary instance a T n 1 2 n T 2 d e D E A 8 / 22
32 Candidate 2 What s wrong with candidate 1? We didn t take into account the degree spectrum of a tree. 9 / 22
33 Candidate 2 What s wrong with candidate 1? We didn t take into account the degree spectrum of a tree. Definition Degree spectrum of tree is the sequence of non-negative integers {d j }, where d j is the number of vertices that have j children. 9 / 22
34 Candidate 2 What s wrong with candidate 1? We didn t take into account the degree spectrum of a tree. Definition Degree spectrum of tree is the sequence of non-negative integers {d j }, where d j is the number of vertices that have j children. Conjecture Two trees are isomorphic if and only if they have the same degree spectrum. 9 / 22
35 Candidate 2 (part 2) Observation Since a tree isomorphism preserves longest paths from the root, the number of levels in a tree is a tree isomorphism invariant. 10 / 22
36 Candidate 2 (part 2) Observation Since a tree isomorphism preserves longest paths from the root, the number of levels in a tree is a tree isomorphism invariant. Contrary instance a A b c B C d e 1 D E T 1 1. T 2. n n 10 / 22
37 Candidate 3 Conjecture Two trees are isomorphic if and only if they have the same degree spectrum at each level. 11 / 22
38 Candidate 3 Conjecture Two trees are isomorphic if and only if they have the same degree spectrum at each level. If two trees have the same degree spectrum at each level, then they must automatically have the same number of levels, the same number of vertices at each level, and the same global degree spectrum! 11 / 22
39 Candidate 3 Conjecture Two trees are isomorphic if and only if they have the same degree spectrum at each level. If two trees have the same degree spectrum at each level, then they must automatically have the same number of levels, the same number of vertices at each level, and the same global degree spectrum! Observation The number of leaf descendants of a vertex and the level number of a vertex are both tree isomorphism invariants. 11 / 22
40 Candidate 3 (part 2) Contrary instance a A level degree spectrum (0, 0, 1, 0,...) b c B C d e f D E F T 1 g 1 G 1 T 2 (0, 1, 1, 0,...) (2, 0, 1, 0,...) (1, 1, 0, 0,...). n. n. (1, 0, 0, 0,...) 12 / 22
41 Algorithm by Aho, Hopcroft and Ullman Determine tree isomorphism in time O( V ). AHU algorithm Uses complete history of degree spectrum of the vertex descendants as a complete invariant. 13 / 22
42 Algorithm by Aho, Hopcroft and Ullman Determine tree isomorphism in time O( V ). AHU algorithm Uses complete history of degree spectrum of the vertex descendants as a complete invariant. The idea of AHU algorithm The AHU algorithm associates with each vertex a tuple that describes the complete history of its descendants. 13 / 22
43 Algorithm by Aho, Hopcroft and Ullman Determine tree isomorphism in time O( V ). AHU algorithm Uses complete history of degree spectrum of the vertex descendants as a complete invariant. The idea of AHU algorithm The AHU algorithm associates with each vertex a tuple that describes the complete history of its descendants. Hard question Why our previous invariants are not complete? 13 / 22
44 Algorithm by Aho, Hopcroft and Ullman Determine tree isomorphism in time O( V ). AHU algorithm Uses complete history of degree spectrum of the vertex descendants as a complete invariant. The idea of AHU algorithm The AHU algorithm associates with each vertex a tuple that describes the complete history of its descendants. Hard question Why our previous invariants are not complete? Let s discuss AHU algorithm. We start from O( V 2 ) version and then I tell how to make it faster (O( V )). 13 / 22
45 Understanding AHU algorithm Knuth tuples Let s assign parenthetical tuples to all tree vertices. 14 / 22
46 Understanding AHU algorithm Knuth tuples Let s assign parenthetical tuples to all tree vertices. Knuth tuples example A B C D E F G 14 / 22
47 Understanding AHU algorithm Knuth tuples Let s assign parenthetical tuples to all tree vertices. Knuth tuples example A B C D (0) E (0) F (0) G (0) 14 / 22
48 Understanding AHU algorithm Knuth tuples Let s assign parenthetical tuples to all tree vertices. Knuth tuples example A B ((0)) C D (0) E (0) F (0) G (0) 14 / 22
49 Understanding AHU algorithm Knuth tuples Let s assign parenthetical tuples to all tree vertices. Knuth tuples example A B ((0)) C ( (0) (0) ) D (0) E (0) F (0) G (0) 14 / 22
50 Understanding AHU algorithm Knuth tuples Let s assign parenthetical tuples to all tree vertices. Knuth tuples example A B ((0)) C ( (0) (0) ) D (0) E (0) F (0) G (0) 14 / 22
51 Understanding AHU algorithm Knuth tuples Let s assign parenthetical tuples to all tree vertices. Knuth tuples example A B ((0)) C ( (0) (0) ) D (0) E (0) F (0) G (0) 14 / 22
52 Understanding AHU algorithm Knuth tuples Let s assign parenthetical tuples to all tree vertices. Knuth tuples example A ( ((0)) ((0)(0)) (0) ) B ((0)) C ( (0) (0) ) D (0) E (0) F (0) G (0) 14 / 22
53 Understanding AHU algorithm Knuth tuples Let s assign parenthetical tuples to all tree vertices. Knuth tuples example A ( ((0)) ((0)(0)) (0) ) B ((0)) C ( (0) (0) ) D (0) E (0) F (0) G (0) 14 / 22
54 Understanding AHU algorithm Knuth tuples Let s assign parenthetical tuples to all tree vertices. Knuth tuples example A ( ((0)) ((0)(0)) (0) ) B ((0)) C ( (0) (0) ) D (0) E (0) F (0) G (0) 14 / 22
55 Understanding AHU algorithm Knuth tuples Let s assign parenthetical tuples to all tree vertices. Knuth tuples example A ( ((0)) ((0)(0)) (0) ) B ((0)) C ( (0) (0) ) D (0) E (0) F (0) G (0) 14 / 22
56 Understanding AHU algorithm (part 2) There is algorithm Assign-Knuth-Tuples that visits every vertex once or twice. Assign-Knuth-Tuples(v) 1: if v is a leaf then 2: Give v the tuple name (0) 3: else 4: for all child w of v do 5: Assign-Knuth-Tuples(w) 6: end for 7: end if 8: Concatenate the names of all children of v to temp 9: Give v the tuple name temp 15 / 22
57 Understanding AHU algorithm (part 3) Observation There is no order on parenthetical tuples. 16 / 22
58 Understanding AHU algorithm (part 3) Observation There is no order on parenthetical tuples. Example A ( (0) ((0)) ) a ( ((0)) (0) ) B (0) C ((0)) b ((0)) c (0) D (0) d (0) 16 / 22
59 Understanding AHU algorithm (part 3) Observation There is no order on parenthetical tuples. Example A ( (0) ((0)) ) a ( ((0)) (0) ) B (0) C ((0)) b ((0)) c (0) D (0) d (0) Let s convert parenthetical tuples to canonical names. We should drop all 0 -s and replace ( and ) with 1 and 0 respectively. 16 / 22
60 Understanding AHU algorithm (part 3) Observation There is no order on parenthetical tuples. Example A a B 10 C 1100 b 1100 c 10 D 10 d 10 Let s convert parenthetical tuples to canonical names. We should drop all 0 -s and replace ( and ) with 1 and 0 respectively. 16 / 22
61 Understanding AHU algorithm (part 3) Observation There is no order on parenthetical tuples. Example A a B 10 C 1100 b 1100 c 10 D 10 d 10 Let s convert parenthetical tuples to canonical names. We should drop all 0 -s and replace ( and ) with 1 and 0 respectively. 16 / 22
62 Understanding AHU algorithm (part 4) Assign-Canonical-Names(v) 1: if v is a leaf then 2: Give v the tuple name 10 3: else 4: for all child w of v do 5: Assign-Canonical-Names(v) 6: end for 7: end if 8: Sort the names of the children of v 9: Concatenate the names of all children of v to temp 10: Give v the name 1temp0 17 / 22
63 Understanding AHU algorithm (part 5) We should discuss some important questions. 18 / 22
64 Understanding AHU algorithm (part 5) We should discuss some important questions. Invariant? Is canonical name of a root a tree isomorphism invariant? 18 / 22
65 Understanding AHU algorithm (part 5) We should discuss some important questions. Invariant? Is canonical name of a root a tree isomorphism invariant? Complete invariant? Is canonical name of a root a complete tree isomorphism invariant? 18 / 22
66 Understanding AHU algorithm (part 5) We should discuss some important questions. Invariant? Is canonical name of a root a tree isomorphism invariant? Complete invariant? Is canonical name of a root a complete tree isomorphism invariant? AHU-Tree-Isomorphism(T 1, T 2 ) 1: r 1 root(t 1 ) 2: r 2 root(t 2 ) 3: Assign-Canonical-Names(r 1 ) 4: Assign-Canonical-Names(r 2 ) 5: if name(r 1 ) = name(r 2 ) then 6: return True 7: else 8: return False 9: end if 18 / 22
67 AHU algorithm improvement Observation To compute the root name of a tree of n vertices in one long strand, takes time proportional to n, which is Ω(n 2 ). 19 / 22
68 AHU algorithm improvement Observation To compute the root name of a tree of n vertices in one long strand, takes time proportional to n, which is Ω(n 2 ). Observation For all levels i, the canonical name of level i is a tree isomorphism invariant. 19 / 22
69 AHU algorithm improvement Observation To compute the root name of a tree of n vertices in one long strand, takes time proportional to n, which is Ω(n 2 ). Observation For all levels i, the canonical name of level i is a tree isomorphism invariant. Observation Two trees T 1 and T 2 are isomorphic if and only if for all levels i canonical level names of T 1 and T 2 are identical. 19 / 22
70 AHU algorithm improvement Observation To compute the root name of a tree of n vertices in one long strand, takes time proportional to n, which is Ω(n 2 ). Observation For all levels i, the canonical name of level i is a tree isomorphism invariant. Observation Two trees T 1 and T 2 are isomorphic if and only if for all levels i canonical level names of T 1 and T 2 are identical. The idea 1 Assign canonical names for level, sort by level, and check by level that the canonical level names agree. 19 / 22
71 AHU algorithm improvement Observation To compute the root name of a tree of n vertices in one long strand, takes time proportional to n, which is Ω(n 2 ). Observation For all levels i, the canonical name of level i is a tree isomorphism invariant. Observation Two trees T 1 and T 2 are isomorphic if and only if for all levels i canonical level names of T 1 and T 2 are identical. The idea 1 Assign canonical names for level, sort by level, and check by level that the canonical level names agree. The idea 2 Assign canonical names for level and if canonical level names agree than replace canonical names with integers. 19 / 22
72 AHU algorithm example Example A a B C b c 0 D E d e 1 20 / 22
73 AHU algorithm example Example A a B C b c 0 D 0 E 0 d 0 e / 22
74 AHU algorithm example Example A a B C b c 0 D 1 E 1 d 1 e 1 1,1 1 1,1 20 / 22
75 AHU algorithm example Example A a B C b c 0 D 1 E 1 d 1 e 1 1,1 1 1,1 20 / 22
76 AHU algorithm example Example A a B 0 C 1 1 b 1 1 c 0 D 1 E 1 d 1 e / 22
77 AHU algorithm example Example A a B 1 C 2 b 2 c 1 D 1 E 1 d 1 e 1 1,2 1 1,2 20 / 22
78 AHU algorithm example Example A a B 1 C 2 b 2 c 1 D 1 E 1 d 1 e 1 1,2 1 1,2 20 / 22
79 AHU algorithm example Example A 1 2 a 1 2 B 1 C 2 b 2 c 1 D 1 E 1 d 1 e / 22
80 AHU algorithm example Example A 1 a 1 B 1 C 2 b 2 c 1 D 1 E 1 d 1 e / 22
81 AHU algorithm example Example A 1 a 1 B 1 C 2 b 2 c 1 D 1 E 1 d 1 e / 22
82 AHU algorithm example Example A 1 a 1 B 1 C 2 b 2 c 1 D 1 E 1 d 1 e 1 OK 1 20 / 22
83 Resume Resume We have made three unsuccessful attempts to construct complete tree isomorphism invariant. We discussed O( V 2 ) version of AHU algorithm. We discussed ways of AHU algorithm improvement to make it work in O( V ) time. 21 / 22
84 Thank you for your attention! Any questions? 22 / 22
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