Graph Isomorphism

Graph Isomorphism

Algorithms and networks

Graph Isomorphism 2

Today

• Graph isomorphism: definition • Complexity: isomorphism completeness • The refinement heuristic • Isomorphism for trees

– Rooted trees – Unrooted trees

Graph Isomorphism 3

Graph Isomorphism

• Two graphs G=(V,E) and H=(W,F) are isomorphic if there is a bijective function f: V → W such that for all v, w ∈ V: {v,w} ∈ E ⇔ {f(v),f(w)} ∈ F

Applications

• Chemistry: databases of molecules (etc.) – Actually needed: canonical form of molecule

structure / graph • Design verification • Software plagiarism detection • Speeding up algorithms for highly

symmetric graphs

Graph Isomorphism 4

Graph Isomorphism 5

Variant for labeled graphs

• Let G = (V,E), H=(W,F) be graphs with vertex labelings l: V → L, l’ → L.

• G and H are isomorphic labeled graphs, if there is a bijective function f: V → W such that – For all v, w ∈ V: {v,w} ∈ E ⇔ {f(v),f(w)} ∈ F – For all v ∈ V: l(v) = l’(f(v)).

• Application: organic chemistry: – determining if two molecules are identical.

Graph Isomorphism 6

Complexity of graph isomorphism

• Problem is in NP, but – No NP-completeness proof is known – No polynomial time algorithm is known

• If GI is NP-complete, then “strange things happen” – “Polynomial time hierarchy collapses to a finite level”

NP

NP-complete

P

Graph isomorphism

If P ≠ NP ?

Graph Isomorphism 7

Isomorphism-complete

• Problems are isomorphism-complete, if they are `equally hard’ as graph isomorphism – Isomorphism of bipartite graphs – Isomorphism of labeled graphs – Automorphism of graphs

• Given: a graph G=(V,E) • Question: is there a non-trivial

automorphism

Automorphism • An automorphism is a bijective

function f: V → V with for all v,w∈V: {v,w} ∈ E, if and only if {f(v),f(w)} ∈ E.

A non-trivial automorphism is an automorphism that is not the identity

G1 has 6 automorphisms, and 5 non-trivial automorphisms

G2 has 2 automorphisms, and 1 non-trivial automorphism

Graph Isomorphism 8

w

v

G1

G2

Graph Isomorphism 9

More isomorphism complete problems

• Finding a graph isomorphism f • Isomorphism of semi-groups • Isomorphism of finite automata • Isomorphism of finite algebra’s • Isomorphism of

– Connected graphs – Directed graphs – Regular graphs – Perfect graphs – Chordal graphs – Graphs that are isomorphic with their complement

Graph Isomorphism 10

Special cases are easier

• Polynomial time algorithms for – Graphs of bounded degree – Planar graphs – Trees – Bounded treewidth

• Expected polynomial time for random graphs

This course


An equivalence relation on vertices

• Say v ~ w, if and only if there is an automorphism mapping v to w.

• ~ is an equivalence relation • Partitions the vertices in automorphism

classes • Tells on structure of graph


Iterative vertex partition heuristic: the idea

• Partition the vertices of G and H in classes • Each class for G has a corresponding class

for H. • Property: vertices in class must be mapped

to vertices in corresponding class • Refine classes as long as possible • When no refinement possible, check all

possible ways that `remain’.

Iterative vertex partition heuristic skeleton

• Partition the vertices of G and H in classes – If v and w are in different classes, there is no

isomorphism or automorphism mapping v to w • Repeat

– Refine the classes Until … we do not find refinements

• Solve



Iterative vertex partition heuristic

• If |V| ≠ |W|, or |E| ≠ |F|, output: no. Done. • Otherwise, we partition the vertices of G and H

into classes, such that – Each class for G has a corresponding class for H – If f is an isomorphism from G to H, then f(v) belongs to

the class, corresponding to the class of v. • First try: vertices belong to the same class, when

they have the same degree. – If f is an isomorphism, then the degree of f(v) equals

the degree of v for each vertex v.


Scheme

• Start with sequence SG = (A1) of subsets of G with A1=V, and sequence SH = (B1) of subsets of H with B1=W.

• Repeat until … – Replace Ai in SG by Ai1,…,Air and replace Bi in SH by

Bi1,…,Bir. • Ai1,…,Air is partition of Ai

• Bi1,…,Bir is partition of Bi

• For each isormorphism f: v in Aij if and only if f(v) in Bij


Possible refinement

• Count for each vertex in Ai and Bi how many neighbors they have in Aj and Bj for some i, j.

• Set Ais = {v in Ai | v has s neighbors in Aj}. • Set Bis = {v in Bi | v has s neighbors in Bj}. • Invariant: for all v in the ith set in SG, f(v) in the

ith set in SH. • If some |Ai| ≠ |Bi|, then stop: no isomorphism.


Other refinements

• Partition upon other characteristics of vertices – Label – Number of vertices at distance d (in a set Ai). – …


After refining

• If each Ai contains one vertex: check the only possible isomorphism.

• Otherwise, cleverly enumerate all functions that are still possible, and check these.

• Works well in practice!


Isomorphism on trees

• Linear time algorithm to test if two (labeled) trees are isomorphic. (Aho, Hopcroft, Ullman, 1974)

• Algorithm to test if two rooted trees are isomorphic.

• Used as a subroutine for unrooted trees.


Rooted tree isomorphism

• For a vertex v in T, let T(v) be the subtree of T with v as root.

• Level of vertex: distance to root. • If T1 and T2 have different number of levels:

not isomorphic, and we stop. Otherwise, we continue:


Structure of algorithm

• Tree is processed level by level, from bottom to root

• Processing a level: – A long label for each vertex is computed – This is transformed to a short label

• Vertices in the same layer whose subtrees are isomorphic get the same labels: – If v and w on the same level, then

• L(v)=L(w), if and only if T(v) and T(w) are isomorphic with an isomorphism that maps v to w.


Labeling procedure • For each vertex, get the set of labels assigned to its

children. • Sort these sets.

– Bucketsort the pairs (L(w), v) for T1, w child of v

– Bucketsort the pairs (L(w), v) for T2, w child of v • For each v, we now have a long label LL(v) which

is the sorted set of labels of the children. • Use bucketsort to sort the vertices in T1 and T2

such that vertices with same long label are consecutive in ordering.


On sorting w.r.t. the long lists (1)

• Preliminary work: – Sort the nodes is the layer on the number of

children they have. • Linear time. (Counting sort / Radix sort.)

– Make a set of pairs (j,i) with (j,i) in the set when the jth number in a long list is i.

– Radix sort this set of pairs.


On sorting w.r.t. the long lists (2)

• Let q be the maximum length of a long list • Repeat

– Distribute among buckets the nodes with at least q children, with respect to the qth label in their long list

• Nodes distributed in buckets in earlier round are taken here in the order as they appear in these buckets.

• The sorted list of pairs (j,i) is used to skip empty buckets in this step.

– q --; – Until q=0.


After vertices are sorted with respect to long label

• Give vertices with same long label same short label (start counting at 0), and repeat at next level.

• If we see that the set of labels for a level of T1 is not equal to the set for the same level of T2, stop: not isomorphic.


Time

• One layer with n1 nodes with n2 nodes in next layer costs O(n1 + n2) time.

• Total time: O(n).


Unrooted trees

• Center of a tree – A vertex v with the property that the maximum distance

to any other vertex in T is as small as possible. – Each tree has a center of one or two vertices.

• Finding the center: – Repeat until we have a vertex or a single edge:

• Remove all leaves from T. – O(n) time: each vertex maintains current degree in

variable. Variables are updated when vertices are removed, and vertices put in set of leaves when their degree becomes 1.


Isomorphism of unrooted trees

• Note: the center must be mapped to the center • If T1 and T2 both have a center of size 1:

– Use those vertices as root. • If T1 and T2 both have a center of size 2:

– Try the two different ways of mapping the centers – Or: subdivide the edge between the two centers and

take the new vertices as root • Otherwise: not isomorphic. • 1 or 2 calls to isomorphism of rooted trees: O(n).


Conclusions

• Similar methods work for finding automorphisms

• We saw: heuristic for arbitrary graphs, algorithm for trees

• There are algorithms for several special graph classes (e.g., planar graphs, graphs of bounded degree,…)

• The related Subgraph Isomorphism problem is NP-complete

Documents

Graph Isomorphism