Graph Isomorphism Algorithms and networks
Today Graph isomorphism: definition Complexity: isomorphism completeness The refinement heuristic Isomorphism for trees Rooted trees Unrooted trees Graph Isomorphism 2
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 Graph Isomorphism 3
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
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 5
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 If P NP NP-complete? Graph isomorphism NP P Graph Isomorphism 6
Algorithmic bound Theorem (Babai, 2017) Graph Isomorphism can be solved in time 2 O((logn) for some constant c. c ) Quasipolynomial time Better than anything known for NP-hard problems Graph Isomorphism 7
Isomorphism-complete Problems are isomorphismcomplete, 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 Graph Isomorphism 8
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 G 1 has 6 automorphisms, and 5 nontrivial automorphisms G 2 has 2 automorphisms, and 1 nontrivial automorphism v G 2 w G 1 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 This course Expected polynomial time for random graphs Graph Isomorphism 11
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 Graph Isomorphism 12
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. Graph Isomorphism 13
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 Graph Isomorphism 14
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. Graph Isomorphism 15
Scheme Start with sequence SG = (A 1 ) of subsets of G with A 1 =V, and sequence SH = (B 1 ) of subsets of H with B 1 =W. Repeat until Replace A i in SG by A i1,,a ir and replace B i in SH by B i1,,b ir. A i1,,a ir is partition of A i B i1,,b ir is partition of B i For each isormorphism f: v in A ij if and only if f(v) in B ij Graph Isomorphism 16
Possible refinement Count for each vertex in A i and B i how many neighbors they have in A j and B j for some i, j. Set A is = {v in A i v has s neighbors in A j }. Set B is = {v in B i v has s neighbors in B j }. Invariant: for all v in the ith set in SG, f(v) in the ith set in SH. If some A i B i, then stop: no isomorphism. Graph Isomorphism 17
Other refinements Partition upon other characteristics of vertices Label Number of vertices at distance d (in a set A i ). Graph Isomorphism 18
After refining If each A i contains one vertex: check the only possible isomorphism. Otherwise, cleverly enumerate all functions that are still possible, and check these. Works well in practice! Graph Isomorphism 19
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. Graph Isomorphism 20
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 T 1 and T 2 have different number of levels: not isomorphic, and we stop. Otherwise, we continue: Graph Isomorphism 21
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. Graph Isomorphism 22
Labeling procedure For each vertex, get the set of labels assigned to its children. Sort these sets. Bucketsort the pairs (L(w), v) for T 1, w child of v Bucketsort the pairs (L(w), v) for T 2, 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 T 1 and T 2 such that vertices with same long label are consecutive in ordering. Graph Isomorphism 23
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. Graph Isomorphism 24
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. Graph Isomorphism 25
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 T 1 is not equal to the set for the same level of T 2, stop: not isomorphic. Graph Isomorphism 26
Time One layer with n 1 nodes with n 2 nodes in next layer costs O(n 1 + n 2 ) time. Total time: O(n). Graph Isomorphism 27
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. Graph Isomorphism 28
Isomorphism of unrooted trees Note: the center must be mapped to the center If T 1 and T 2 both have a center of size 1: Use those vertices as root. If T 1 and T 2 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). Graph Isomorphism 29
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 Graph Isomorphism 30