Practical Graph Isomorphism, II - Semantic Scholar
Practical Graph Isomorphism, II Brendan McKay Australian National University
Adolfo Piperno University of Rome
practical graph isomorphism ii — 1
Isomorphism and Automorphism We have some finite objects consisting of some finite sets and some relations on them. An isomorphism between two objects is a bijection between their sets so that the relations are preserved. Hopefully, isomorphism is an equivalence relation on the set of all objects.
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Isomorphism and Automorphism We have some finite objects consisting of some finite sets and some relations on them. An isomorphism between two objects is a bijection between their sets so that the relations are preserved. Hopefully, isomorphism is an equivalence relation on the set of all objects. Isomorphisms from an object to itself are automorphisms. If we didn’t do anything silly, the set of automorphisms forms a group under composition. Of course we call it the automorphism group. 8
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1
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Isomorphism and Automorphism We have some finite objects consisting of some finite sets and some relations on them. An isomorphism between two objects is a bijection between their sets so that the relations are preserved. Hopefully, isomorphism is an equivalence relation on the set of all objects. Isomorphisms from an object to itself are automorphisms. If we didn’t do anything silly, the set of automorphisms forms a group under composition. Of course we call it the automorphism group. 8
7
6
5
(1) (1 4)(2 3)(5 8)(6 7) (1 8)(2 7)(3 6)(4 5) (1 5)(2 6)(3 7)(4 8)
1
2
3
4
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Ubiquity of graph isomorphism Very many isomorphism problems can be efficiently expressed as graph isomorphism problems. For convenience, we use coloured graphs, in which the vertices (and possibly the edges) are coloured. Isomorphisms must preserve vertex and edge colour.
1 3 5 4 1 5 4 3 3 4 5 1
Say two matrices are isomorphic if one can be obtained from the other by permuting rows and columns.
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Ubiquity of graph isomorphism Very many isomorphism problems can be efficiently expressed as graph isomorphism problems. For convenience, we use coloured graphs, in which the vertices (and possibly the edges) are coloured. Isomorphisms must preserve vertex and edge colour.
1 3 5 4 1 5 4 3 3 4 5 1
Say two matrices are isomorphic if one can be obtained from the other by permuting rows and columns.
=1 =3 =4 =5
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Hadamard equivalence
1 −1 1 −1 −1 −1
1 1 1
Say isomorphism of two ±1 matrices allows permutation and negation of rows and columns.
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Hadamard equivalence
1 −1 1 −1 −1 −1
1 1 1
Say isomorphism of two ±1 matrices allows permutation and negation of rows and columns.
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Isotopy
1 3 2 2 1 3 3 2 1
Say isomorphism of two matrices allows permutation of rows, columns and symbols.
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Isotopy
1
Ro 1 3 2 2 1 3 3 2 1
1
Say isomorphism of two matrices allows permutation of rows, columns and symbols.
mn
2
2
lu Co
w
3
3
3
2
Symbol
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1
Isotopy
1
Ro 1 3 2 2 1 3 3 2 1
1
Say isomorphism of two matrices allows permutation of rows, columns and symbols.
mn
2
2
lu Co
w
3
3
3
2
Symbol
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1
Group isomorphism
e g1 g2 g3
g1 e g3 g2 g g e g 1 2 3
g3 g2 g1 e
The group elements are edge colours and vertices. The black vertex is the identity element. Preserve vertex colours but allow permutation of edge colours.
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Group isomorphism
e g1 g2 g3
g1 e g3 g2 g g e g 1 2 3
g3 g2 g1 e
The group elements are edge colours and vertices. The black vertex is the identity element. Preserve vertex colours but allow permutation of edge colours.
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Permutation equivalence of linear codes
0 0 1 1 0
1 1 0 1 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 0 1 1 0 1 1 0 1 0 0 0 0 1 0
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Permutation equivalence of linear codes
0 0 1 1 0
1 1 0 1 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 0 1 1 0 1 1 0 1 0 0 0 0 1 0
?
Suppose isomorphism of two 0-1 matrices means that their columns can be permutated so that the rows generate the same vector space over GF (2). Nobody knows how to make an equivalent graph problem of size which is polynomial in the size of the generator matrix.
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Theoretical complexity The graph isomorphism problem is:
GI: Given two graphs, determine whether they are isomorphic. O(
• The fastest known algorithm for GI is 2
√
(n log n))
time; no polynomial time
algorithm is known (despite many claims).
• GI is not known to be NP-complete. • GI is not known to be in co-NP. • There are polynomial time algorithms for many special classes of graphs (bounded degree, bounded genus, classes with excluded minors or topological subgraphs). Few of these are practical, planar graphs being a notable exception. It is possible that group isomorphism is easier than GI and that permutation equivalence of linear codes is harder than GI. But nobody can prove it.
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Canonical labelling Given a class of objects, and a definition of “isomorphism”, we can choose one member of each isomorphism class and call it the canonical member of the class. The process of finding the canonical member of the isomorphism class containing a given object is called “canonical labelling”.
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Canonical labelling Given a class of objects, and a definition of “isomorphism”, we can choose one member of each isomorphism class and call it the canonical member of the class. The process of finding the canonical member of the isomorphism class containing a given object is called “canonical labelling”. Two labelled objects which are isomorphic become identical when they are canonically labelled. Since identity of objects is usually far easier to check than isomorphism, canonical labelling is the preferred method of sorting a collection of objects into isomorphism classes.
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Canonical labelling Given a class of objects, and a definition of “isomorphism”, we can choose one member of each isomorphism class and call it the canonical member of the class. The process of finding the canonical member of the isomorphism class containing a given object is called “canonical labelling”. Two labelled objects which are isomorphic become identical when they are canonically labelled. Since identity of objects is usually far easier to check than isomorphism, canonical labelling is the preferred method of sorting a collection of objects into isomorphism classes. A possible definition of the canonical member of an isomorphism class would be the member which maximises some linear representation (such as a list of edges). In practice, most programs compute a canonical member which is easy for computers to find rather than easy for humans to define.
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Software for graph isomorphism This task was already described as a disease in 1970, and programs still appear regularly. The most successful programs still supported are:
• nauty (McKay, 1976–) canonical label and automorphism group • VF2 (Cordella, Foggia, Sansone and Vento, 1999–) comparison of two graphs • saucy (Darga, Sakallah and Markov, 2004–) automorphism group • bliss (Juntilla and Kaski, 2007–) canonical label and automorphism group • Traces (Piperno, 2008–) canonical label and automorphism group • conauto (L´ opez-Presa, Anta and Chiroque, 2009–) automorphism group and comparison of two graphs Many similar principles appear in these programs. saucy and bliss began life as reimplementations of nauty but have since diverged.
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Partition refinement A key concept used by nauty is partition refinement. (“partition” = “colouring”) An equitable partition is one where every two vertices of the same colour are adjacent to the same number of vertices of each colour. Two examples of equitable partitions:
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Refinement of a partition means to subdivide its cells. Given a partition (colouring) π , there is a unique equitable partition that is a refinement of π and has the least number of colours.
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Refinement of a partition means to subdivide its cells. Given a partition (colouring) π , there is a unique equitable partition that is a refinement of π and has the least number of colours.
Initial
Count green neighbours
Count pink neighbours
Count yellow neighbours
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Search tree All competitive isomorphism programs generate a tree based on partition refinement. The nodes of the tree correspond to partitions (colourings). The root of the tree corresponds to the initial colouring, refined. If a node corresponds to a discrete partition (each vertex with a different colour), it has no children and is a leaf. Otherwise we choose a colour used more than once (the target cell), individualize one of those vertices by giving it a new unique colour, and refine to get a child.
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Search tree All competitive isomorphism programs generate a tree based on partition refinement. The nodes of the tree correspond to partitions (colourings). The root of the tree corresponds to the initial colouring, refined. If a node corresponds to a discrete partition (each vertex with a different colour), it has no children and is a leaf. Otherwise we choose a colour used more than once (the target cell), individualize one of those vertices by giving it a new unique colour, and refine to get a child. Each leaf lists the vertices in some order (since colours have a predefined order), so it corresponds to a labelling of the graph.
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Search tree All competitive isomorphism programs generate a tree based on partition refinement. The nodes of the tree correspond to partitions (colourings). The root of the tree corresponds to the initial colouring, refined. If a node corresponds to a discrete partition (each vertex with a different colour), it has no children and is a leaf. Otherwise we choose a colour used more than once (the target cell), individualize one of those vertices by giving it a new unique colour, and refine to get a child. Each leaf lists the vertices in some order (since colours have a predefined order), so it corresponds to a labelling of the graph. If we have two leaves giving the same labelled graph, we have found an automorphism. If we define an order on labelled graphs, such as lexicographic order, then the greatest labelled graph corresponding to a leaf is a canonical graph.
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Search tree All competitive isomorphism programs generate a tree based on partition refinement. The nodes of the tree correspond to partitions (colourings). The root of the tree corresponds to the initial colouring, refined. If a node corresponds to a discrete partition (each vertex with a different colour), it has no children and is a leaf. Otherwise we choose a colour used more than once (the target cell), individualize one of those vertices by giving it a new unique colour, and refine to get a child. Each leaf lists the vertices in some order (since colours have a predefined order), so it corresponds to a labelling of the graph. If we have two leaves giving the same labelled graph, we have found an automorphism. If we define an order on labelled graphs, such as lexicographic order, then the greatest labelled graph corresponding to a leaf is a canonical graph. However the tree can be much too large to generate completely.
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[1458|2367]
[ ... ]
[ ... ]
[ 4 |1 8 | 5 | 2 7 | 6 | 3 ]
[ 1 |4 5 | 8 | 3 6 | 7 | 2 ]
[ ... ] [ ... ]
[1|4|5 |8|6|3|7|2]
[1|5|4 |8|3|6|7|2]
[4|1|8|5|7|2|6|3]
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(1)
(3 6)(4 5)
(1 4)(2 3)(5 8)(6 7)
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[ ... ] [ ... ]
[ ... ]
Node invariants A node invariant is a value φ(ν ) attached to each node in the tree that depends only on the combinatorial properties of ν and its ancestors in the search tree (not on the labelling of the graph). Moreover, the order of node invariants must be preserved by descendants: Let ν1 , ν2 be nodes at the same level in the search tree. If ν10 is a descendant of ν1 , and ν20 is a descendant of ν2 , then
φ(ν1 ) < φ(ν2 ) =⇒ φ(ν10 ) < φ(ν20 ).
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Node invariants A node invariant is a value φ(ν ) attached to each node in the tree that depends only on the combinatorial properties of ν and its ancestors in the search tree (not on the labelling of the graph). Moreover, the order of node invariants must be preserved by descendants: Let ν1 , ν2 be nodes at the same level in the search tree. If ν10 is a descendant of ν1 , and ν20 is a descendant of ν2 , then
φ(ν1 ) < φ(ν2 ) =⇒ φ(ν10 ) < φ(ν20 ). For example,
φ(ν ) = c (ν 0 ), c (ν 00 ), . . . , c (ν )
is a node invariant with lexicographic ordering, where ν 0 , ν 00 , . . . , ν is the path from the root of the tree to ν , and c ( ) is the number of colours.
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Node invariants A node invariant is a value φ(ν ) attached to each node in the tree that depends only on the combinatorial properties of ν and its ancestors in the search tree (not on the labelling of the graph). Moreover, the order of node invariants must be preserved by descendants: Let ν1 , ν2 be nodes at the same level in the search tree. If ν10 is a descendant of ν1 , and ν20 is a descendant of ν2 , then
φ(ν1 ) < φ(ν2 ) =⇒ φ(ν10 ) < φ(ν20 ). For example,
φ(ν ) = c (ν 0 ), c (ν 00 ), . . . , c (ν )
is a node invariant with lexicographic ordering, where ν 0 , ν 00 , . . . , ν is the path from the root of the tree to ν , and c ( ) is the number of colours. Let φ∗ be the greatest node invariant of a leaf. Then the lexicographically greatest labelled graph corresponding to a leaf ν with φ(ν ) = φ∗ is a canonical graph.
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Pruning operations Node invariants, together with automorphisms, allow us to remove parts of the search tree without generating them. 1. If ν1 , ν2 are nodes at the same level in the search tree and φ(ν1 ) < φ(ν2 ), then no canonical labelling is descended from ν1 . 2. If ν1 , ν2 are nodes at the same level in the search tree and φ(ν1 ) 6= φ(ν2 ), then no labelled graph descended from ν1 is the same as one descended from ν2 . g
3. If ν1 , ν2 are nodes with ν2 = ν1 for an automorphism g , then g maps the subtree descended from ν1 onto the subtree descended from ν2 .
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Pruning operations Node invariants, together with automorphisms, allow us to remove parts of the search tree without generating them. 1. If ν1 , ν2 are nodes at the same level in the search tree and φ(ν1 ) < φ(ν2 ), then no canonical labelling is descended from ν1 . 2. If ν1 , ν2 are nodes at the same level in the search tree and φ(ν1 ) 6= φ(ν2 ), then no labelled graph descended from ν1 is the same as one descended from ν2 . g
3. If ν1 , ν2 are nodes with ν2 = ν1 for an automorphism g , then g maps the subtree descended from ν1 onto the subtree descended from ν2 .
Each of these observations enables us to prune parts of the search tree. The hope is that the remaining parts will be small enough to compute.
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Variations between programs The competing programs vary from each other in ways that include: 1. Data structures 2. Strength of the refinement procedure 3. Order of traversal of the search tree 4. Means of discovering automorphisms 5. Processing of automorphisms
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Stronger refinement —
nauty “invariants”
Sometimes refinement to equitable partition is insufficient to separate vertices with clearly different combinatorial properties. Those properties can be used to make the refinement stronger.
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Stronger refinement —
nauty “invariants”
Sometimes refinement to equitable partition is insufficient to separate vertices with clearly different combinatorial properties. Those properties can be used to make the refinement stronger. For example, we can count the number of 3-cycles and 4-cycles through each vertex:
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Stronger refinement —
nauty “invariants”
Sometimes refinement to equitable partition is insufficient to separate vertices with clearly different combinatorial properties. Those properties can be used to make the refinement stronger. For example, we can count the number of 3-cycles and 4-cycles through each vertex:
practical graph isomorphism ii — 18
Stronger refinement —
nauty “invariants”
Sometimes refinement to equitable partition is insufficient to separate vertices with clearly different combinatorial properties. Those properties can be used to make the refinement stronger. For example, we can count the number of 3-cycles and 4-cycles through each vertex:
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Example of stronger refinement performance Consider random quartic graphs with 1000 vertices.
• nauty (dense data structures)
5.6 seconds
• nauty (sparse data structures)
0.5 seconds
• nauty (count vertices at distance 2)
0.0014 seconds
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Example of stronger refinement performance Consider random quartic graphs with 1000 vertices.
• nauty (dense data structures)
5.6 seconds
• nauty (sparse data structures)
0.5 seconds
• nauty (count vertices at distance 2)
0.0014 seconds
The big problem with these methods is that the best refinement technique varies from one graph class to another. The user has to know which one to choose, which few do.
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Tree traversal order — a
Traces innovation
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Tree traversal order — a
Traces innovation
practical graph isomorphism ii — 20
Tree traversal order — a
Traces innovation
Classical order: depth-first search
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Tree traversal order — a
Traces innovation
Traces order: breadth-first search
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Tree traversal order — a
Traces innovation
Traces order: breadth-first search The problem with BFS is that automorphisms are discovered at leaves.
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Tree traversal order — a
Traces innovation
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Tree traversal order — a
Traces innovation
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Tree traversal order — a
Traces innovation
Traces: experimental paths Experimental paths allow automorphism detection during breadth-first search.
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Refinement traces – a
Traces innovation
In all programs, for almost all graphs, most of the work involves partition refinement. However, the result of partition refinement is often thrown away because the node invariant of the resulting node is unsuitable.
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Refinement traces – a
Traces innovation
In all programs, for almost all graphs, most of the work involves partition refinement. However, the result of partition refinement is often thrown away because the node invariant of the resulting node is unsuitable. To alleviate this problem, Traces defines the node invariant to be a vector whose components are computed incrementally during the refinement. Then the refinement can often be aborted early. (This extends a technique introduced by bliss.)
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Refinement traces – a
Traces innovation
In all programs, for almost all graphs, most of the work involves partition refinement. However, the result of partition refinement is often thrown away because the node invariant of the resulting node is unsuitable. To alleviate this problem, Traces defines the node invariant to be a vector whose components are computed incrementally during the refinement. Then the refinement can often be aborted early. (This extends a technique introduced by bliss.) Initial
Count pink neighbours
Count green neighbours
Count yellow neighbours
A possible trace (not Traces’ trace) is the number of colours at each step: (2,3,5,6).
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Sparse automorphism detection — a
v
saucy innovation
x
equitable partition (no other blue or green) w
y
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Sparse automorphism detection — a
v
saucy innovation
x
equitable partition (no other blue or green) y
w
individualize v
individualize w
v
x
v
x
w
y
w
y
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Sparse automorphism detection — a
v
saucy innovation
x
equitable partition (no other blue or green) y
w
individualize v
individualize w
v
x
v
x
w
y
w
y
This discovers the automorphism (v w )(x y ) without comparing leaves.
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Automorphism group handling Programs using depth-first search naturally produce automorphisms in the form of a base and strong generating set. Other programs, like Traces, produce less predictable generators. In order to perform automorphism-based pruning of the search tree, we need to efficiently (usually) determine if two sequences of vertices are equivalent under the group generated by the automorphisms found so far.
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Automorphism group handling Programs using depth-first search naturally produce automorphisms in the form of a base and strong generating set. Other programs, like Traces, produce less predictable generators. In order to perform automorphism-based pruning of the search tree, we need to efficiently (usually) determine if two sequences of vertices are equivalent under the group generated by the automorphisms found so far.
Traces uses a technique called the random Schreier method to perform this computation probabiliistically.
nauty also uses this in circumstances where the basic group handling is insufficient.
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Base and strong generating set
5 6
8
2
1
12
7
11
3
10
15
4
16
14 13
9
Fixing 1,5,9,13 : everything fixed Fixing 1,5,9. Orbit of 13 is {13, 15}. (13 15) Fixing 1,5. Orbit of 9 is {9, 11}. (9 11) Fixing 1. Orbit of 5 is {5, 7, 13, 15}. (5 7) (2 4)(5 13)(6 16)(7 15)(8 14)(10 12) Fixing nothing. Orbit of 1 is {1, 3, 5, 7, 9, 11, 13, 15}. (1 3) (1 5)(2 8)(3 7)(4 6)(9 13)(10 16)(11 15)(12 14) Group size = 2 × 2 × 4 × 8 = 128.
This is a base and strong generating set. There are at most n − 1 generators, even though the group size can be up to n! .
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Bliss 7.2
Automorphism group
10−1
Nauty 2.5
Saucy 2.1
Traces 2.0
Traces 2.0
10−2 10−3 10−4 10−5 101
Bliss 7.2
Conauto 2.0
Time (sec) (sec)
Time (sec) (sec)
Nauty 2.5
102 103 Vertices
Canonical label
10−1 10−2 10−3 10−4 10−5 101
Figure 1: Random graphs with p =
practical graph isomorphism ii — 26
102 103 Vertices 1 2
Bliss 7.2 Nauty 2.5
Nauty 2.5
Saucy 2.1
Traces 2.0
Traces 2.0
10−2
10−2 −3
10
10−4
10−3 10−4 10−5
10−5 101
Canonical label
Bliss 7.2
Conauto 2.0
Time (sec)
Time (sec)
10−1
Automorphism group
102 103 Vertices
104
101
102 103 Vertices
Figure 2: Random graphs with average degree
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√
n
104
Bliss 7.2 Nauty 2.5
Automorphism group
Bliss 7.2
Conauto 2.0 Saucy 2.1
102
Traces 2.0
> 600s
102
Traces 2.0
> 600s
101 Time (sec)
101 Time (sec)
Canonical label
Nauty 2.5
100 10−1 10−2
100 10−1 10−2
10−3
10−3
10−4
10−4 102
103 104 Vertices
105
102
Figure 3: Random cubic graphs
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103 104 Vertices
105
Bliss 7.2 Nauty 2.5
Automorphism group
Bliss 7.2
Saucy 2.1
Canonical label
Nauty 2.5
Conauto 2.0
Traces 2.0
> 600s
> 600s
Traces 2.0
101 Time (sec)
Time (sec)
101 10−1 10−3 10−5 101
10−1 10−3 10−5
102
103 104 Vertices
105
101
Figure 4: Random trees
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102
103 104 Vertices
105
Bliss 7.2
Automorphism group
Nauty 2.5
Bliss 7.2
Conauto 2.0
Nauty 2.5
Saucy 2.1
Traces 2.0
Canonical label
Traces 2.0
101
> 400s
Time (sec)
Time (sec)
101 10−1 10−3 10−5
10−1 10−3 10−5
101
102
103 104 Vertices
105
106
101
Figure 5: Hypercubes
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102
103 104 Vertices
105
106
Bliss 7.2 Nauty 2.5
Automorphism group
Nauty 2.5
Saucy 2.1
Traces 2.0
100
Traces 2.0
10−1
Time (sec)
Time (sec)
100
Canonical label
Bliss 7.2
Conauto 2.0
10−2 10−3 10−4
10−1 10−2 10−3 10−4
10−5
10−5 101
102
103 Vertices
104
101
102
103 Vertices
Figure 6: Miscellaneous vertex-transitive graphs
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104
Bliss 7.2
Automorphism group
Nauty 2.5
Nauty 2.5 Traces 2.0
Saucy 2.1 Traces 2.0
10
10−1
Time (sec)
Time (sec)
100
Canonical label
Bliss 7.2
Conauto 2.0
10−2 10−3 10−4
−1
10−2 10−3 10−4 10−5
10−5 101
102 103 Vertices
104
101
102 103 Vertices
Figure 7: 2-dimensional and 3-dimensional grids
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104
Bliss 7.2 Nauty 2.5
10
−1
Automorphism group
Bliss 7.2
Conauto 2.0
10
Saucy 2.1
−1
Canonical label
Nauty 2.5 Traces 2.0
Traces 2.0
10−2 Time (sec)
Time (sec)
10−2 10−3 10−4 10−5
10−3 10−4 10−5
0
200
400 600 Vertices
800
0
Figure 8: Latin square graphs
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200
400 600 Vertices
800
Bliss 7.2 Nauty 2.5
Automorphism group
Bliss 7.2
Conauto 2.0
101
Saucy 2.1
Traces 2.0
101
Traces 2.0
Time (sec)
100 Time (sec)
Canonical label
Nauty 2.5
−1
10
−2
10
−3
10
100 10−1 10−2 10−3
−4
10
10−4
10−5
10−5 10
1
2
10 Vertices
10
3
101
Figure 9: Strongly regular graphs
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102 Vertices
103
Bliss 7.2
Automorphism group
Bliss 7.2
Canonical label
Nauty 2.5
102
Conauto 2.0
Nauty 2.5 Traces 2.0
Traces 2.0
101 Time (sec)
Time (sec)
101 10−1 10−3
100 10−1 10−2 10−3 10−4
10−5
10−5 0
200
400 600 Vertices
800 1,000
101
Figure 10: Hadamard matrix graphs
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102 Vertices
103
Bliss 7.2 Bliss 7.2
Nauty 2.5 Conauto 2.0 Traces 2.0
Nauty 2.5
Automorphism group
> 600s
> 600s
102
102 Time (sec)
Time (sec)
Canonical label
Traces 2.0
101 100
101 100
10−1 10−1 104
105 Group Size
104
105 Group Size
Figure 11: Projective planes of order 16
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Bliss 7.2
Automorphism group
Bliss 7.2
Canonical label
Nauty 2.5 Nauty 2.5
100
Conauto 2.0 Traces 2.0 Traces 2.0
Time (sec)
Time (sec)
10 10−1
−2
10
10−3
1
100 10−1 10−2 10−3
102.4 102.6 102.8 103 Vertices
103.2
102.4 102.6 102.8 103 Vertices
Figure 12: Cai-F¨ urer-Immermann graphs
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103.2
Bliss 7.2 Nauty 2.5
10
2
Automorphism group
Bliss 7.2
Conauto 2.0
10
Saucy 2.1
2
Canonical label
Nauty 2.5 Traces 2.0
Traces 2.0
101
> 100s
Time (sec)
Time (sec)
101 100 10−1 10−2 10−3
> 100s
100 10−1 10−2 10−3
10−4 102
103
102
Vertices
103 Vertices
Figure 13: Miyazaki graphs
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Adolfo Piperno University of Rome
practical graph isomorphism ii — 1
Isomorphism and Automorphism We have some finite objects consisting of some finite sets and some relations on them. An isomorphism between two objects is a bijection between their sets so that the relations are preserved. Hopefully, isomorphism is an equivalence relation on the set of all objects.
practical graph isomorphism ii — 2
Isomorphism and Automorphism We have some finite objects consisting of some finite sets and some relations on them. An isomorphism between two objects is a bijection between their sets so that the relations are preserved. Hopefully, isomorphism is an equivalence relation on the set of all objects. Isomorphisms from an object to itself are automorphisms. If we didn’t do anything silly, the set of automorphisms forms a group under composition. Of course we call it the automorphism group. 8
7
6
5
1
2
3
4
practical graph isomorphism ii — 2
Isomorphism and Automorphism We have some finite objects consisting of some finite sets and some relations on them. An isomorphism between two objects is a bijection between their sets so that the relations are preserved. Hopefully, isomorphism is an equivalence relation on the set of all objects. Isomorphisms from an object to itself are automorphisms. If we didn’t do anything silly, the set of automorphisms forms a group under composition. Of course we call it the automorphism group. 8
7
6
5
(1) (1 4)(2 3)(5 8)(6 7) (1 8)(2 7)(3 6)(4 5) (1 5)(2 6)(3 7)(4 8)
1
2
3
4
practical graph isomorphism ii — 2
Ubiquity of graph isomorphism Very many isomorphism problems can be efficiently expressed as graph isomorphism problems. For convenience, we use coloured graphs, in which the vertices (and possibly the edges) are coloured. Isomorphisms must preserve vertex and edge colour.
1 3 5 4 1 5 4 3 3 4 5 1
Say two matrices are isomorphic if one can be obtained from the other by permuting rows and columns.
practical graph isomorphism ii — 3
Ubiquity of graph isomorphism Very many isomorphism problems can be efficiently expressed as graph isomorphism problems. For convenience, we use coloured graphs, in which the vertices (and possibly the edges) are coloured. Isomorphisms must preserve vertex and edge colour.
1 3 5 4 1 5 4 3 3 4 5 1
Say two matrices are isomorphic if one can be obtained from the other by permuting rows and columns.
=1 =3 =4 =5
practical graph isomorphism ii — 3
Hadamard equivalence
1 −1 1 −1 −1 −1
1 1 1
Say isomorphism of two ±1 matrices allows permutation and negation of rows and columns.
practical graph isomorphism ii — 4
Hadamard equivalence
1 −1 1 −1 −1 −1
1 1 1
Say isomorphism of two ±1 matrices allows permutation and negation of rows and columns.
practical graph isomorphism ii — 4
Isotopy
1 3 2 2 1 3 3 2 1
Say isomorphism of two matrices allows permutation of rows, columns and symbols.
practical graph isomorphism ii — 5
Isotopy
1
Ro 1 3 2 2 1 3 3 2 1
1
Say isomorphism of two matrices allows permutation of rows, columns and symbols.
mn
2
2
lu Co
w
3
3
3
2
Symbol
practical graph isomorphism ii — 5
1
Isotopy
1
Ro 1 3 2 2 1 3 3 2 1
1
Say isomorphism of two matrices allows permutation of rows, columns and symbols.
mn
2
2
lu Co
w
3
3
3
2
Symbol
practical graph isomorphism ii — 5
1
Group isomorphism
e g1 g2 g3
g1 e g3 g2 g g e g 1 2 3
g3 g2 g1 e
The group elements are edge colours and vertices. The black vertex is the identity element. Preserve vertex colours but allow permutation of edge colours.
practical graph isomorphism ii — 6
Group isomorphism
e g1 g2 g3
g1 e g3 g2 g g e g 1 2 3
g3 g2 g1 e
The group elements are edge colours and vertices. The black vertex is the identity element. Preserve vertex colours but allow permutation of edge colours.
practical graph isomorphism ii — 6
Permutation equivalence of linear codes
0 0 1 1 0
1 1 0 1 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 0 1 1 0 1 1 0 1 0 0 0 0 1 0
practical graph isomorphism ii — 7
Permutation equivalence of linear codes
0 0 1 1 0
1 1 0 1 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 0 1 1 0 1 1 0 1 0 0 0 0 1 0
?
Suppose isomorphism of two 0-1 matrices means that their columns can be permutated so that the rows generate the same vector space over GF (2). Nobody knows how to make an equivalent graph problem of size which is polynomial in the size of the generator matrix.
practical graph isomorphism ii — 7
Theoretical complexity The graph isomorphism problem is:
GI: Given two graphs, determine whether they are isomorphic. O(
• The fastest known algorithm for GI is 2
√
(n log n))
time; no polynomial time
algorithm is known (despite many claims).
• GI is not known to be NP-complete. • GI is not known to be in co-NP. • There are polynomial time algorithms for many special classes of graphs (bounded degree, bounded genus, classes with excluded minors or topological subgraphs). Few of these are practical, planar graphs being a notable exception. It is possible that group isomorphism is easier than GI and that permutation equivalence of linear codes is harder than GI. But nobody can prove it.
practical graph isomorphism ii — 8
Canonical labelling Given a class of objects, and a definition of “isomorphism”, we can choose one member of each isomorphism class and call it the canonical member of the class. The process of finding the canonical member of the isomorphism class containing a given object is called “canonical labelling”.
practical graph isomorphism ii — 9
Canonical labelling Given a class of objects, and a definition of “isomorphism”, we can choose one member of each isomorphism class and call it the canonical member of the class. The process of finding the canonical member of the isomorphism class containing a given object is called “canonical labelling”. Two labelled objects which are isomorphic become identical when they are canonically labelled. Since identity of objects is usually far easier to check than isomorphism, canonical labelling is the preferred method of sorting a collection of objects into isomorphism classes.
practical graph isomorphism ii — 9
Canonical labelling Given a class of objects, and a definition of “isomorphism”, we can choose one member of each isomorphism class and call it the canonical member of the class. The process of finding the canonical member of the isomorphism class containing a given object is called “canonical labelling”. Two labelled objects which are isomorphic become identical when they are canonically labelled. Since identity of objects is usually far easier to check than isomorphism, canonical labelling is the preferred method of sorting a collection of objects into isomorphism classes. A possible definition of the canonical member of an isomorphism class would be the member which maximises some linear representation (such as a list of edges). In practice, most programs compute a canonical member which is easy for computers to find rather than easy for humans to define.
practical graph isomorphism ii — 9
Software for graph isomorphism This task was already described as a disease in 1970, and programs still appear regularly. The most successful programs still supported are:
• nauty (McKay, 1976–) canonical label and automorphism group • VF2 (Cordella, Foggia, Sansone and Vento, 1999–) comparison of two graphs • saucy (Darga, Sakallah and Markov, 2004–) automorphism group • bliss (Juntilla and Kaski, 2007–) canonical label and automorphism group • Traces (Piperno, 2008–) canonical label and automorphism group • conauto (L´ opez-Presa, Anta and Chiroque, 2009–) automorphism group and comparison of two graphs Many similar principles appear in these programs. saucy and bliss began life as reimplementations of nauty but have since diverged.
practical graph isomorphism ii — 10
Partition refinement A key concept used by nauty is partition refinement. (“partition” = “colouring”) An equitable partition is one where every two vertices of the same colour are adjacent to the same number of vertices of each colour. Two examples of equitable partitions:
practical graph isomorphism ii — 11
Refinement of a partition means to subdivide its cells. Given a partition (colouring) π , there is a unique equitable partition that is a refinement of π and has the least number of colours.
practical graph isomorphism ii — 12
Refinement of a partition means to subdivide its cells. Given a partition (colouring) π , there is a unique equitable partition that is a refinement of π and has the least number of colours.
Initial
Count green neighbours
Count pink neighbours
Count yellow neighbours
practical graph isomorphism ii — 12
Search tree All competitive isomorphism programs generate a tree based on partition refinement. The nodes of the tree correspond to partitions (colourings). The root of the tree corresponds to the initial colouring, refined. If a node corresponds to a discrete partition (each vertex with a different colour), it has no children and is a leaf. Otherwise we choose a colour used more than once (the target cell), individualize one of those vertices by giving it a new unique colour, and refine to get a child.
practical graph isomorphism ii — 13
Search tree All competitive isomorphism programs generate a tree based on partition refinement. The nodes of the tree correspond to partitions (colourings). The root of the tree corresponds to the initial colouring, refined. If a node corresponds to a discrete partition (each vertex with a different colour), it has no children and is a leaf. Otherwise we choose a colour used more than once (the target cell), individualize one of those vertices by giving it a new unique colour, and refine to get a child. Each leaf lists the vertices in some order (since colours have a predefined order), so it corresponds to a labelling of the graph.
practical graph isomorphism ii — 13
Search tree All competitive isomorphism programs generate a tree based on partition refinement. The nodes of the tree correspond to partitions (colourings). The root of the tree corresponds to the initial colouring, refined. If a node corresponds to a discrete partition (each vertex with a different colour), it has no children and is a leaf. Otherwise we choose a colour used more than once (the target cell), individualize one of those vertices by giving it a new unique colour, and refine to get a child. Each leaf lists the vertices in some order (since colours have a predefined order), so it corresponds to a labelling of the graph. If we have two leaves giving the same labelled graph, we have found an automorphism. If we define an order on labelled graphs, such as lexicographic order, then the greatest labelled graph corresponding to a leaf is a canonical graph.
practical graph isomorphism ii — 13
Search tree All competitive isomorphism programs generate a tree based on partition refinement. The nodes of the tree correspond to partitions (colourings). The root of the tree corresponds to the initial colouring, refined. If a node corresponds to a discrete partition (each vertex with a different colour), it has no children and is a leaf. Otherwise we choose a colour used more than once (the target cell), individualize one of those vertices by giving it a new unique colour, and refine to get a child. Each leaf lists the vertices in some order (since colours have a predefined order), so it corresponds to a labelling of the graph. If we have two leaves giving the same labelled graph, we have found an automorphism. If we define an order on labelled graphs, such as lexicographic order, then the greatest labelled graph corresponding to a leaf is a canonical graph. However the tree can be much too large to generate completely.
practical graph isomorphism ii — 13
1
2
3
4
8
7
6
5
[1458|2367]
[ ... ]
[ ... ]
[ 4 |1 8 | 5 | 2 7 | 6 | 3 ]
[ 1 |4 5 | 8 | 3 6 | 7 | 2 ]
[ ... ] [ ... ]
[1|4|5 |8|6|3|7|2]
[1|5|4 |8|3|6|7|2]
[4|1|8|5|7|2|6|3]
1
8
6
2
1
8
5
3
2
6
8
1
4
7
5
3
4
7
6
2
3
5
7
4
(1)
(3 6)(4 5)
(1 4)(2 3)(5 8)(6 7)
practical graph isomorphism ii — 14
[ ... ] [ ... ]
[ ... ]
Node invariants A node invariant is a value φ(ν ) attached to each node in the tree that depends only on the combinatorial properties of ν and its ancestors in the search tree (not on the labelling of the graph). Moreover, the order of node invariants must be preserved by descendants: Let ν1 , ν2 be nodes at the same level in the search tree. If ν10 is a descendant of ν1 , and ν20 is a descendant of ν2 , then
φ(ν1 ) < φ(ν2 ) =⇒ φ(ν10 ) < φ(ν20 ).
practical graph isomorphism ii — 15
Node invariants A node invariant is a value φ(ν ) attached to each node in the tree that depends only on the combinatorial properties of ν and its ancestors in the search tree (not on the labelling of the graph). Moreover, the order of node invariants must be preserved by descendants: Let ν1 , ν2 be nodes at the same level in the search tree. If ν10 is a descendant of ν1 , and ν20 is a descendant of ν2 , then
φ(ν1 ) < φ(ν2 ) =⇒ φ(ν10 ) < φ(ν20 ). For example,
φ(ν ) = c (ν 0 ), c (ν 00 ), . . . , c (ν )
is a node invariant with lexicographic ordering, where ν 0 , ν 00 , . . . , ν is the path from the root of the tree to ν , and c ( ) is the number of colours.
practical graph isomorphism ii — 15
Node invariants A node invariant is a value φ(ν ) attached to each node in the tree that depends only on the combinatorial properties of ν and its ancestors in the search tree (not on the labelling of the graph). Moreover, the order of node invariants must be preserved by descendants: Let ν1 , ν2 be nodes at the same level in the search tree. If ν10 is a descendant of ν1 , and ν20 is a descendant of ν2 , then
φ(ν1 ) < φ(ν2 ) =⇒ φ(ν10 ) < φ(ν20 ). For example,
φ(ν ) = c (ν 0 ), c (ν 00 ), . . . , c (ν )
is a node invariant with lexicographic ordering, where ν 0 , ν 00 , . . . , ν is the path from the root of the tree to ν , and c ( ) is the number of colours. Let φ∗ be the greatest node invariant of a leaf. Then the lexicographically greatest labelled graph corresponding to a leaf ν with φ(ν ) = φ∗ is a canonical graph.
practical graph isomorphism ii — 15
Pruning operations Node invariants, together with automorphisms, allow us to remove parts of the search tree without generating them. 1. If ν1 , ν2 are nodes at the same level in the search tree and φ(ν1 ) < φ(ν2 ), then no canonical labelling is descended from ν1 . 2. If ν1 , ν2 are nodes at the same level in the search tree and φ(ν1 ) 6= φ(ν2 ), then no labelled graph descended from ν1 is the same as one descended from ν2 . g
3. If ν1 , ν2 are nodes with ν2 = ν1 for an automorphism g , then g maps the subtree descended from ν1 onto the subtree descended from ν2 .
practical graph isomorphism ii — 16
Pruning operations Node invariants, together with automorphisms, allow us to remove parts of the search tree without generating them. 1. If ν1 , ν2 are nodes at the same level in the search tree and φ(ν1 ) < φ(ν2 ), then no canonical labelling is descended from ν1 . 2. If ν1 , ν2 are nodes at the same level in the search tree and φ(ν1 ) 6= φ(ν2 ), then no labelled graph descended from ν1 is the same as one descended from ν2 . g
3. If ν1 , ν2 are nodes with ν2 = ν1 for an automorphism g , then g maps the subtree descended from ν1 onto the subtree descended from ν2 .
Each of these observations enables us to prune parts of the search tree. The hope is that the remaining parts will be small enough to compute.
practical graph isomorphism ii — 16
Variations between programs The competing programs vary from each other in ways that include: 1. Data structures 2. Strength of the refinement procedure 3. Order of traversal of the search tree 4. Means of discovering automorphisms 5. Processing of automorphisms
practical graph isomorphism ii — 17
Stronger refinement —
nauty “invariants”
Sometimes refinement to equitable partition is insufficient to separate vertices with clearly different combinatorial properties. Those properties can be used to make the refinement stronger.
practical graph isomorphism ii — 18
Stronger refinement —
nauty “invariants”
Sometimes refinement to equitable partition is insufficient to separate vertices with clearly different combinatorial properties. Those properties can be used to make the refinement stronger. For example, we can count the number of 3-cycles and 4-cycles through each vertex:
practical graph isomorphism ii — 18
Stronger refinement —
nauty “invariants”
Sometimes refinement to equitable partition is insufficient to separate vertices with clearly different combinatorial properties. Those properties can be used to make the refinement stronger. For example, we can count the number of 3-cycles and 4-cycles through each vertex:
practical graph isomorphism ii — 18
Stronger refinement —
nauty “invariants”
Sometimes refinement to equitable partition is insufficient to separate vertices with clearly different combinatorial properties. Those properties can be used to make the refinement stronger. For example, we can count the number of 3-cycles and 4-cycles through each vertex:
practical graph isomorphism ii — 18
Example of stronger refinement performance Consider random quartic graphs with 1000 vertices.
• nauty (dense data structures)
5.6 seconds
• nauty (sparse data structures)
0.5 seconds
• nauty (count vertices at distance 2)
0.0014 seconds
practical graph isomorphism ii — 19
Example of stronger refinement performance Consider random quartic graphs with 1000 vertices.
• nauty (dense data structures)
5.6 seconds
• nauty (sparse data structures)
0.5 seconds
• nauty (count vertices at distance 2)
0.0014 seconds
The big problem with these methods is that the best refinement technique varies from one graph class to another. The user has to know which one to choose, which few do.
practical graph isomorphism ii — 19
Tree traversal order — a
Traces innovation
practical graph isomorphism ii — 20
Tree traversal order — a
Traces innovation
practical graph isomorphism ii — 20
Tree traversal order — a
Traces innovation
Classical order: depth-first search
practical graph isomorphism ii — 20
Tree traversal order — a
Traces innovation
Traces order: breadth-first search
practical graph isomorphism ii — 20
Tree traversal order — a
Traces innovation
Traces order: breadth-first search The problem with BFS is that automorphisms are discovered at leaves.
practical graph isomorphism ii — 20
Tree traversal order — a
Traces innovation
practical graph isomorphism ii — 21
Tree traversal order — a
Traces innovation
practical graph isomorphism ii — 21
Tree traversal order — a
Traces innovation
Traces: experimental paths Experimental paths allow automorphism detection during breadth-first search.
practical graph isomorphism ii — 21
Refinement traces – a
Traces innovation
In all programs, for almost all graphs, most of the work involves partition refinement. However, the result of partition refinement is often thrown away because the node invariant of the resulting node is unsuitable.
practical graph isomorphism ii — 22
Refinement traces – a
Traces innovation
In all programs, for almost all graphs, most of the work involves partition refinement. However, the result of partition refinement is often thrown away because the node invariant of the resulting node is unsuitable. To alleviate this problem, Traces defines the node invariant to be a vector whose components are computed incrementally during the refinement. Then the refinement can often be aborted early. (This extends a technique introduced by bliss.)
practical graph isomorphism ii — 22
Refinement traces – a
Traces innovation
In all programs, for almost all graphs, most of the work involves partition refinement. However, the result of partition refinement is often thrown away because the node invariant of the resulting node is unsuitable. To alleviate this problem, Traces defines the node invariant to be a vector whose components are computed incrementally during the refinement. Then the refinement can often be aborted early. (This extends a technique introduced by bliss.) Initial
Count pink neighbours
Count green neighbours
Count yellow neighbours
A possible trace (not Traces’ trace) is the number of colours at each step: (2,3,5,6).
practical graph isomorphism ii — 22
Sparse automorphism detection — a
v
saucy innovation
x
equitable partition (no other blue or green) w
y
practical graph isomorphism ii — 23
Sparse automorphism detection — a
v
saucy innovation
x
equitable partition (no other blue or green) y
w
individualize v
individualize w
v
x
v
x
w
y
w
y
practical graph isomorphism ii — 23
Sparse automorphism detection — a
v
saucy innovation
x
equitable partition (no other blue or green) y
w
individualize v
individualize w
v
x
v
x
w
y
w
y
This discovers the automorphism (v w )(x y ) without comparing leaves.
practical graph isomorphism ii — 23
Automorphism group handling Programs using depth-first search naturally produce automorphisms in the form of a base and strong generating set. Other programs, like Traces, produce less predictable generators. In order to perform automorphism-based pruning of the search tree, we need to efficiently (usually) determine if two sequences of vertices are equivalent under the group generated by the automorphisms found so far.
practical graph isomorphism ii — 24
Automorphism group handling Programs using depth-first search naturally produce automorphisms in the form of a base and strong generating set. Other programs, like Traces, produce less predictable generators. In order to perform automorphism-based pruning of the search tree, we need to efficiently (usually) determine if two sequences of vertices are equivalent under the group generated by the automorphisms found so far.
Traces uses a technique called the random Schreier method to perform this computation probabiliistically.
nauty also uses this in circumstances where the basic group handling is insufficient.
practical graph isomorphism ii — 24
Base and strong generating set
5 6
8
2
1
12
7
11
3
10
15
4
16
14 13
9
Fixing 1,5,9,13 : everything fixed Fixing 1,5,9. Orbit of 13 is {13, 15}. (13 15) Fixing 1,5. Orbit of 9 is {9, 11}. (9 11) Fixing 1. Orbit of 5 is {5, 7, 13, 15}. (5 7) (2 4)(5 13)(6 16)(7 15)(8 14)(10 12) Fixing nothing. Orbit of 1 is {1, 3, 5, 7, 9, 11, 13, 15}. (1 3) (1 5)(2 8)(3 7)(4 6)(9 13)(10 16)(11 15)(12 14) Group size = 2 × 2 × 4 × 8 = 128.
This is a base and strong generating set. There are at most n − 1 generators, even though the group size can be up to n! .
practical graph isomorphism ii — 25
Bliss 7.2
Automorphism group
10−1
Nauty 2.5
Saucy 2.1
Traces 2.0
Traces 2.0
10−2 10−3 10−4 10−5 101
Bliss 7.2
Conauto 2.0
Time (sec) (sec)
Time (sec) (sec)
Nauty 2.5
102 103 Vertices
Canonical label
10−1 10−2 10−3 10−4 10−5 101
Figure 1: Random graphs with p =
practical graph isomorphism ii — 26
102 103 Vertices 1 2
Bliss 7.2 Nauty 2.5
Nauty 2.5
Saucy 2.1
Traces 2.0
Traces 2.0
10−2
10−2 −3
10
10−4
10−3 10−4 10−5
10−5 101
Canonical label
Bliss 7.2
Conauto 2.0
Time (sec)
Time (sec)
10−1
Automorphism group
102 103 Vertices
104
101
102 103 Vertices
Figure 2: Random graphs with average degree
practical graph isomorphism ii — 27
√
n
104
Bliss 7.2 Nauty 2.5
Automorphism group
Bliss 7.2
Conauto 2.0 Saucy 2.1
102
Traces 2.0
> 600s
102
Traces 2.0
> 600s
101 Time (sec)
101 Time (sec)
Canonical label
Nauty 2.5
100 10−1 10−2
100 10−1 10−2
10−3
10−3
10−4
10−4 102
103 104 Vertices
105
102
Figure 3: Random cubic graphs
practical graph isomorphism ii — 28
103 104 Vertices
105
Bliss 7.2 Nauty 2.5
Automorphism group
Bliss 7.2
Saucy 2.1
Canonical label
Nauty 2.5
Conauto 2.0
Traces 2.0
> 600s
> 600s
Traces 2.0
101 Time (sec)
Time (sec)
101 10−1 10−3 10−5 101
10−1 10−3 10−5
102
103 104 Vertices
105
101
Figure 4: Random trees
practical graph isomorphism ii — 29
102
103 104 Vertices
105
Bliss 7.2
Automorphism group
Nauty 2.5
Bliss 7.2
Conauto 2.0
Nauty 2.5
Saucy 2.1
Traces 2.0
Canonical label
Traces 2.0
101
> 400s
Time (sec)
Time (sec)
101 10−1 10−3 10−5
10−1 10−3 10−5
101
102
103 104 Vertices
105
106
101
Figure 5: Hypercubes
practical graph isomorphism ii — 30
102
103 104 Vertices
105
106
Bliss 7.2 Nauty 2.5
Automorphism group
Nauty 2.5
Saucy 2.1
Traces 2.0
100
Traces 2.0
10−1
Time (sec)
Time (sec)
100
Canonical label
Bliss 7.2
Conauto 2.0
10−2 10−3 10−4
10−1 10−2 10−3 10−4
10−5
10−5 101
102
103 Vertices
104
101
102
103 Vertices
Figure 6: Miscellaneous vertex-transitive graphs
practical graph isomorphism ii — 31
104
Bliss 7.2
Automorphism group
Nauty 2.5
Nauty 2.5 Traces 2.0
Saucy 2.1 Traces 2.0
10
10−1
Time (sec)
Time (sec)
100
Canonical label
Bliss 7.2
Conauto 2.0
10−2 10−3 10−4
−1
10−2 10−3 10−4 10−5
10−5 101
102 103 Vertices
104
101
102 103 Vertices
Figure 7: 2-dimensional and 3-dimensional grids
practical graph isomorphism ii — 32
104
Bliss 7.2 Nauty 2.5
10
−1
Automorphism group
Bliss 7.2
Conauto 2.0
10
Saucy 2.1
−1
Canonical label
Nauty 2.5 Traces 2.0
Traces 2.0
10−2 Time (sec)
Time (sec)
10−2 10−3 10−4 10−5
10−3 10−4 10−5
0
200
400 600 Vertices
800
0
Figure 8: Latin square graphs
practical graph isomorphism ii — 33
200
400 600 Vertices
800
Bliss 7.2 Nauty 2.5
Automorphism group
Bliss 7.2
Conauto 2.0
101
Saucy 2.1
Traces 2.0
101
Traces 2.0
Time (sec)
100 Time (sec)
Canonical label
Nauty 2.5
−1
10
−2
10
−3
10
100 10−1 10−2 10−3
−4
10
10−4
10−5
10−5 10
1
2
10 Vertices
10
3
101
Figure 9: Strongly regular graphs
practical graph isomorphism ii — 34
102 Vertices
103
Bliss 7.2
Automorphism group
Bliss 7.2
Canonical label
Nauty 2.5
102
Conauto 2.0
Nauty 2.5 Traces 2.0
Traces 2.0
101 Time (sec)
Time (sec)
101 10−1 10−3
100 10−1 10−2 10−3 10−4
10−5
10−5 0
200
400 600 Vertices
800 1,000
101
Figure 10: Hadamard matrix graphs
practical graph isomorphism ii — 35
102 Vertices
103
Bliss 7.2 Bliss 7.2
Nauty 2.5 Conauto 2.0 Traces 2.0
Nauty 2.5
Automorphism group
> 600s
> 600s
102
102 Time (sec)
Time (sec)
Canonical label
Traces 2.0
101 100
101 100
10−1 10−1 104
105 Group Size
104
105 Group Size
Figure 11: Projective planes of order 16
practical graph isomorphism ii — 36
Bliss 7.2
Automorphism group
Bliss 7.2
Canonical label
Nauty 2.5 Nauty 2.5
100
Conauto 2.0 Traces 2.0 Traces 2.0
Time (sec)
Time (sec)
10 10−1
−2
10
10−3
1
100 10−1 10−2 10−3
102.4 102.6 102.8 103 Vertices
103.2
102.4 102.6 102.8 103 Vertices
Figure 12: Cai-F¨ urer-Immermann graphs
practical graph isomorphism ii — 37
103.2
Bliss 7.2 Nauty 2.5
10
2
Automorphism group
Bliss 7.2
Conauto 2.0
10
Saucy 2.1
2
Canonical label
Nauty 2.5 Traces 2.0
Traces 2.0
101
> 100s
Time (sec)
Time (sec)
101 100 10−1 10−2 10−3
> 100s
100 10−1 10−2 10−3
10−4 102
103
102
Vertices
103 Vertices
Figure 13: Miyazaki graphs
practical graph isomorphism ii — 38
