Accepted Manuscript

Fast Computation of Bipartite Graph Matching

Francesc Serratosa

PII: S0167-8655(14)00133-0DOI: http://dx.doi.org/10.1016/j.patrec.2014.04.015Reference: PATREC 6004

To appear in: Pattern Recognition Letters

Received Date: 13 December 2013

Please cite this article as: Serratosa, F., Fast Computation of Bipartite Graph Matching, Pattern RecognitionLetters (2014), doi: http://dx.doi.org/10.1016/j.patrec.2014.04.015

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Fast Computation of Bipartite Graph

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Francesc Serratosa

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Pattern Recognition Letters j ou r n a l h om ep a ge : w w w .e ls e v ie r .c om

Fast Computation of Bipartite Graph Matching

Francesc Serratosaa

a Universitat Rovira i Virgili Tarragona, Catalonia, Spain.

francesc.serratosa@urv.cat

1. Introduction

Attributed Graphs have been of crucial importance in pattern recognition throughout more than 3 decades [1]. Graphs have been used to model several kinds of problems in some pattern recognition fields such as object recognition [2] and [3], scene view alignment [4], [5], [6], [7], multiple object alignment [8], [9], object characterization [10], [11] among a great amount of other applications. Interesting reviews of techniques and applications are [12], [13] and [14]. If elements in pattern recognition are modelled through attributed graphs, error-tolerant graph-matching algorithms are needed that aim to compute a matching between nodes of two attributed graphs that minimizes some kind of objective function. Unfortunately, the time and space complexity to compute the minimum of these objective functions is very high. For this reason, a lot of research has been done on trying to reduce as much as possible the run time of the graph-matching algorithms through sub-optimal techniques. Since its presentation, Bipartite algorithm [15] has been considered one of the best graph-matching algorithms due to it obtains a sub-optimal distance value almost near to the optimal

one but with a considerable decrease on the run time.

This paper presents a variant of the Bipartite algorithm [15] that obtains exactly the same distance value but with a reduced run time. Experimental validation shows a Speed up of 5 on well-known databases and higher Speed up on synthetically generated graphs. In fact, higher is the order of the graphs, higher is also the Speed up of our algorithm. This property is interesting since in the next years, we will see a need on representing the objects (social nets, scenes, proteins…) on larger structures.

The outline of the paper is as follows, in the next section, we define the attributed graphs and the graph-edit distance. On section 3, we explain how to compute the graph edit distance and we concretize on the Bipartite algorithm. Finally, on section 4,

we present our new method and we schematically show our algorithm. On section 5, we show the experimental validation and we finish the article with some conclusions.

2. Attributed Graphs and Graph Edit Distance

In this section, we first define the Attributed Graphs, Cliques

and Graph matching and then we explain the Graph Edit Distance.

2.1. Definitions

Attributed Graph and Cliques: Let ��and ��denote the domains of possible values for attributed vertices and arcs, respectively. An attributed graph (over �� and ��) is defined by a tuple � � ���, ��, �, ��, where �� � �� | � �,… , �� is the set of vertices (or nodes), �� � ��� �| , � ∈ �,… , �� is the set of arcs (or edges), �: �� → �� assigns attribute values to vertices and �: �� → �� assigns attribute values to arcs. The order of graph � is �.

We define a clique � on an attributed graph � as a local structure composed of a node and its outgoing edges � ��� , ��� �|� ∈ �,… , ��, �, ��. Error correcting graph isomorphism between graphs: Let

G� � �Σ��, Σ��, γ��, γ��� and G � �Σ� , Σ� , γ� , γ� � be two attributed

graphs of initial order n and m. To allow maximum flexibility in

the matching process, graphs are extended with null nodes [16] to

be of order n# m. We will refer to null nodes of G� and G by

Σ$�� ⊆ Σ�� and Σ$� ⊆ Σ� respectively. We assume null nodes have

indices & ∈ '( # 1,… , ( #*� and + ∈ '* # 1,… , ( #*� for

graphs G� and G , respectively. Let T be a set of all possible

bijections between two vertex sets �� and � . Bijection

f �, : Σ�� → Σ� , assigns one vertex of G� to only one vertex of G .

ARTIC LE INFO ABSTRAC T

Article history:

Received

Received in revised form

Accepted

Available online

We present a new algorithm to compute the Graph Edit Distance in a sub-optimal way. We

demonstrate that the distance value is exactly the same than the one obtained by the algorithm

called Bipartite but with a reduced run time. The only restriction we impose is that the edit costs

have to be defined such that the Graph Edit Distance can be really defined as a distance function,

that is, the cost of insertion plus deletion of nodes (or arcs) have to be lower or equal than the

cost of substitution of nodes (or arcs). Empirical validation shows that higher is the order of the

graphs, higher is the obtained Speed up.

2012 Elsevier Ltd. All rights reserved.

2 The bijection between arcs, denoted by f��, , is defined accordingly to the bijection of their terminal nodes. In other

words:

./0,123450 6 � 3781 ⇒ .0,1�:40� � :71 ∧ .0,12:506 � :81 (1)

:40, :50 ∈ Σ�� − Σ$�� and :71 , :81 ∈ Σ� − Σ$�

We define the non-existent or null edges by Σ$�� ⊆ Σ��and

Σ$� ⊆ Σ� .

2.2. Graph Edit Distance between two graphs

Graph Edit Distance [17, 18, 19] is the most used method to

solve the error-tolerant graph matching. It is based on defining a

distance between attributed graphs through the minimum modifications that are required to transform one attributed graph

into the other. To do so, it is needed to define these

modifications, which are called edit operations. Basically, six

different edit operations have been defined: insertion, deletion and substitution of both nodes and edges. In this way, for every

pair of attributed graphs (=0 and =1), there is an edit path

3>+?@&?ℎ�=0, =1� � �BC, … , BD� (where each B7 denotes an edit operation) that transforms one graph into the other. Given two

graphs, there is a set of edit paths, which we name E, that each of them transforms one of the graphs into the other. Figure 1 shows

the edit path that transforms =0 into =1. It is composed of the following 5 edit operations: Delete edge, Delete node, Insert

node, Insert edge and Substitute Node. The substitution operation is needed since the attributes of both nodes are different.

Edit cost functions have been introduced to quantitatively evaluate which edit path is the best. The aim of these functions is

to assign a penalty cost to each edit operation according to the amount of modification that it introduces in the transformation

sequence.

Figure 1. One of the edit paths that transforms =0 into =1.

Given two attributed graphs to be compared, we can define a

graph bijection .0,1 ∈ F between them and also we can relate this

bijection with the edit path, 3>+?@&?ℎ�=0, =1� ∈ E. To do so, we

can assume the edit operation called Substitution simply

represents node-to-node assignations. Moreover, the edit operations Deletion and Insertion are transformed to mappings of

non-null nodes of the first or second graph to null nodes of the

second or first graph. Using this transformation, given two

graphs, =0 and =1, and a bijection between their nodes, .0,1, the graph edit cost is given by (Definition 7 of [20]):

G>+?HIJ?�=0,=1, .0,1� � K HLM2:40, :716LNO∈PQORSTQOLUV∈SQVRSTQV

#

∑ HLX2:40, :716LNO∈PQORSTQOLUV∈STQV

#∑ HL72:40, :716LNO∈STQOLUV∈SQVRSTQV

# (2)

K H/M23450 , 3781 6/NYO ∈PZORSTZO/U[V∈SZVRSTZV

# K H/X23450 , 3781 6/NYO ∈PZORSTZO/U[V∈STZV

# K H/723450 , 3781 6/NYO ∈STZO

/U[V∈SZVRSTZV

.0,1�:40� � :71 and ./0,123470 6 � 3781

where the edit costs are: HLM: Cost of substituting node :40 of =0

for node .0,1�:40� of =1. HLX: Cost of deleting node :40 of =0.

HL7: Cost of inserting node :71 of =1. And for edges, H/M: Cost of

substituting edge 3450 of graph =0 for edge ./0,123450 6 of =1. H/X:

Cost of assigning edge 3450 of =0 to a non-existing edge of =1.

H/7: Cost of assigning edge 3451 of =1 to a non-existing edge of

=0. The cost of mapping two null nodes or two null arcs is always defined to be zero. For this reason, we have not

considered this case in the equation. Figure 2 shows the obtained

labelling given the edit path presented in figure 1. The cost of

this labelling is: G>+?HIJ?�=0, =1 , .0,1� � H/X #HLX # HL7 #H/7 # HLM.

Figure 2. Labelling .0,1 between =0 and =1 given the edit path of fig. 2.

Finally, the Graph Edit Distance is defined as the minimum cost

under any bijection in F:

G>+?\+J?&(]3�=0, =1� � min_O,V∈` G>+?HIJ?�=0, =1 , .0,1� (3)

Using this definition, the Graph Edit Distance directly

depends on parameters or functions HLM, HLX, HL7 , H/M, H/X and

H/7. Several definitions of these functions exist, if we focus first

on the definition of functionsHLM and H/M, the most common approaches are the following. The first and simplest approach

considers HLM�:40, :71� � aLM if >+J?2bL0�:40�, bL1�:71�6 >Fℎd3JℎIe> otherwise HLM � 0, >+J?�·� is defined as a distance

function over the domain of the attributes. Specific examples of

this cost can be found in fingerprint verification [21] whereHLM ∈�0,1� or in [20, 22]. The second and most frequently used

approach corresponds to the case whereHLM2:40, :51 , hL6 ∈ ℝ. In

this case, node substitution cost depends on the attributes of the

nodes and possibly on some other parameters hL as shown in [23,

24] and [4], among others. Similar approaches can be used to

defineH/M. With regard to HLX , HL7, H/X andH/7, these functions usually simply assign a constant cost. However, they can also

depend on node or edge attributes [25, 26] and [27].

We say the optimal bijection, .0,1∗, is the one that obtains the

minimum cost,

.0,1∗ � argmin_O,V∈` G>+?HIJ?�=0, =1 , .0,1� (4)

We define the distance and the optimal bijection between two

cliques in a similar way as the distance between two graphs since they are local structures of graphs. We name the cost of

substituting clique Ko� by Kp

as q4,7. The cost of deleting clique

Ko� as q4,r and the cost of inserting clique Kp

as qr,7.

3 Independently of the definition of the edit costs, if we wish the Edit Distance to be defined as a distance function, it is needed to

assure the following four properties [28]:

1) HLM ≤ HLX #HL7 and H/M ≤ H/X # H/7. 2) HLX � HL7 and H/X � H/7. (5)

3) HLM � 0 and H/M � 0 if same attributes.

4) All costs have to be non-negative.

3. Edit Distance Computation

In this section, we first comment the algorithms presented to

compute an optimal and sub-optimal distance value and then we

explain more deeply the Bipartite algorithm.

3.1. Optimal and Suboptimal Edit Distance Algorithms

The A* algorithm [29] is the most widely used method to

compute the exact value of the graph edit distance. It is a classical tree search algorithm that is based on exploring the

space of all possible mappings of nodes and edges of both

attributed graphs. The computational cost of this algorithm is

exponential in the number of nodes in its worst case. To alleviate this problem, some heuristic functions have been published to

reduce the space search. At the beginnings of the 80s decade, the firsts sub-optimal algorithms to solve the error-tolerant graph-

matching problem appeared [17, 30]. Nowadays, other techniques and surveys have been published [12, 13, 31, 32] and

[18]. Recently, some sub-optimal methods to compute the edit

distance have been presented. The main idea is to optimise local criteria instead of global [33, 34]. New methods have been

presented such as the ones based on Dominant Sets [35] or the Hausdorff distance [24]. And some other methods have been

proposed for specific graph definitions such as planar graphs [37,

38], bounded-valence graphs [39], unique vertex labels [40] or image registration [41, 42].

Bipartite [15] is one of the newest methods presented to solve the Edit Distance. Experimental validation shows that, nowadays,

it is one of the best sub-optimal algorithms since it obtains a good approximation of the distance value in cubic computational cost.

The method we present obtains exactly the same distance value but with a reduced computational time.

3.2. Edit Distance Computation by Bipartite algorithm (BP)

The assignment problem considers the task of finding an

optimal assignment of the elements of a set t to the elements of

another set u, where both sets have the same cardinality ( �|t| � |u|. Let us assume there is a (v( cost matrix H. The

matrix elements H7,8 correspond to the cost of assigning the i-th

element of t to the j-th element of u. An optimal assignment is the one that minimises the sum of the assignment costs and so,

the assignment problem can be stated as finding the

permutationp that minimises ∑ Cp,��p�ypzC . Munkres’ algorithm

[43] solves the assignation problem in O�(|�, in the worst case. It is a refinement of an earlier version by Kuhn [44] and is also

referred to as Kuhn-Munkres or Hungarian algorithm. The

algorithm repeatedly finds the maximum number of independent

optimal assignments and in the worst case the maximum number

of operations needed by the algorithm is O�(|�. Due to this local exploration is performed through columns (or rows), in cases the

cost matrix is not symmetric, the real run time drastically

depends on the order of exploration (columns or rows). Later, an algorithm to solve this problem applied to non-square matrices

where presented [45].

Bipartite, or BP for short [15], is an efficient algorithm for edit distance computation for general graphs that

use the Munkres’ algorithm. That is, they generalised the original

Munkres’ algorithm that solve the assignment problem to the

computation of the graph edit distance by defining a specific cost matrix. In experiments on artificial and real-world data, authors

demonstrate BP obtains an important speed-up of the

computation respect other methods while at the same time the

accuracy of the approximated distances is not much affected [15]. For this reason, since its publication, it has become one of the

most used graph-matching algorithms.

Given attributed graphs G� and G , the �n #m�}�n# m� cost matrix C is defined as follows,

Where q4,7 denotes the cost of substituting clique Ko� by Kp ,

q4,r denotes the cost of deleting clique Ko� and qr,7 denotes the

cost of inserting clique Kp . On the basis of this cost matrix

definition, Munkres’ algorithm can be executed to find the minimum cost for all permutations. Obviously, as described in

[15], this minimum cost is a sub-optimal Edit Distance value

between the involved graphs since cost matrix rows are related to

cliques of graph G� and columns are related to cliques of G . Moreover, it is considered a correct permutation the one that

∑ Co,��o�y~�ozC < ∞. That is, all costs are assigned to non-infinitive

values.

Munkres’ algorithm has been demonstrated to be optimal for solving the assignation problem and it has a polynomial

computational cost: O��( #*�|�. Nevertheless, it is important to note that it is suboptimal for solving the graph edit distance. This

is because the second order information is not completely considered since cliques are evaluated individually. For this

reason, the computed distance values are equal or smaller than the distance values obtained in an optimal method, which have to

be computed in an exponential cost method. Finally, in [46], a preliminary version of the BP algorithm

was presented that had a computational cost of O�*&}�(,*�|�. It had the advantage that the cost matrix had not to be extended

but the deletion and insertion costs were not considered. The variant of the BP we present has the same computational cost

than [47] but obtains the same distance value than [15].

4. Fast Bipartite (FBP)

In this section, after a short introduction, we present a new definition of the Edit Distance and then we depict the algorithm

to compute the graph matching.

4.1. Introduction

In this paper, we propose a new method that obtains exactly

the same distance value and mapping between nodes than the BP

but with a reduced computational time. We refer to it as Fast

4 Bipartite or FBP for short. We also used the Munkres’ method but we propose to use a different and smaller matrix cost in the

cases that the costs are defined such that the edit distance is really

a distance function (equation 5). The computational cost of our

method is O��max�(,*��|�. Our method needs the edit distance to properly be defined as

a distance function, which means that the four distance properties

described on equation (5) have to hold. In this case we can

demonstrate the following Lemma:

Lemma 1. Suppose we have two graphs G� and G of order

( #*. These graphs originally had ( and * nodes but have been

enlarged with * and ( null nodes, respectively. If equation (5)

holds then:

1) If ( ≥ * then the optimal bijection .0,1∗ does not

involve any node deletion of ��.

2) If ( ≤ * then the optimal bijection .0,1∗ does not

involve any node insertion of Σ� .

The aim of Lemma 1 is to show that the whole nodes of the graph

with initial lower or equal order are substituted if the Edit Distance is defined as a distance function. Therefore, it is not

possible to have an insertion operation and deletion operation at

the same time in an optimal bijection.

Proof. If C�� ≤ C�� # C�p then the optimal labelling never has

these two operations together:C��2vo�, vp 6 and C�p2v��, v� 6 where

the initial nodes are vo� ∈ Σ�� − Σ$�� and v� ∈ Σ� − Σ$� and the

extended nodes (null nodes) are v�� ∈ Σ$�� and vp ∈ Σ$� . This is

because, for sure the labelling C��2vo�, v� 6 has a lower value and

the cost of substituting the two null nodes v�� and vp is zero. The

same holds with the edges case∎

4.2. New Edit Distance definition

We define the following edit cost,

G>+?HIJ?′�=0, =1, .0,1� � G>+?HIJ?�=0, =1 , .0,1� −HL/_X7�=0, =1� (6)

Where,

HL/_X7�=0, =1� � K HLX2:40, :�16LNO∈PQO

# K HL72:�0, :716LUV∈SQV

#

∑ H/X23450 , 3�16/NYO ∈PZO #∑ H/723�0 , 3781 6/U[

V∈SZV (7)

:�1 ∈ Σ$L1, :�0 ∈ Σ$L0, 3�1 ∈ Σ$/1 and 3�0 ∈ Σ$/0

The new edit cost definition subtracts from the original edit

cost the costs of deleting nodes and arcs from G� and inserting

nodes and arcs from G . Nodes v� and v�� (or arcs e� and e�� )

represent any extended node (or arc). Note that in cases that nodes or arcs where the extended ones (null nodes o arcs), that is,

vo� ∈ Σ$��, vp ∈ Σ$� , eo�� ∈ Σ$�� or ep� ∈ Σ$� , then their corresponding

cost is zero by definition. Moreover, the subtracted cost C��_�p does not depend on any bijection f �, . In a similar way than the original edit cost, the optimal bijection,

.′0,1∗, is the one that obtains the minimum cost,

.′0,1∗ � argmin_O,V∈` G>+?HIJ?′�=0, =1 , .0,1� (8)

Theorem 1: The equality .′0,1∗ � .0,1∗ holds for all pair of

graphs =0 and =1.

Proof. Cost C��_�p�G�, G � is independent of any bijection f �,

therefore it is constant throughout all the exploration space and

does not affect the point where the function minimises ∎

Theorem 1 shows as that minimisingG>+?HIJ?′ is similar than

minimising G>+?HIJ?. In the next section, we describe an

algorithm to minimise G>+?HIJ?′ with lower computational time

than the Bipartite algorithm.

4.3. Edit Distance Computation by Fast Bipartite algorithm

We define D as an n #m vector. The first n positions are filled

with the costs of deleting cliques Ko� that we named H4,r , and the

other * positions are filled with zeros: D � �HC,r ,…H�,r , 0… 0� . Moreover, we define I as an m# n vector. The first * positions

are filled with the costs of inserting cliques Kp that we named

Hr,7, and the other ( positions are filled with zeros: I ��C�,C, …C�,� , 0… 0�. Note that zeros in both vectors represent the

cost of deleting or inserting null cliques. Besides, it is easy to

demonstrate that C��_�p�G�,G � � ⟨1�, �Io # Do�⟩. With these two vectors, we define two �n #m�}�m# n� matrices, \T and ��. The first one is obtained by the replication of

vector D through columns and the second one is obtained by the

replication of vector I through rows.

We are ready to define our cost matrix C′ as follows, C′=C-

2\T # ��6,

H � �

������������CC,C − 2CC,� # C�,C6 …

⋮

… CC,� − 2CC,� # C�,�6 ⋮

⋮ Cy,C − 2Cy,� # C�,C6 …

⋮ … Cy,� − 2Cy,� # C�,�6

0 ∞ …∞ ⋱ ⋮ 0

∞

∞

0 ⋮ ⋱ ∞… ∞ 0

0 ∞ …∞ ⋱ ⋮ 0

∞

∞

0 ⋮ ⋱ ∞… ∞ 0

0

 ¡¡¡¡¡¡¡¡¡¡¢

Similarly to cost matrix H, this matrix is composed of four

quadrants. The dimensions of each quadrant is: HQ1� � nXm,

HQ2� � nXn,HQ3� � mXm andHQ4� � mXn. Cells in the first one are filled with the value of substituting the cliques (as in

H) but the cost of deleting and inserting the respective cliques is subtracted. The second and third quadrants are composed of

infinitive values except at the diagonal that is filled with zeros. All cells on the fourth quadrant have a zero.

On the basis of the new cost matrix C′ defined above, Munkres’

algorithm [43] can be executed and it finds the optimal

permutation p that minimises ∑ C′o,��o�y~�ozC . Note that any correct

5 permutation on p is equivalent to a bijection f �, between nodes

of graphs G� and G .

Lemma 2. In the case that ( ≤ * (or * ≤ ( ), all correct

permutations hold that there are a minimum of * (or ()

assignments with null value, C′o,��o� � 0. Proof. In the case that ( ≤ * (or * ≤ ( ), matrix C′ has * rows

(or ( columns) with only 0 or infinitive values. Due to a correct permutation does not select matrix cells with value infinitive, the

only option is to select * (or () cells with null value∎

Theorem 2. The value of EditCost′ is equal to a correct

permutation cost of C′, formally:

If ( ≤ * then EditCost′�G�, G , f�, � � ∑ C′o,py~�ozC � ∑ C′o,pyozC .

If * ≤ ( then EditCost′�G�, G , f�, � � ∑ C′o,p�~ypzC � ∑ C′o,p�7zC .

Where f�, �a� � i. Proof. Suppose that ( ≤ *. Moreover, without of loosing

generality suppose that nodes :40: 1 ≤ & ≤ ( are mapped to

nodes :71: 1 ≤ + ≤ ( (if this was not the case, we always can

reorder the nodes of G such that this supposition holds).

Therefore by Lemma 2 we have that :71: ( # 1 ≤ + ≤ * have to

be inserted and there is not any :40 that has to be deleted. By

definition,EditCost� � EditCost − C��_�p �

∑ Co,p�4zC #∑ C�,p®7z�~C #∑ 0�~®4z®~C¯°°°°°°°°°°±°°°°°°°°°°²G>+?HIJ?

−∑ Co,��4zC #∑ C�,p®7zC°̄°°°°±°°°°°²

C��_�p.

Rearranging the terms, EditCost� � ∑ Co,p�4zC −∑ Co,��4zC −∑ C�,p�7zC . Applying the associative property, EditCost� �∑ ³Co,p − 2Co,� # C�,o6´�4zC � ∑ C′o,p�4zC . Finally, by Lemma 2 we

can conclude that ∑ C′o,p�4zC � ∑ C′o,p�~®4zC . The case * < ( is

similar than ( ≤ *∎

Theorem 2 gives us a way to compute the G>+?HIJ?′ through the

permutation matrix C′. Moreover, Theorem 1 showed us that the

optimal bijection of G>+?HIJ?′ is also an optimal bijection of

G>+?HIJ?. And finally, equations (6) and (7) make possible to

obtain the G>+?HIJ? with G>+?HIJ?′. Considering this three facts,

we could implement and algorithm that obtains the G>+?HIJ? through applying the Munkres’ algorithm to matrix C′. Nevertheless, due to the dimensions of C′ are the same than the

original C, the computational cost would be equivalent. If we

restrict the cost such that equation (5) holds then by Lemma 1,

we can assure the Munkres’ algorithm only needs to explore the

sub-matrix composed by the first quadrant composed of the first

n rows and m columns, that we call it HQ1�. Algorithm 1

computes the µ&J?u+¶&d?+?3.

Algorithm 1. Fast Bipartite

HQ1� � Computation_Cost�G�, G �. // HQ1� is the (v* first quadrant of cost matrix C�

P � Munkres�HQ1�� // P is the (v* permutation matrix

EditCost� � Sum2¼½*�@.∗ HQ1��6. // .∗ represents the multiplication element by element

EditCost � EditCost� # C��_�p // Final distance value

End.

As commented, the Munkres algorithm was initially implemented

to find the permutation of a quadratic matrix. In case * ≠ (,

matrix HQ1�can be extended with negative values (lower than

any original cost). Nevertheless, it is usual the implemented

functions of the Munkres’ algorithm to automatically enlarge the

cost matrix. The worst computational cost of Fast Bipartite is the

cost of the Munkres’ algorithm, that is: ¿�max�*,(�|�. The

cost of Bipartite algorithm [15] is ¿��* # (�|�.

5. Experimental Validation

The goodness of the Bipartite algorithm has been tested in

several papers, for this reason, we only want to present the Speed

up of our method respect the classical one [15]. We do not present new recognition-ratio tests or correlation tests between

the sub-optimal distance and the optimal one since we obtain

exactly the same distance value (as described above). We have performed two types of experiments. The first ones are

performed on artificially created graphs. The aim of these experiments is twofold. On the one hand, we want to show the

real computational time of the Bipartite algorithm. On the other hand, we want to show the speed up of the Fast Bipartite respect

of the order of the graphs. The second experiments are performed on well-know graph datasets. The aim is to execute again the

tests published in paper [15] where the Bipartite algorithm was presented and show the Speed up of our method on these largely

used databases.

5.1. Speed up respect graph order

In this first set of experiments, graphs have been randomly created. Nodes have one attribute and arcs do not have attributes

and are directed. The substitution cost between nodes is the absolute value of the difference between attribute values,

C�� � Àγ�2vo�6 − γ�2vp 6À. The substitution cost between arcs is 0

if both exist and 1 otherwise. Cost insertion and cost deletion of

nodes and arcs has been set to C�p � C�� � C�p � C�� � 0.5.

Each experiment has been run 100 times to not be influenced on

the random data. Source code on Matlab can be downloaded from [48].

Figure 3 shows the total execution time in seconds of 100

executions of the Bipartite algorithm tÂÃ and the Fast Bipartite

algorithm tÄÂÃ respect the order of the graphs 5, 10, 15, 20, 25, 30, 35, 40, 45 and 50 (MacPro, processor I7). Axes represent the

order of the graph presented on the first parameter and the second

parameter of the algorithm. We realise that both figures are not

symmetric and it is less costly the computation of the distance when the first graph has lower order than the second graph. In the

Bipartite case, when the order of both graphs is exactly the same,

the increase of the run time respect the order of the graphs is

really lower than in the cases that there is a small difference between both graph orders. The figure of the Fast Bipartite is

completely different since the higher is the order of the first

graph, the higher the run time. From these figures, we conclude

that (using our implementation of the Munkres’ algorithm) it is worth to present on the first parameter of the Munkres’ function

the graph with lower order and on the second parameter, the

graph of higher order.

6 (a) Bipartite algorithm.

(b) Fast Bipartite algorithm.

Figure 3. Run time tÂÃ and tÄÂÃ respect the order of the graphs.

Figure 4 shows the Speed up of our algorithm: ÅÆyÇp���ÈÉÊ�ÅÆyÇp���ÉÊ� . In

all the cases the Speed up is higher than one. It is important to note that the highest Speed up appears in the cases that the first

graph has the lowest order, that is, the desired situation that we

have realised both algorithms have the lowest run time.

Moreover, the larger the difference between graph orders, the better the Speed up.

Figure 4. Speed up (ratio between Bipartite and Fast Bipartite)

5.2. Speed up respect well-known graph datasets

The purpose of the experiments in this section is to empirically

verify the Speed up of our algorithm in some well-known graph databases known as IAM graph database repository. These

databases have been used during some years to test different a

types of algorithms related to graphs such as, classification, clustering, prototyping or graph embedding. We do not want to

add any comment to these databases since a lot of literature has been written talking about them. The first explanations of these

databases can be found in [47] and also they have been commented in [15]. They can be downloaded from the IAPR-

TC15 web page [49]. In this set of experiments we have compared all graphs on the

test set respect all graphs on the reference set as authors did in [15]. Then, from this large number of comparisons, we have

extracted the mean computational times t̅ÉÊ and t̅ÈÉÊ and the

Speed up (table 1). The source code in Matlab can be

downloaded from [48]. Moreover, the graph with the lowest order have been presented as the first parameter and the one with

the highest order as the second parameter since now we know

that this combination is the one that obtains lower run times on

both algorithms.

Table 1. Mean computational time of Bipartite and Fast Bipartite and Speed Up of Fast Bipartite respect Bipartite

Letter (L) Letter (M) Letter (H) COIL GREC Fingerprint Molecules Proteins

Mean Order 4.5 4.6 4.7 3.0 12.9 5.4 9.5 32.6

Max Order 8 10 9 11 39 26 85 126

Ì̅ÍÎ 0.0016 0.0021 0.0018 0.0022 0.0293 0.0098 0.3913 1.4606

Ì̅ÏÍÎ 0.0014 0.0018 0.0016 0.0019 0.0131 0.0062 0.0997 0.278

Speed up 1.0888 1.1344 1.1381 1.1779 2.2319 1.5927 3.9249 5.2544

The Speed up is higher than one on the whole experiments therefore it is always worth to use the Fast Bipartite instead of the

Bipartite algorithm. Moreover, the higher is the mean and the

maximum number of nodes, the higher is the Speed up. Figure 5

shows graphically the run time of both algorithms.

Figure 5. Normalised run time of FBP and BP.

6. Conclusions

This paper presents a new algorithm called Fast Bipartite to

compute the Graph Edit Distance. We have shown that under the

restriction that the edit costs are defined such as the Edit Distance

is really a distance function, we obtain exactly the same distance

value than the Bipartite algorithm but with a reduced run time. Moreover, we have also realised the run time of the Bipartite

algorithm depends on the order of presentation of both graphs.

More precisely, it is important to consider the number of nodes of

graphs. Empirical evaluation shows the Fast Bipartite is always faster than the Bipartite algorithm and higher is the order of both

graphs, better is the Speed up we obtain.

Acknowledgments

This research is supported by the CICYT projects DPI2013-

42458-P.

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9

- A sub-optimal algorithm is presented to compute the

graph edit distance.

- It obtains exactly the same distance value than the

Bipartite algorithm.

- The computational cost is lower than the Bipartite

one.

- Validation shows an important decrease on run time.