SEKE 2012 Proceedings - Knowledge Systems Institute

III.
GRAPH REPRESENTATION OF CLASS
DIAGRAMS
UML class diagrams can be c onverted to labeled directed
graphs in which the classes are represe nted by nodes, and the
relationships between the classes are re presented as edges of
the graph. In addition, edges contain extra information which
specify whether they represent dependencies, generalization,
association, and so on. With this representation in m ind, the
problem of m atching a quer y class diagram to another class
diagram in the repository becomes that of graph m atching. In
particular, since the graphs to be compared usually have
different numbers of nodes a nd edges, the problem is referred
to as inexact g raph matching [11]. Fig. 1 s hows how a class
diagram is converted to a directed graph. An adjacency matrix
representation of the graph is al so shown in Table I. Rath er
than containing zeros and ones, the entries of the matrix s how
the types of Class relationships represented by the edges of the
graph.
IV. SIMILARITY METRIC
We propose a similarity metric which is com posed of two
parts; name (semantic) similarity and str ucture (topology)
similarity [4], [12]. Nam e similarity measures the semantic
relatedness of the concepts (classes) in the class diagram s to be
compared, while structure similarity measures how closely the
relationships between classes match one another. In other
words, name similarity determines how t he corresponding
nodes of tw o graphs are r elated, while structure similarity
measures the level of similar ity between corresponding edges
[13], [9].
A. Name Similarity
One way of measuring the sem antic relatedness of class
names is to use domain ontology [4]. Another possibility is to
utilize a lexical database such as WordNet.
Corporate
Customer
Customer
Personal
Customer
Figure 1. A Class Diagram and its corresponding directed graph. Nodes 1, 2,
3, 4 and 5 in the graph represent the Customer, Order, Corporate Customer,
Personal Customer and Item Classes
TABLE I.
Order
Item
Generalization
ADJACENCY MATRIX
1 2 3 4 5
1 None None None None None
2 Association None None None None
3 Generalization None None None None
4 Generalization None None None None
5 None Aggregation None None None
1
Association
Generalization
3 4 5
2
Aggregation
We propose using WordNet as is done in [3]. The Name
Similarity of two class diagrams A and B having equal num ber
of classes is given in (1).
n
cns(
Ai , Bi
)
i1
NS(
A,
B)
n
A i is the na me of the i th class in class diagram A, B i is the
name of the i th class in class diagram B while n is the number
of classes contained in both diagram s. cns (Class Na me
Similarity) is a function that returns a semantic relatedness
value between zero and one . Zero and one denote maximal
relatedness and un-relatedne ss of conce pts, respectively. The
division by n in equation 1 ensures that the value of nam e
similarity (NS) always lies between 0 and 1.
B. Structure Similarity
In order to measure the stru cture similarity between two
matrices, we define a squa re matrix Diff, whose entries
represent the level of dissimilarity between the various types of
class relationships. The (i, j)th en try of the matrix is a measure
of the dissimilarity between the i th type of relationship and the
j th type of relationship. A value of 1 indi cates that the two
relationships are extremely dissimilar, while 0 indicates that the
relationships are the sa me (hence the diagonal entries of the
matrix are all zeros). The entries of this matrix can be filled by
gathering information from UML experts, or by appl ying
ontology as in [4]. Table II (adapted from [4]) shows a sample
Difference matrix (Diff). Since the main objective of retrieving
class diagrams is to reuse the m, the entries in Diff shoul d be
proportional to the am ount of effo rt required to con vert one
type of relationship to another, after retrieving a class diagram
from the repository. The last row labeled ‘None’ shows the
level of dissimilarity between having no relationship between
two classes (that is no edge connecting the vertices) and having
a relationship between the two classes.
Let A and B be two directed graphs (representing class
diagrams) each having n nodes. In addition, let the n x n
matrices AdjA and AdjB be the adjacency matrices of A and B,
respectively. The Structure Similarity (SS) between A and B is
computed as shown in (2). nm is the number of times the edges
in both graphs match exactly, while nu is th e number of times
the edges do not match.
Diff ( AdjA(
i,
j),
AdjB(
i,
j))
i,
j
SS( A,
B)
nm nu
The overall similarity metric between two class diagrams
represented by graphs A and B is a weighted sum of the Name
Similarity and Structure Similarity as shown in (3). is a value
between 0 and 1 that determ ines the relative im portance of SS
and NS.
S( A,
B)
* SS(
A,
B)
(1 ) * NS(
A,
B)
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