Unformatted text preview: class graph object graph
e1 objects: e2 e3 Figure 2: A path in an object graph. If there is a path in the graph from o1 to o2 ,
then class o1
C
class o2 .
¨ ¨
"¨ Lemma 3 Let c1 and c2 be classes. Then there exists an object graph O of C
c1 and class o2
c2 and O o1 o2 iff
and objects o1 , o2 such that class o1
c1 class o1
C
class o2
c2 .
© ¨
P ¨ ¨ ¨
"¨ ¨ Proof: Immediate.
Theorem 1
C
"¨ e c
"¨ ec 0£ ( &
'R £
$ FIRST c c
© ¨ Proof: We must show that there exists an object graph O of C and objects o
and o such that
£
£ C c.
£
"¨ e £ ¨
"¨ iff c 3. o eO o c
£ ¨ 2. class o c $
S 1. class o The forward direction is immediate from the deﬁnition of an object graph of C.
For the reverse deﬁnition, consider a sequence of classes such that
e2 ¨22
"4332
"¨ c2 en
T e1 1
"¨ c1 cn
e c
£
"¨ c Then construct an object graph with n 2 objects, of classes c, c1 , . . . ,cn, c ,
with a link labelled e from the cobject to the c1 object, a link labelled e1 from the
£ U 8 c1 object to the c2 object, etc. Let o be the ﬁrst object in this sequence and o be
the last. This object graph satisﬁes the requirements. QED
£ Similarly, to ﬁnd all the edges from an object of class c that might lead to an
edge e, we compute
£ e C
"¨
"¨ c ¨ £ c H¨
WV( £ &
'S
$ © ¨ 7 e ec 0
5 £ FIRST c e A Simple Example We illustrate automated traversal generation with a simple example, written in Java
using the DJ library.
Consider an equation system that contains equations with a lefthand side and
a righthand side. A typical set of equations might be
piX
q8hgE5db`X
f
e Xc
a
Y
ppYX
8wv'u1sgE5dg1X
t
i Xc
f
r Xc
a
e
ppr
8Gxu5sgE5dx8X
t
yc
f
r Xc
a
i A possible class diagram for these structures is shown in Figure 3, using notation similar to that of our previous ﬁgures. The asterisks indicate onemany relations; they are included for UML compatibility but are not part of our model; in
our model hasaspart relations are always possibly onemany.
¨
1
8 c2 ©222
333©
EE1w c1 cK paths using the search criterion
In DJ, we search for all minimal c1
cK
1E DJ traversal speciﬁcations allow several extensions of this notation. The criterion
VE1w c2
88 c1
1
E1Ew c0 cK
E has the same behavior as the search above, but fails if its starting point is not of
type c0 . DJ also allows edge speciﬁcations of the form
e. This can be
implemented by extending FIRST c1 c2 to FIRST c e as deﬁned in section 6 and
modifying the algorithm in section 5 correspondingly...
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 Fall '09

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