<s>
In	O
mathematics	O
,	O
computer	B-General_Concept
science	I-General_Concept
and	O
digital	O
electronics	O
,	O
a	O
dependency	B-Application
graph	O
is	O
a	O
directed	O
graph	O
representing	O
dependencies	B-Application
of	O
several	O
objects	O
towards	O
each	O
other	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
derive	O
an	O
evaluation	O
order	O
or	O
the	O
absence	O
of	O
an	O
evaluation	O
order	O
that	O
respects	O
the	O
given	O
dependencies	B-Application
from	O
the	O
dependency	B-Application
graph	O
.	O
</s>
<s>
Given	O
a	O
set	O
of	O
objects	O
and	O
a	O
transitive	O
relation	O
with	O
modeling	O
a	O
dependency	B-Application
"	O
a	O
depends	O
on	O
b	O
"	O
(	O
"	O
a	O
needs	O
b	O
evaluated	O
first	O
"	O
)	O
,	O
the	O
dependency	B-Application
graph	O
is	O
a	O
graph	O
with	O
the	O
transitive	O
reduction	O
of	O
R	O
.	O
</s>
<s>
In	O
a	O
dependency	B-Application
graph	O
,	O
the	O
cycles	O
of	O
dependencies	B-Application
(	O
also	O
called	O
circular	B-Language
dependencies	I-Language
)	O
lead	O
to	O
a	O
situation	O
in	O
which	O
no	O
valid	O
evaluation	O
order	O
exists	O
,	O
because	O
none	O
of	O
the	O
objects	O
in	O
the	O
cycle	O
may	O
be	O
evaluated	O
first	O
.	O
</s>
<s>
If	O
a	O
dependency	B-Application
graph	O
does	O
not	O
have	O
any	O
circular	B-Language
dependencies	I-Language
,	O
it	O
forms	O
a	O
directed	O
acyclic	O
graph	O
,	O
and	O
an	O
evaluation	O
order	O
may	O
be	O
found	O
by	O
topological	B-Algorithm
sorting	I-Algorithm
.	O
</s>
<s>
Most	O
topological	B-Algorithm
sorting	I-Algorithm
algorithms	O
are	O
also	O
capable	O
of	O
detecting	O
cycles	O
in	O
their	O
inputs	O
;	O
however	O
,	O
it	O
may	O
be	O
desirable	O
to	O
perform	O
cycle	O
detection	O
separately	O
from	O
topological	B-Algorithm
sorting	I-Algorithm
in	O
order	O
to	O
provide	O
appropriate	O
handling	O
for	O
the	O
detected	O
cycles	O
.	O
</s>
<s>
The	O
equation	O
system	O
"	O
A	O
=	O
B	O
;	O
B	O
=D	O
+C	O
;	O
C	O
=D	O
+A	O
;	O
D=	O
12	O
;	O
"	O
contains	O
a	O
circular	B-Language
dependency	I-Language
formed	O
by	O
A	O
,	O
B	O
and	O
C	O
,	O
as	O
B	O
must	O
be	O
evaluated	O
before	O
A	O
,	O
C	O
must	O
be	O
evaluated	O
before	O
B	O
,	O
and	O
A	O
must	O
be	O
evaluated	O
before	O
C	O
.	O
</s>
<s>
A	O
correct	O
evaluation	O
order	O
is	O
a	O
numbering	O
of	O
the	O
objects	O
that	O
form	O
the	O
nodes	O
of	O
the	O
dependency	B-Application
graph	O
so	O
that	O
the	O
following	O
equation	O
holds	O
:	O
with	O
.	O
</s>
<s>
In	O
fact	O
,	O
a	O
correct	O
numbering	O
is	O
a	O
topological	B-Algorithm
order	I-Algorithm
,	O
and	O
any	O
topological	B-Algorithm
order	I-Algorithm
is	O
a	O
correct	O
numbering	O
.	O
</s>
<s>
Thus	O
,	O
any	O
algorithm	O
that	O
derives	O
a	O
correct	O
topological	B-Algorithm
order	I-Algorithm
derives	O
a	O
correct	O
evaluation	O
order	O
.	O
</s>
<s>
An	O
acyclic	O
dependency	B-Application
graph	O
corresponds	O
to	O
a	O
trace	O
of	O
a	O
trace	O
monoid	O
as	O
follows	O
:	O
</s>
<s>
There	O
is	O
an	O
edge	O
or	O
if	O
and	O
only	O
if	O
is	O
in	O
the	O
dependency	B-Application
relation	O
.	O
</s>
<s>
The	O
monoidal	O
operation	O
takes	O
the	O
disjoint	O
union	O
of	O
two	O
graphs	O
 '	O
vertex	O
sets	O
,	O
preserves	O
the	O
existing	O
edges	O
in	O
each	O
graph	O
,	O
and	O
draws	O
new	O
edges	O
from	O
the	O
first	O
to	O
the	O
second	O
where	O
the	O
dependency	B-Application
relation	O
allows	O
,	O
</s>
<s>
Dependency	B-Application
graphs	O
are	O
used	O
in	O
:	O
</s>
<s>
Automated	O
software	O
installers	B-Operating_System
:	O
They	O
walk	O
the	O
graph	O
looking	O
for	O
software	B-Application
packages	I-Application
that	O
are	O
required	O
but	O
not	O
yet	O
installed	O
.	O
</s>
<s>
The	O
dependency	B-Application
is	O
given	O
by	O
the	O
coupling	B-Application
of	O
the	O
packages	O
.	O
</s>
<s>
Software	O
build	O
scripts	O
such	O
as	O
Unix	B-Application
Make	B-Application
,	O
Node	B-Language
npm	O
install	O
,	O
php	O
composer	O
,	O
Twitter	B-Application
bower	O
install	O
,	O
or	O
Apache	B-Application
Ant	I-Application
.	O
</s>
<s>
They	O
need	O
to	O
know	O
what	O
files	O
have	O
changed	O
so	O
only	O
the	O
correct	O
files	O
need	O
to	O
be	O
recompiled	B-Language
.	O
</s>
<s>
In	O
compiler	B-Language
technology	O
and	O
formal	O
language	O
implementation	O
:	O
</s>
<s>
Instruction	O
scheduling	O
:	O
Dependency	B-Application
graphs	O
are	O
computed	O
for	O
the	O
operands	O
of	O
assembly	B-Language
or	O
intermediate	O
instructions	O
and	O
used	O
to	O
determine	O
an	O
optimal	O
order	O
for	O
the	O
instructions	O
.	O
</s>
<s>
Dynamic	O
graph	O
analytics	O
:	O
GraphBolt	O
and	O
KickStarter	O
capture	O
value	O
dependencies	B-Application
for	O
incremental	O
computing	O
when	O
graph	O
structure	O
changes	O
.	O
</s>
<s>
Spreadsheet	B-Application
calculators	O
.	O
</s>
<s>
Web	O
Forms	O
standards	O
such	O
as	O
XForms	B-Application
to	O
know	O
what	O
visual	O
elements	O
to	O
update	O
if	O
data	O
in	O
the	O
model	O
changes	O
.	O
</s>
<s>
Dependency	B-Application
graphs	O
are	O
one	O
aspect	O
of	O
:	O
</s>
<s>
Job	B-Algorithm
shop	I-Algorithm
scheduling	I-Algorithm
:	O
A	O
collection	O
of	O
related	O
theoretical	O
problems	O
in	O
computer	B-General_Concept
science	I-General_Concept
.	O
</s>
