<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
a	O
graph	O
is	O
an	O
abstract	O
data	O
type	O
that	O
is	O
meant	O
to	O
implement	O
the	O
undirected	O
graph	O
and	O
directed	O
graph	O
concepts	O
from	O
the	O
field	O
of	O
graph	O
theory	O
within	O
mathematics	O
.	O
</s>
<s>
A	O
graph	B-Application
data	I-Application
structure	I-Application
consists	O
of	O
a	O
finite	O
(	O
and	O
possibly	O
mutable	O
)	O
set	O
of	O
vertices	O
(	O
also	O
called	O
nodes	O
or	O
points	O
)	O
,	O
together	O
with	O
a	O
set	O
of	O
unordered	O
pairs	O
of	O
these	O
vertices	O
for	O
an	O
undirected	O
graph	O
or	O
a	O
set	O
of	O
ordered	O
pairs	O
for	O
a	O
directed	O
graph	O
.	O
</s>
<s>
A	O
graph	B-Application
data	I-Application
structure	I-Application
may	O
also	O
associate	O
to	O
each	O
edge	O
some	O
edge	O
value	O
,	O
such	O
as	O
a	O
symbolic	O
label	O
or	O
a	O
numeric	O
attribute	O
(	O
cost	O
,	O
capacity	O
,	O
length	O
,	O
etc	O
.	O
</s>
<s>
The	O
basic	O
operations	O
provided	O
by	O
a	O
graph	B-Application
data	I-Application
structure	I-Application
G	O
usually	O
include	O
:	O
</s>
<s>
The	O
entries	O
indicate	O
the	O
incidence	B-Algorithm
relation	I-Algorithm
between	O
the	O
vertex	O
at	O
a	O
row	O
and	O
edge	O
at	O
a	O
column	O
.	O
</s>
<s>
Adjacency	B-Data_Structure
list	I-Data_Structure
Adjacency	B-Algorithm
matrix	I-Algorithm
Incidence	B-Algorithm
matrix	I-Algorithm
Store	O
graph	O
Add	O
vertex	O
Add	O
edge	O
Remove	O
vertex	O
Remove	O
edge	O
Are	O
vertices	O
x	O
and	O
y	O
adjacent	O
(	O
assuming	O
that	O
their	O
storage	O
positions	O
are	O
known	O
)	O
?	O
</s>
<s>
Adjacency	B-Data_Structure
lists	I-Data_Structure
are	O
generally	O
preferred	O
for	O
the	O
representation	O
of	O
sparse	O
graphs	O
,	O
while	O
an	O
adjacency	B-Algorithm
matrix	I-Algorithm
is	O
preferred	O
if	O
the	O
graph	O
is	O
dense	O
;	O
that	O
is	O
,	O
the	O
number	O
of	O
edges	O
|E|	O
is	O
close	O
to	O
the	O
number	O
of	O
vertices	O
squared	O
,	O
|V|2	O
,	O
or	O
if	O
one	O
must	O
be	O
able	O
to	O
quickly	O
look	O
up	O
if	O
there	O
is	O
an	O
edge	O
connecting	O
two	O
vertices	O
.	O
</s>
<s>
The	O
graph	B-Application
representation	I-Application
used	O
for	O
parallel	O
architectures	O
plays	O
a	O
significant	O
role	O
in	O
facing	O
those	O
challenges	O
.	O
</s>
<s>
Poorly	O
chosen	O
representations	O
may	O
unnecessarily	O
drive	O
up	O
the	O
communication	O
cost	O
of	O
the	O
algorithm	O
,	O
which	O
will	O
decrease	O
its	O
scalability	B-Architecture
.	O
</s>
<s>
In	O
the	O
following	O
,	O
shared	O
and	O
distributed	B-Operating_System
memory	I-Operating_System
architectures	O
are	O
considered	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
a	O
shared	B-Operating_System
memory	I-Operating_System
model	O
,	O
the	O
graph	B-Application
representations	I-Application
used	O
for	O
parallel	O
processing	O
are	O
the	O
same	O
as	O
in	O
the	O
sequential	O
case	O
,	O
since	O
parallel	O
read-only	O
access	O
to	O
the	O
graph	B-Application
representation	I-Application
(	O
e.g.	O
</s>
<s>
an	O
adjacency	B-Data_Structure
list	I-Data_Structure
)	O
is	O
efficient	O
in	O
shared	B-Operating_System
memory	I-Operating_System
.	O
</s>
<s>
In	O
the	O
distributed	B-Operating_System
memory	I-Operating_System
model	O
,	O
the	O
usual	O
approach	O
is	O
to	O
partition	O
the	O
vertex	O
set	O
of	O
the	O
graph	O
into	O
sets	O
.	O
</s>
<s>
For	O
standard	O
communication	O
interfaces	O
like	O
MPI	B-Application
,	O
the	O
ID	O
of	O
the	O
PE	O
owning	O
the	O
other	O
endpoint	O
has	O
to	O
be	O
identifiable	O
.	O
</s>
<s>
This	O
can	O
be	O
understood	O
as	O
a	O
row-wise	O
or	O
column-wise	O
decomposition	O
of	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
2D	O
partitioning	O
:	O
Every	O
processor	O
gets	O
a	O
submatrix	O
of	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Then	O
each	O
processor	O
gets	O
a	O
submatrix	O
of	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
dimension	O
.	O
</s>
<s>
Compressed	B-General_Concept
graph	B-Application
representations	I-Application
have	O
been	O
developed	O
to	O
reduce	O
I/O	O
and	O
memory	O
requirements	O
.	O
</s>
<s>
General	O
techniques	O
such	O
as	O
Huffman	B-General_Concept
coding	I-General_Concept
are	O
applicable	O
,	O
but	O
the	O
adjacency	B-Data_Structure
list	I-Data_Structure
or	O
adjacency	B-Algorithm
matrix	I-Algorithm
can	O
be	O
processed	O
in	O
specific	O
ways	O
to	O
increase	O
efficiency	O
.	O
</s>
