<s>
As	O
sequential	B-Algorithm
algorithms	I-Algorithm
for	O
this	O
problem	O
often	O
yield	O
long	O
runtimes	O
,	O
parallelization	O
has	O
shown	O
to	O
be	O
beneficial	O
in	O
this	O
field	O
.	O
</s>
<s>
Another	O
variation	O
of	O
the	O
problem	O
is	O
the	O
single-source-shortest-paths	O
(	O
SSSP	O
)	O
problem	O
,	O
which	O
also	O
has	O
parallel	O
approaches	O
:	O
Parallel	B-Algorithm
single-source	I-Algorithm
shortest	I-Algorithm
path	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
In	O
the	O
remainder	O
of	O
the	O
article	O
it	O
is	O
assumed	O
that	O
the	O
graph	O
is	O
represented	O
using	O
an	O
adjacency	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
Floyd	B-Algorithm
algorithm	I-Algorithm
presented	O
later	O
can	O
handle	O
negative	O
edge	O
weights	O
,	O
whereas	O
the	O
Dijkstra	B-Algorithm
algorithm	I-Algorithm
requires	O
all	O
edges	O
to	O
have	O
a	O
positive	O
weight	O
.	O
</s>
<s>
The	O
Dijkstra	B-Algorithm
algorithm	I-Algorithm
originally	O
was	O
proposed	O
as	O
a	O
solver	O
for	O
the	O
single-source-shortest-paths	O
problem	O
.	O
</s>
<s>
The	O
runtime	O
of	O
DijkstraSSSP	O
is	O
as	O
we	O
expect	O
the	O
graph	O
to	O
be	O
represented	O
using	O
an	O
adjacency	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
However	O
,	O
it	O
is	O
required	O
that	O
every	O
processor	O
has	O
enough	O
local	O
memory	O
to	O
store	O
the	O
entire	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
the	O
graph	O
.	O
</s>
<s>
The	O
main	O
benefit	O
of	O
this	O
parallelization	O
is	O
that	O
it	O
is	O
not	O
required	O
anymore	O
that	O
every	O
processor	O
stores	O
the	O
entire	O
adjacency	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Instead	O
,	O
it	O
is	O
sufficient	O
when	O
each	O
processor	O
within	O
a	O
partition	O
only	O
stores	O
the	O
columns	O
of	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
the	O
nodes	O
for	O
which	O
he	O
is	O
responsible	O
.	O
</s>
<s>
Given	O
a	O
partition	O
size	O
of	O
,	O
each	O
processor	O
only	O
has	O
to	O
store	O
columns	O
of	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
Floyd	B-Algorithm
algorithm	I-Algorithm
solves	O
the	O
All-Pair-Shortest-Paths	O
problem	O
for	O
directed	O
graphs	O
.	O
</s>
<s>
With	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
a	O
graph	O
as	O
input	O
,	O
it	O
calculates	O
shorter	O
paths	O
iterative	O
.	O
</s>
<s>
Where	O
A	O
is	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
,	O
n	O
=	O
|V|	O
the	O
number	O
of	O
nodes	O
and	O
D	O
the	O
distance	O
matrix	O
.	O
</s>
<s>
For	O
a	O
more	O
detailed	O
description	O
of	O
the	O
sequential	B-Algorithm
algorithm	I-Algorithm
look	O
up	O
Floyd	B-Algorithm
–	I-Algorithm
Warshall	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
runtime	O
of	O
the	O
sequential	B-Algorithm
algorithm	I-Algorithm
is	O
determined	O
by	O
the	O
triple	O
nested	O
for	O
loop	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
runtime	O
of	O
the	O
sequential	B-Algorithm
algorithm	I-Algorithm
is	O
.	O
</s>
