<s>
There	O
are	O
classical	O
sequential	B-Algorithm
algorithms	I-Algorithm
which	O
solve	O
this	O
problem	O
,	O
such	O
as	O
Dijkstra	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
In	O
this	O
article	O
,	O
however	O
,	O
we	O
present	O
two	O
parallel	B-Operating_System
algorithms	I-Operating_System
solving	O
this	O
problem	O
.	O
</s>
<s>
Another	O
variation	O
of	O
the	O
problem	O
is	O
the	O
all-pairs-shortest-paths	O
(	O
APSP	O
)	O
problem	O
,	O
which	O
also	O
has	O
parallel	O
approaches	O
:	O
Parallel	B-Algorithm
all-pairs	I-Algorithm
shortest	I-Algorithm
path	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
For	O
all	O
parallel	B-Operating_System
algorithms	I-Operating_System
we	O
will	O
assume	O
a	O
PRAM	B-Operating_System
model	O
with	O
concurrent	O
reads	O
and	O
concurrent	O
writes	O
.	O
</s>
<s>
The	O
delta	B-Algorithm
stepping	I-Algorithm
algorithm	O
is	O
a	O
label-correcting	O
algorithm	O
,	O
which	O
means	O
the	O
tentative	O
distance	O
of	O
a	O
vertex	O
can	O
be	O
corrected	O
several	O
times	O
via	O
edge	O
relaxations	O
until	O
the	O
last	O
step	O
of	O
the	O
algorithm	O
,	O
when	O
all	O
tentative	O
distances	O
are	O
fixed	O
.	O
</s>
<s>
Following	O
is	O
the	O
delta	B-Algorithm
stepping	I-Algorithm
algorithm	O
in	O
pseudocode	O
:	O
</s>
<s>
The	O
third	O
computational	O
kernel	O
of	O
the	O
Graph	B-Device
500	I-Device
benchmark	O
runs	O
a	O
single-source	O
shortest	O
path	O
computation	O
.	O
</s>
<s>
The	O
reference	O
implementation	O
of	O
the	O
Graph	B-Device
500	I-Device
benchmark	O
uses	O
the	O
delta	B-Algorithm
stepping	I-Algorithm
algorithm	O
for	O
this	O
computation	O
.	O
</s>
<s>
For	O
all	O
,	O
define	O
to	O
be	O
the	O
neighbor	O
set	O
of	O
S	O
.	O
During	O
the	O
execution	O
of	O
standard	O
breadth-first	B-Algorithm
search	I-Algorithm
or	O
Dijkstra	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
the	O
frontier	O
is	O
the	O
neighbor	O
set	O
of	O
all	O
visited	O
vertices	O
.	O
</s>
<s>
Lines	O
5-9	O
then	O
run	O
the	O
Bellman-Ford	B-Algorithm
substeps	O
until	O
all	O
vertices	O
with	O
radius	O
less	O
than	O
are	O
settled	O
.	O
</s>
