<s>
In	O
theoretical	O
computer	O
science	O
and	O
network	B-Protocol
routing	I-Protocol
,	O
Suurballe	B-Protocol
's	I-Protocol
algorithm	I-Protocol
is	O
an	O
algorithm	O
for	O
finding	O
two	O
disjoint	O
paths	O
in	O
a	O
nonnegatively-weighted	O
directed	O
graph	O
,	O
so	O
that	O
both	O
paths	O
connect	O
the	O
same	O
pair	O
of	O
vertices	O
and	O
have	O
minimum	O
total	O
length	O
.	O
</s>
<s>
The	O
main	O
idea	O
of	O
Suurballe	B-Protocol
's	I-Protocol
algorithm	I-Protocol
is	O
to	O
use	O
Dijkstra	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
to	O
find	O
one	O
path	O
,	O
to	O
modify	O
the	O
weights	O
of	O
the	O
graph	O
edges	O
,	O
and	O
then	O
to	O
run	O
Dijkstra	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
a	O
second	O
time	O
.	O
</s>
<s>
The	O
modification	O
to	O
the	O
weights	O
is	O
similar	O
to	O
the	O
weight	O
modification	O
in	O
Johnson	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
and	O
preserves	O
the	O
non-negativity	O
of	O
the	O
weights	O
while	O
allowing	O
the	O
second	O
instance	O
of	O
Dijkstra	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
to	O
find	O
the	O
correct	O
second	O
path	O
.	O
</s>
<s>
The	O
problem	O
of	O
finding	O
two	O
disjoint	O
paths	O
of	O
minimum	O
weight	O
can	O
be	O
seen	O
as	O
a	O
special	O
case	O
of	O
a	O
minimum	B-Algorithm
cost	I-Algorithm
flow	I-Algorithm
problem	I-Algorithm
,	O
where	O
in	O
this	O
case	O
there	O
are	O
two	O
units	O
of	O
"	O
flow	O
"	O
and	O
nodes	O
have	O
unit	O
"	O
capacity	O
"	O
.	O
</s>
<s>
Suurballe	B-Protocol
's	I-Protocol
algorithm	I-Protocol
,	O
also	O
,	O
can	O
be	O
seen	O
as	O
a	O
special	O
case	O
of	O
a	O
minimum	B-Algorithm
cost	I-Algorithm
flow	I-Algorithm
algorithm	O
that	O
repeatedly	O
pushes	O
the	O
maximum	O
possible	O
amount	O
of	O
flow	O
along	O
a	O
shortest	O
augmenting	O
path	O
.	O
</s>
<s>
The	O
first	O
path	O
found	O
by	O
Suurballe	B-Protocol
's	I-Protocol
algorithm	I-Protocol
is	O
the	O
shortest	O
augmenting	O
path	O
for	O
the	O
initial	O
(	O
zero	O
)	O
flow	O
,	O
and	O
the	O
second	O
path	O
found	O
by	O
Suurballe	B-Protocol
's	I-Protocol
algorithm	I-Protocol
is	O
the	O
shortest	O
augmenting	O
path	O
for	O
the	O
residual	O
graph	O
left	O
after	O
pushing	O
one	O
unit	O
of	O
flow	O
along	O
the	O
first	O
path	O
.	O
</s>
<s>
Suurballe	B-Protocol
's	I-Protocol
algorithm	I-Protocol
performs	O
the	O
following	O
steps	O
:	O
</s>
<s>
Find	O
the	O
shortest	O
path	O
tree	O
rooted	O
at	O
node	O
by	O
running	O
Dijkstra	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
(	O
figure	O
C	O
)	O
.	O
</s>
<s>
Find	O
the	O
shortest	O
path	O
in	O
the	O
residual	O
graph	O
by	O
running	O
Dijkstra	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
(	O
figure	O
E	O
)	O
.	O
</s>
<s>
The	O
following	O
example	O
shows	O
how	O
Suurballe	B-Protocol
's	I-Protocol
algorithm	I-Protocol
finds	O
the	O
shortest	O
pair	O
of	O
disjoint	O
paths	O
from	O
A	O
to	O
F	O
.	O
</s>
<s>
Suurballe	B-Protocol
's	I-Protocol
algorithm	I-Protocol
may	O
be	O
seen	O
as	O
a	O
special	O
case	O
of	O
the	O
successive	O
shortest	O
paths	O
method	O
for	O
finding	O
a	O
minimum	B-Algorithm
cost	I-Algorithm
flow	I-Algorithm
with	O
total	O
flow	O
amount	O
two	O
from	O
to	O
.	O
</s>
<s>
This	O
algorithm	O
requires	O
two	O
iterations	O
of	O
Dijkstra	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Using	O
Fibonacci	B-Application
heaps	I-Application
,	O
both	O
iterations	O
can	O
be	O
performed	O
in	O
time	O
where	O
and	O
are	O
the	O
number	O
of	O
vertices	O
and	O
edges	O
respectively	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
same	O
time	O
bound	O
applies	O
to	O
Suurballe	B-Protocol
's	I-Protocol
algorithm	I-Protocol
.	O
</s>
<s>
The	O
version	O
of	O
Suurballe	B-Protocol
's	I-Protocol
algorithm	I-Protocol
as	O
described	O
above	O
finds	O
paths	O
that	O
have	O
disjoint	O
edges	O
,	O
but	O
that	O
may	O
share	O
vertices	O
.	O
</s>
<s>
Two	O
edge-disjoint	O
paths	O
in	O
this	O
modified	O
graph	O
necessarily	O
correspond	O
to	O
two	O
vertex-disjoint	O
paths	O
in	O
the	O
original	O
graph	O
,	O
and	O
vice	O
versa	O
,	O
so	O
applying	O
Suurballe	B-Protocol
's	I-Protocol
algorithm	I-Protocol
to	O
the	O
modified	O
graph	O
results	O
in	O
the	O
construction	O
of	O
two	O
vertex-disjoint	O
paths	O
in	O
the	O
original	O
graph	O
.	O
</s>
<s>
By	O
using	O
a	O
modified	O
version	O
of	O
Dijkstra	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
that	O
simultaneously	O
computes	O
the	O
distances	O
to	O
each	O
vertex	O
in	O
the	O
graphs	O
,	O
it	O
is	O
also	O
possible	O
to	O
find	O
the	O
total	O
lengths	O
of	O
the	O
shortest	O
pairs	O
of	O
paths	O
from	O
a	O
given	O
source	O
vertex	O
to	O
every	O
other	O
vertex	O
in	O
the	O
graph	O
,	O
in	O
an	O
amount	O
of	O
time	O
that	O
is	O
proportional	O
to	O
a	O
single	O
instance	O
of	O
Dijkstra	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
