<s>
Bidirectional	B-Algorithm
search	I-Algorithm
is	O
a	O
graph	O
search	O
algorithm	O
that	O
finds	O
a	O
shortest	O
path	O
from	O
an	O
initial	O
vertex	O
to	O
a	O
goal	O
vertex	O
in	O
a	O
directed	O
graph	O
.	O
</s>
<s>
The	O
reason	O
for	O
this	O
approach	O
is	O
that	O
in	O
many	O
cases	O
it	O
is	O
faster	O
:	O
for	O
instance	O
,	O
in	O
a	O
simplified	O
model	O
of	O
search	O
problem	O
complexity	O
in	O
which	O
both	O
searches	O
expand	O
a	O
tree	O
with	O
branching	B-Data_Structure
factor	I-Data_Structure
b	O
,	O
and	O
the	O
distance	O
from	O
start	O
to	O
goal	O
is	O
d	O
,	O
each	O
of	O
the	O
two	O
searches	O
has	O
complexity	O
O( 	O
bd/2	O
)	O
(	O
in	O
Big	O
O	O
notation	O
)	O
,	O
and	O
the	O
sum	O
of	O
these	O
two	O
search	O
times	O
is	O
much	O
less	O
than	O
the	O
O(bd )	O
complexity	O
that	O
would	O
result	O
from	O
a	O
single	O
search	O
from	O
the	O
beginning	O
to	O
the	O
goal	O
.	O
</s>
<s>
Andrew	O
Goldberg	O
and	O
others	O
explained	O
the	O
correct	O
termination	O
conditions	O
for	O
the	O
bidirectional	O
version	O
of	O
Dijkstra	B-Algorithm
’s	I-Algorithm
Algorithm	I-Algorithm
.	O
</s>
<s>
As	O
in	O
A*	B-Protocol
search	I-Protocol
,	O
bi-directional	B-Algorithm
search	I-Algorithm
can	O
be	O
guided	O
by	O
a	O
heuristic	B-Algorithm
estimate	O
of	O
the	O
remaining	O
distance	O
to	O
the	O
goal	O
(	O
in	O
the	O
forward	O
tree	O
)	O
or	O
from	O
the	O
start	O
(	O
in	O
the	O
backward	O
tree	O
)	O
.	O
</s>
<s>
was	O
the	O
first	O
one	O
to	O
design	O
and	O
implement	O
a	O
bi-directional	O
heuristic	B-Algorithm
search	O
algorithm	O
.	O
</s>
<s>
A	O
solution	O
found	O
by	O
the	O
uni-directional	O
A*	B-Protocol
algorithm	I-Protocol
using	O
an	O
admissible	O
heuristic	B-Algorithm
has	O
a	O
shortest	O
path	O
length	O
;	O
the	O
same	O
property	O
holds	O
for	O
the	O
BHFFA2	O
bidirectional	O
heuristic	B-Algorithm
version	O
described	O
in	O
de	O
Champeaux	O
(	O
1983	O
)	O
.	O
</s>
<s>
A	O
Bidirectional	O
Heuristic	B-Algorithm
Search	O
is	O
a	O
state	B-Algorithm
space	I-Algorithm
search	I-Algorithm
from	O
some	O
state	O
to	O
another	O
state	O
,	O
searching	O
from	O
to	O
and	O
from	O
to	O
simultaneously	O
.	O
</s>
<s>
In	O
bidirectional	B-Algorithm
search	I-Algorithm
,	O
these	O
are	O
sometimes	O
called	O
the	O
search	O
'	O
frontiers	O
 '	O
or	O
'	O
wavefronts	O
 '	O
,	O
referring	O
to	O
how	O
they	O
appear	O
when	O
a	O
search	O
is	O
represented	O
graphically	O
.	O
</s>
<s>
These	O
differ	O
by	O
the	O
function	O
used	O
to	O
calculate	O
the	O
heuristic	B-Algorithm
.	O
</s>
<s>
Front-to-Back	O
algorithms	O
calculate	O
the	O
value	O
of	O
a	O
node	O
by	O
using	O
the	O
heuristic	B-Algorithm
estimate	O
between	O
and	O
the	O
root	O
of	O
the	O
opposite	O
search	O
tree	O
,	O
or	O
.	O
</s>
<s>
Front-to-Front	O
algorithms	O
calculate	O
the	O
value	O
of	O
a	O
node	O
by	O
using	O
the	O
heuristic	B-Algorithm
estimate	O
between	O
and	O
some	O
subset	O
of	O
.	O
</s>
<s>
The	O
canonical	O
example	O
is	O
that	O
of	O
the	O
BHFFA	O
(	O
Bidirectional	O
Heuristic	B-Algorithm
Front-to-Front	O
Algorithm	O
)	O
,	O
where	O
the	O
function	O
is	O
defined	O
as	O
the	O
minimum	O
of	O
all	O
heuristic	B-Algorithm
estimates	O
between	O
the	O
current	O
node	O
and	O
the	O
nodes	O
on	O
the	O
opposing	O
front	O
.	O
</s>
<s>
not	O
overestimating	O
)	O
heuristic	B-Algorithm
estimate	O
of	O
the	O
distance	O
between	O
nodes	O
and	O
.	O
</s>
<s>
This	O
involves	O
calculating	O
a	O
heuristic	B-Algorithm
estimate	O
from	O
to	O
every	O
node	O
in	O
the	O
opposing	O
set	O
,	O
as	O
described	O
above	O
.	O
</s>
