<s>
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
a	O
mathematical	O
algorithm	O
in	O
group	O
theory	O
for	O
solving	O
the	O
automorphic	O
equivalence	O
problem	O
in	O
the	O
finite	O
rank	O
free	O
group	O
Fn	O
.	O
</s>
<s>
It	O
is	O
still	O
unknown	O
(	O
except	O
for	O
the	O
case	O
n	O
=	O
2	O
)	O
if	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
has	O
polynomial	O
time	O
complexity	O
.	O
</s>
<s>
The	O
automorphism	O
problem	O
for	O
was	O
algorithmically	O
solved	O
by	O
J	O
.	O
H	O
.	O
C	O
.	O
Whitehead	O
in	O
a	O
classic	O
1936	O
paper	O
,	O
and	O
his	O
solution	O
came	O
to	O
be	O
known	O
as	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Subsequently	O
,	O
Rapaport	O
,	O
and	O
later	O
,	O
based	O
on	O
her	O
work	O
,	O
Higgins	O
and	O
Lyndon	O
,	O
gave	O
a	O
purely	O
combinatorial	O
and	O
algebraic	O
re-interpretation	O
of	O
Whitehead	O
's	O
work	O
and	O
of	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
exposition	O
of	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
in	O
the	O
book	O
of	O
Lyndon	O
and	O
Schupp	O
is	O
based	O
on	O
this	O
combinatorial	O
approach	O
.	O
</s>
<s>
Culler	O
and	O
Vogtmann	O
,	O
in	O
their	O
1986	O
paper	O
that	O
introduced	O
the	O
Outer	O
space	O
,	O
gave	O
a	O
hybrid	O
approach	O
to	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
presented	O
in	O
combinatorial	O
terms	O
but	O
closely	O
following	O
Whitehead	O
's	O
original	O
ideas	O
.	O
</s>
<s>
Our	O
exposition	O
regarding	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
mostly	O
follows	O
Ch.I.4	O
in	O
the	O
book	O
of	O
Lyndon	O
and	O
Schupp	O
,	O
as	O
well	O
as	O
.	O
</s>
<s>
The	O
automorphism	B-Algorithm
group	I-Algorithm
has	O
a	O
particularly	O
useful	O
finite	O
generating	O
set	O
of	O
Whitehead	O
automorphisms	O
or	O
Whitehead	O
moves	O
.	O
</s>
<s>
Given	O
the	O
first	O
part	O
of	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
consists	O
of	O
iteratively	O
applying	O
Whitehead	O
moves	O
to	O
to	O
take	O
each	O
of	O
them	O
to	O
an	O
``automorphically	O
minimal	O
"	O
form	O
,	O
where	O
the	O
cyclically	O
reduced	O
length	O
strictly	O
decreases	O
at	O
each	O
step	O
.	O
</s>
<s>
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
also	O
solves	O
the	O
search	O
automorphism	O
problem	O
for	O
.	O
</s>
<s>
Namely	O
,	O
given	O
,	O
if	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
concludes	O
that	O
,	O
the	O
algorithm	O
also	O
outputs	O
an	O
automorphism	O
such	O
that	O
.	O
</s>
<s>
Often	O
,	O
for	O
a	O
Whitehead	O
automorphism	O
,	O
the	O
corresponding	O
outer	B-Algorithm
automorphism	I-Algorithm
in	O
is	O
also	O
called	O
a	O
Whitehead	O
automorphism	O
or	O
a	O
Whitehead	O
move	O
.	O
</s>
<s>
Then	O
is	O
actually	O
an	O
inner	B-Algorithm
automorphism	I-Algorithm
of	O
given	O
by	O
conjugation	O
by	O
,	O
and	O
,	O
moreover	O
,	O
is	O
a	O
Whitehead	O
automorphism	O
of	O
the	O
second	O
kind	O
,	O
with	O
the	O
multiplier	O
.	O
</s>
<s>
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
for	O
the	O
automorphic	O
equivalence	O
problem	O
,	O
given	O
decides	O
whether	O
or	O
not	O
.	O
</s>
<s>
More	O
specifically	O
,	O
using	O
the	O
breadth-first	B-Algorithm
approach	I-Algorithm
,	O
one	O
constructs	O
the	O
connected	O
components	O
of	O
the	O
automorphism	O
graph	O
and	O
checks	O
if	O
.	O
</s>
<s>
The	O
Peak	O
Reduction	O
Lemma	O
implies	O
that	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
correctly	O
solves	O
the	O
automorphic	O
equivalence	O
problem	O
in	O
.	O
</s>
<s>
For	O
that	O
reason	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
for	O
deciding	O
,	O
given	O
,	O
whether	O
or	O
not	O
,	O
runs	O
in	O
at	O
most	O
exponential	O
time	O
in	O
.	O
</s>
<s>
Consequently	O
,	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
for	O
the	O
automorphic	O
equivalence	O
problem	O
in	O
,	O
given	O
runs	O
in	O
quadratic	O
time	O
in	O
.	O
</s>
<s>
McCool	O
used	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
and	O
the	O
peak	O
reduction	O
to	O
prove	O
that	O
for	O
any	O
the	O
stabilizer	O
is	O
finitely	O
presentable	O
,	O
and	O
obtained	O
a	O
similar	O
results	O
for	O
-stabilizers	O
of	O
m-tuples	O
of	O
conjugacy	O
classes	O
in	O
.	O
</s>
<s>
Gersten	O
obtained	O
a	O
variation	O
of	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
for	O
deciding	O
,	O
given	O
two	O
finite	O
subsets	O
,	O
whether	O
the	O
subgroups	O
are	O
automorphically	O
equivalent	O
,	O
that	O
is	O
,	O
whether	O
there	O
exists	O
such	O
that	O
.	O
</s>
<s>
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
and	O
peak	O
reduction	O
play	O
a	O
key	O
role	O
in	O
the	O
proof	O
by	O
Culler	O
and	O
Vogtmann	O
that	O
the	O
Culler	O
–	O
Vogtmann	O
Outer	O
space	O
is	O
contractible	O
.	O
</s>
<s>
Collins	O
and	O
Zieschang	O
obtained	O
analogs	O
of	O
Whitehead	O
's	O
peak	O
reduction	O
and	O
of	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
for	O
automorphic	O
equivalence	O
in	O
free	O
products	O
of	O
groups	O
.	O
</s>
<s>
Gilbert	O
used	O
a	O
version	O
of	O
a	O
peak	O
reduction	O
lemma	O
to	O
construct	O
a	O
presentation	O
for	O
the	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
a	O
free	O
product	O
.	O
</s>
<s>
Consequently	O
,	O
if	O
are	O
two	O
such	O
``generic	O
"	O
elements	O
,	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
decides	O
whether	O
are	O
automorphically	O
equivalent	O
in	O
linear	O
time	O
in	O
.	O
</s>
<s>
Consequently	O
,	O
if	O
are	O
such	O
that	O
is	O
automorphically	O
equivalent	O
to	O
some	O
with	O
the	O
above	O
property	O
,	O
then	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
decides	O
whether	O
are	O
automorphically	O
equivalent	O
in	O
time	O
.	O
</s>
<s>
The	O
Garside	O
algorithm	O
for	O
solving	O
the	O
conjugacy	O
problem	O
in	O
braid	B-Application
groups	I-Application
has	O
a	O
similar	O
general	O
structure	O
to	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
with	O
"	O
cycling	O
moves	O
"	O
playing	O
the	O
role	O
of	O
Whitehead	O
moves	O
.	O
</s>
<s>
Day	O
obtained	O
analogs	O
of	O
Whitehead	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
and	O
of	O
Whitehead	O
's	O
peak	O
reduction	O
for	O
automorphic	O
equivalence	O
of	O
elements	O
of	O
right-angled	O
Artin	O
groups	O
.	O
</s>
