<s>
In	O
computational	O
complexity	O
the	O
decision	B-Application
tree	I-Application
model	I-Application
is	O
the	O
model	O
of	O
computation	O
in	O
which	O
an	O
algorithm	O
is	O
considered	O
to	O
be	O
basically	O
a	O
decision	B-Algorithm
tree	I-Algorithm
,	O
i.e.	O
,	O
a	O
sequence	O
of	O
queries	O
or	O
tests	O
that	O
are	O
done	O
adaptively	O
,	O
so	O
the	O
outcome	O
of	O
previous	O
tests	O
can	O
influence	O
the	O
tests	O
performed	O
next	O
.	O
</s>
<s>
Typically	O
,	O
these	O
tests	O
have	O
a	O
small	O
number	O
of	O
outcomes	O
(	O
such	O
as	O
a	O
yes	O
–	O
no	O
question	O
)	O
and	O
can	O
be	O
performed	O
quickly	O
(	O
say	O
,	O
with	O
unit	O
computational	O
cost	O
)	O
,	O
so	O
the	O
worst-case	O
time	O
complexity	O
of	O
an	O
algorithm	O
in	O
the	O
decision	B-Application
tree	I-Application
model	I-Application
corresponds	O
to	O
the	O
depth	O
of	O
the	O
corresponding	O
decision	B-Algorithm
tree	I-Algorithm
.	O
</s>
<s>
This	O
notion	O
of	O
computational	O
complexity	O
of	O
a	O
problem	O
or	O
an	O
algorithm	O
in	O
the	O
decision	B-Application
tree	I-Application
model	I-Application
is	O
called	O
its	O
decision	B-Application
tree	I-Application
complexity	I-Application
or	O
query	B-Application
complexity	I-Application
.	O
</s>
<s>
Decision	B-Algorithm
trees	I-Algorithm
models	O
are	O
instrumental	O
in	O
establishing	O
lower	O
bounds	O
for	O
complexity	O
theory	O
for	O
certain	O
classes	O
of	O
computational	O
problems	O
and	O
algorithms	O
.	O
</s>
<s>
Several	O
variants	O
of	O
decision	B-Application
tree	I-Application
models	I-Application
have	O
been	O
introduced	O
,	O
depending	O
on	O
the	O
computational	B-Application
model	I-Application
and	O
type	O
of	O
query	O
algorithms	O
are	O
allowed	O
to	O
perform	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
decision	B-Algorithm
tree	I-Algorithm
argument	O
is	O
used	O
to	O
show	O
that	O
a	O
comparison	B-Algorithm
sort	I-Algorithm
of	O
items	O
must	O
take	O
comparisons	O
.	O
</s>
<s>
For	O
comparison	B-Algorithm
sorts	I-Algorithm
,	O
a	O
query	O
is	O
a	O
comparison	O
of	O
two	O
items	O
,	O
with	O
two	O
outcomes	O
(	O
assuming	O
no	O
items	O
are	O
equal	O
)	O
:	O
either	O
or	O
.	O
</s>
<s>
Comparison	B-Algorithm
sorts	I-Algorithm
can	O
be	O
expressed	O
as	O
a	O
decision	B-Algorithm
tree	I-Algorithm
in	O
this	O
model	O
,	O
since	O
such	O
sorting	O
algorithms	O
only	O
perform	O
these	O
types	O
of	O
queries	O
.	O
</s>
<s>
Decision	B-Algorithm
trees	I-Algorithm
are	O
often	O
employed	O
to	O
understand	O
algorithms	O
for	O
sorting	O
and	O
other	O
similar	O
problems	O
;	O
this	O
was	O
first	O
done	O
by	O
Ford	O
and	O
Johnson	O
.	O
</s>
<s>
For	O
example	O
,	O
many	O
sorting	O
algorithms	O
are	O
comparison	B-Algorithm
sorts	I-Algorithm
,	O
which	O
means	O
that	O
they	O
only	O
gain	O
information	O
about	O
an	O
input	O
sequence	O
via	O
local	O
comparisons	O
:	O
testing	O
whether	O
,	O
,	O
or	O
.	O
</s>
<s>
These	O
algorithms	O
can	O
be	O
modeled	O
as	O
binary	O
decision	B-Algorithm
trees	I-Algorithm
,	O
where	O
the	O
queries	O
are	O
comparisons	O
:	O
an	O
internal	O
node	O
corresponds	O
to	O
a	O
query	O
,	O
and	O
the	O
node	O
's	O
children	O
correspond	O
to	O
the	O
next	O
query	O
when	O
the	O
answer	O
to	O
the	O
question	O
is	O
yes	O
or	O
no	O
.	O
</s>
<s>
For	O
leaf	O
nodes	O
,	O
the	O
output	O
corresponds	O
to	O
a	O
permutation	B-Algorithm
that	O
describes	O
how	O
the	O
input	O
sequence	O
was	O
scrambled	O
from	O
the	O
fully	O
ordered	O
list	O
of	O
items	O
.	O
</s>
<s>
(	O
The	O
inverse	O
of	O
this	O
permutation	B-Algorithm
,	O
,	O
re-orders	O
the	O
input	O
sequence	O
.	O
)	O
</s>
<s>
One	O
can	O
show	O
that	O
comparison	B-Algorithm
sorts	I-Algorithm
must	O
use	O
comparisons	O
through	O
a	O
simple	O
argument	O
:	O
for	O
an	O
algorithm	O
to	O
be	O
correct	O
,	O
it	O
must	O
be	O
able	O
to	O
output	O
every	O
possible	O
permutation	B-Algorithm
of	O
elements	O
;	O
otherwise	O
,	O
the	O
algorithm	O
would	O
fail	O
for	O
that	O
particular	O
permutation	B-Algorithm
as	O
input	O
.	O
</s>
<s>
So	O
,	O
its	O
corresponding	O
decision	B-Algorithm
tree	I-Algorithm
must	O
have	O
at	O
least	O
as	O
many	O
leaves	O
as	O
permutations	B-Algorithm
:	O
leaves	O
.	O
</s>
<s>
Any	O
binary	O
tree	O
with	O
at	O
least	O
leaves	O
has	O
depth	O
at	O
least	O
,	O
so	O
this	O
is	O
a	O
lower	O
bound	O
on	O
the	O
run	O
time	O
of	O
a	O
comparison	B-Algorithm
sorting	I-Algorithm
algorithm	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
existence	O
of	O
numerous	O
comparison-sorting	O
algorithms	O
having	O
this	O
time	O
complexity	O
,	O
such	O
as	O
mergesort	B-Algorithm
and	O
heapsort	B-Application
,	O
demonstrates	O
that	O
the	O
bound	O
is	O
tight	O
.	O
</s>
<s>
This	O
argument	O
does	O
not	O
use	O
anything	O
about	O
the	O
type	O
of	O
query	O
,	O
so	O
it	O
in	O
fact	O
proves	O
a	O
lower	O
bound	O
for	O
any	O
sorting	O
algorithm	O
that	O
can	O
be	O
modeled	O
as	O
a	O
binary	O
decision	B-Algorithm
tree	I-Algorithm
.	O
</s>
<s>
As	O
a	O
result	O
,	O
this	O
also	O
works	O
for	O
randomized	O
decision	B-Algorithm
trees	I-Algorithm
as	O
well	O
.	O
</s>
<s>
Other	O
decision	B-Algorithm
tree	I-Algorithm
lower	O
bounds	O
do	O
use	O
that	O
the	O
query	O
is	O
a	O
comparison	O
.	O
</s>
<s>
A	O
similar	O
argument	O
works	O
for	O
general	O
lower	O
bounds	O
for	O
computing	O
order	B-General_Concept
statistics	I-General_Concept
.	O
</s>
<s>
Linear	B-Application
decision	I-Application
trees	I-Application
generalize	O
the	O
above	O
comparison	O
decision	B-Algorithm
trees	I-Algorithm
to	O
computing	O
functions	O
that	O
take	O
real	O
vectors	O
as	O
input	O
.	O
</s>
<s>
The	O
tests	O
in	O
linear	B-Application
decision	I-Application
trees	I-Application
are	O
linear	O
functions	O
:	O
for	O
a	O
particular	O
choice	O
of	O
real	O
numbers	O
,	O
output	O
the	O
sign	O
of	O
.	O
</s>
<s>
Comparison	O
trees	O
are	O
linear	B-Application
decision	I-Application
trees	I-Application
,	O
because	O
the	O
comparison	O
between	O
and	O
corresponds	O
to	O
the	O
linear	O
function	O
.	O
</s>
<s>
From	O
its	O
definition	O
,	O
linear	B-Application
decision	I-Application
trees	I-Application
can	O
only	O
specify	O
functions	O
whose	O
fibers	B-Algorithm
can	O
be	O
constructed	O
by	O
taking	O
unions	O
and	O
intersections	O
of	O
half-spaces	O
.	O
</s>
<s>
Algebraic	B-Application
decision	I-Application
trees	I-Application
are	O
a	O
generalization	O
of	O
linear	B-Application
decision	I-Application
trees	I-Application
that	O
allow	O
the	O
test	O
functions	O
to	O
be	O
polynomials	O
of	O
degree	O
.	O
</s>
<s>
These	O
decision	B-Application
tree	I-Application
models	I-Application
,	O
defined	O
by	O
Rabin	O
and	O
Reingold	O
,	O
are	O
often	O
used	O
for	O
proving	O
lower	O
bounds	O
in	O
computational	O
geometry	O
.	O
</s>
<s>
For	O
example	O
,	O
Ben-Or	O
showed	O
that	O
element	O
uniqueness	O
(	O
the	O
task	O
of	O
computing	O
,	O
where	O
is	O
0	O
if	O
and	O
only	O
if	O
there	O
exist	O
distinct	O
coordinates	O
such	O
that	O
)	O
requires	O
an	O
algebraic	B-Application
decision	I-Application
tree	I-Application
of	O
depth	O
.	O
</s>
<s>
They	O
also	O
show	O
a	O
lower	O
bound	O
for	O
linear	B-Application
decision	I-Application
trees	I-Application
on	O
the	O
knapsack	O
problem	O
,	O
generalized	O
to	O
algebraic	B-Application
decision	I-Application
trees	I-Application
by	O
Steele	O
and	O
Yao	O
.	O
</s>
<s>
For	O
Boolean	O
decision	B-Algorithm
trees	I-Algorithm
,	O
the	O
task	O
is	O
to	O
compute	O
the	O
value	O
of	O
an	O
n-bit	O
Boolean	O
function	O
for	O
an	O
input	O
.	O
</s>
<s>
There	O
are	O
many	O
types	O
of	O
computational	B-Application
models	I-Application
using	O
decision	B-Algorithm
trees	I-Algorithm
that	O
could	O
be	O
considered	O
,	O
admitting	O
multiple	O
complexity	O
notions	O
,	O
called	O
complexity	O
measures	O
.	O
</s>
<s>
If	O
the	O
output	O
of	O
a	O
decision	B-Algorithm
tree	I-Algorithm
is	O
,	O
for	O
all	O
,	O
the	O
decision	B-Algorithm
tree	I-Algorithm
is	O
said	O
to	O
"	O
compute	O
"	O
.	O
</s>
<s>
,	O
the	O
deterministic	O
decision	B-Application
tree	I-Application
complexity	I-Application
of	O
is	O
the	O
smallest	O
depth	O
among	O
all	O
deterministic	O
decision	B-Algorithm
trees	I-Algorithm
that	O
compute	O
.	O
</s>
<s>
One	O
way	O
to	O
define	O
a	O
randomized	O
decision	B-Algorithm
tree	I-Algorithm
is	O
to	O
add	O
additional	O
nodes	O
to	O
the	O
tree	O
,	O
each	O
controlled	O
by	O
a	O
probability	O
.	O
</s>
<s>
Another	O
equivalent	O
definition	O
is	O
to	O
define	O
it	O
as	O
a	O
distribution	O
over	O
deterministic	O
decision	B-Algorithm
trees	I-Algorithm
.	O
</s>
<s>
is	O
defined	O
as	O
the	O
complexity	O
of	O
the	O
lowest-depth	O
randomized	O
decision	B-Algorithm
tree	I-Algorithm
whose	O
result	O
is	O
with	O
probability	O
at	O
least	O
for	O
all	O
(	O
i.e.	O
,	O
with	O
bounded	O
2-sided	O
error	O
)	O
.	O
</s>
<s>
is	O
known	O
as	O
the	O
Monte	O
Carlo	O
randomized	O
decision-tree	B-Algorithm
complexity	O
,	O
because	O
the	O
result	O
is	O
allowed	O
to	O
be	O
incorrect	O
with	O
bounded	O
two-sided	O
error	O
.	O
</s>
<s>
The	O
Las	O
Vegas	O
decision-tree	B-Algorithm
complexity	O
measures	O
the	O
expected	O
depth	O
of	O
a	O
decision	B-Algorithm
tree	I-Algorithm
that	O
must	O
be	O
correct	O
(	O
i.e.	O
,	O
has	O
zero-error	O
)	O
.	O
</s>
<s>
The	O
nondeterministic	O
decision	B-Application
tree	I-Application
complexity	I-Application
of	O
a	O
function	O
is	O
known	O
more	O
commonly	O
as	O
the	O
certificate	O
complexity	O
of	O
that	O
function	O
.	O
</s>
<s>
The	O
quantum	O
decision	B-Application
tree	I-Application
complexity	I-Application
is	O
the	O
depth	O
of	O
the	O
lowest-depth	O
quantum	O
decision	B-Algorithm
tree	I-Algorithm
that	O
gives	O
the	O
result	O
with	O
probability	O
at	O
least	O
for	O
all	O
.	O
</s>
<s>
Another	O
quantity	O
,	O
,	O
is	O
defined	O
as	O
the	O
depth	O
of	O
the	O
lowest-depth	O
quantum	O
decision	B-Algorithm
tree	I-Algorithm
that	O
gives	O
the	O
result	O
with	O
probability	O
1	O
in	O
all	O
cases	O
(	O
i.e.	O
</s>
<s>
and	O
are	O
more	O
commonly	O
known	O
as	O
quantum	O
query	B-Application
complexities	I-Application
,	O
because	O
the	O
direct	O
definition	O
of	O
a	O
quantum	O
decision	B-Algorithm
tree	I-Algorithm
is	O
more	O
complicated	O
than	O
in	O
the	O
classical	O
case	O
.	O
</s>
<s>
Finding	O
the	O
best	O
upper	O
bounds	O
in	O
the	O
converse	O
direction	O
is	O
a	O
major	O
goal	O
in	O
the	O
field	O
of	O
query	B-Application
complexity	I-Application
.	O
</s>
<s>
All	O
of	O
these	O
types	O
of	O
query	B-Application
complexity	I-Application
are	O
polynomially	O
related	O
.	O
</s>
<s>
Noam	O
Nisan	O
found	O
that	O
the	O
Monte	O
Carlo	O
randomized	O
decision	B-Application
tree	I-Application
complexity	I-Application
is	O
also	O
polynomially	O
related	O
to	O
deterministic	O
decision	B-Application
tree	I-Application
complexity	I-Application
:	O
.	O
</s>
<s>
As	O
for	O
quantum	O
decision	B-Application
tree	I-Application
complexities	I-Application
,	O
,	O
and	O
this	O
bound	O
is	O
tight	O
.	O
</s>
<s>
For	O
partial	B-Algorithm
Boolean	I-Algorithm
functions	I-Algorithm
,	O
that	O
have	O
a	O
domain	O
a	O
subset	O
of	O
,	O
an	O
exponential	O
separation	O
between	O
and	O
is	O
possible	O
;	O
the	O
first	O
example	O
of	O
such	O
a	O
problem	O
was	O
discovered	O
by	O
Deutsch	B-Algorithm
and	I-Algorithm
Jozsa	I-Algorithm
.	O
</s>
<s>
The	O
sensitivity	O
conjecture	O
is	O
the	O
conjecture	O
that	O
sensitivity	O
is	O
polynomially	O
related	O
to	O
query	B-Application
complexity	I-Application
;	O
that	O
is	O
,	O
there	O
exists	O
exponent	O
such	O
that	O
,	O
for	O
all	O
,	O
and	O
.	O
</s>
