<s>
In	O
the	O
analysis	B-General_Concept
of	I-General_Concept
algorithms	I-General_Concept
,	O
the	O
master	B-Algorithm
theorem	I-Algorithm
for	O
divide-and-conquer	O
recurrences	O
provides	O
an	O
asymptotic	O
analysis	B-General_Concept
(	O
using	O
Big	O
O	O
notation	O
)	O
for	O
recurrence	O
relations	O
of	O
types	O
that	O
occur	O
in	O
the	O
analysis	B-General_Concept
of	O
many	O
divide	B-Algorithm
and	I-Algorithm
conquer	I-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
The	O
name	O
"	O
master	B-Algorithm
theorem	I-Algorithm
"	O
was	O
popularized	O
by	O
the	O
widely-used	O
algorithms	O
textbook	O
Introduction	O
to	O
Algorithms	O
by	O
Cormen	O
,	O
Leiserson	O
,	O
Rivest	O
,	O
and	O
Stein	O
.	O
</s>
<s>
The	O
master	B-Algorithm
theorem	I-Algorithm
allows	O
many	O
recurrence	O
relations	O
of	O
this	O
form	O
to	O
be	O
converted	O
to	O
Θ-notation	O
directly	O
,	O
without	O
doing	O
an	O
expansion	O
of	O
the	O
recursive	O
relation	O
.	O
</s>
<s>
The	O
master	B-Algorithm
theorem	I-Algorithm
always	O
yields	O
asymptotically	O
tight	O
bounds	O
to	O
recurrences	O
from	O
divide	B-Algorithm
and	I-Algorithm
conquer	I-Algorithm
algorithms	I-Algorithm
that	O
partition	O
an	O
input	O
into	O
smaller	O
subproblems	O
of	O
equal	O
sizes	O
,	O
solve	O
the	O
subproblems	O
recursively	O
,	O
and	O
then	O
combine	O
the	O
subproblem	O
solutions	O
to	O
give	O
a	O
solution	O
to	O
the	O
original	O
problem	O
.	O
</s>
<s>
Master	B-Algorithm
Theorem	I-Algorithm
boundNotational	O
examples	O
1	O
Work	O
to	O
split/recombine	O
a	O
problem	O
is	O
dwarfed	O
by	O
subproblems	O
.	O
</s>
<s>
So	O
it	O
follows	O
from	O
the	O
second	O
case	O
of	O
the	O
master	B-Algorithm
theorem	I-Algorithm
:	O
</s>
<s>
So	O
it	O
follows	O
from	O
the	O
third	O
case	O
of	O
the	O
master	B-Algorithm
theorem	I-Algorithm
:	O
</s>
<s>
The	O
following	O
equations	O
cannot	O
be	O
solved	O
using	O
the	O
master	B-Algorithm
theorem	I-Algorithm
:	O
</s>
<s>
Therefore	O
,	O
the	O
difference	O
is	O
not	O
polynomial	O
and	O
the	O
basic	O
form	O
of	O
the	O
Master	B-Algorithm
Theorem	I-Algorithm
does	O
not	O
apply	O
.	O
</s>
<s>
Algorithm	O
Recurrence	O
relationship	O
Run	O
time	O
Comment	O
Binary	O
search	O
Apply	O
Master	B-Algorithm
theorem	I-Algorithm
case	O
,	O
where	O
Dartmouth	O
College	O
,	O
</s>
