3
Algorithm Analysis
3.1
Note that
n
is a positive integer.
5
n
log
n
is most ef
f
cient for
n
=1
.
2
n
is most ef
f
cient when
2
≤
n
≤
4
.
10
n
is most ef
f
cient for all
n>
5
.
20
n
and
2
n
are never more
ef
f
cient than the other choices.
3.2
Both
log
3
n
and
log
2
n
will have value
0
when
n
.
Otherwise, 2 is the most ef
f
cient expression for all
1
.
3.3
2
log
3
n
log
2
nn
2
/
3
20
n
4
n
2
3
n
n
!
.
3.4
(a)
n
+6
inputs (an additive amount, independent of
n
).
(b)
8
n
inputs (a multiplicative factor).
(c)
64
n
inputs.
3.5
100
n
.
10
n
.
About
4
.
6
n
(actually,
3
√
100
n
).
n
.
3.6
(a)
These questions are quite hard. If
f
(
n
)=2
n
=
x
, then
f
(2
n
2
n
=
(2
n
)
2
=
x
2
.
(b)
The answer is
2
(
n
log
2
3
)
. Extending from part (a), we need some way to
make the growth rate even higher. In particular, we seek some way to
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 Fall '08
 BELL,D
 Multiplication, Prime number, positive value, 5n log

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