18
Chap. 3 Algorithm Analysis
3.8
Other values for
n
0
and
c
are possible than what is given here.
(a)
The upper bound is
O(
n
)
for
n
0
>
0
and
c
=
c
1
. The lower bound is
Ω(
n
)
for
n
0
>
0
and
c
=
c
1
.
(b)
The upper bound is
O(
n
3
)
for
n
0
>c
3
and
c
=
c
2
+1
. The lower
bound is
Ω(
n
3
)
for
n
0
>c
3
and
c
=
c
2
.
(c)
The upper bound is
O(
n
log
n
)
for
n
0
>c
5
and
c
=
c
4
+1
. The lower
bound is
Ω(
n
log
n
)
for
n
0
>c
5
and
c
=
c
4
.
(d)
The upper bound is
O(2
n
)
for
n
0
>c
7
100
and
c
=
c
6
+1
.T
h
e
lower bound is
Ω(2
n
)
for
n
0
>c
7
100
and
c
=
c
6
. (100 is used for
convenience to insure that
2
n
>n
6
)
3.9
(a)
f
(
n
)=Θ(
g
(
n
))
since
log
n
2
= 2log
n
.
(b)
f
(
n
)
is in
Ω(
g
(
n
))
since
n
c
grows faster than
log
n
c
for any
c
.
(c)
f
(
n
)
is in
Ω(
g
(
n
))
. Dividing both sides by
log
n
, we see that
log
n
grows faster than
1
.
(d)
f
(
n
)
is in
Ω(
g
(
n
))
. If we take both
f
(
n
)
and
g
(
n
)
as exponents for 2,
we get
2
n
on one side and
2
log
2
n
=(2
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 Fall '08
 BELL,D
 NC, upper bound, Logarithm, Binary logarithm, The Loop

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