CMPS 101
Midterm 1
Review Problems
1.
Let
)
(
n
f
and
)
(
n
g
be asymptotically nonnegative functions which are defined on the positive integers.
a.
State the definition of
))
(
(
)
(
n
g
O
n
f
=
.
b.
State the definition of
))
(
(
)
(
n
g
n
f
ω
=
2.
State whether the following assertions are true or false.
If any statements are false, give a related
statement which is true.
a.
))
(
(
)
(
n
g
O
n
f
=
implies
))
(
(
)
(
n
g
o
n
f
=
.
b.
))
(
(
)
(
n
g
O
n
f
=
if and only if
))
(
(
)
(
n
f
n
g
Ω
=
.
c.
))
(
(
)
(
n
g
n
f
Θ
=
if and only if
L
n
g
n
f
n
=
∞
→
))
(
/
)
(
(
lim
, where
∞
<
<
L
0
.
3.
Prove that
))
(
)
(
(
))
(
(
))
(
(
n
g
n
f
n
g
n
f
⋅
Θ
=
Θ
⋅
Θ
.
In other words, if
))
(
(
)
(
1
n
f
n
h
Θ
=
and
))
(
(
)
(
2
n
g
n
h
Θ
=
,
then
))
(
)
(
(
)
(
)
(
2
1
n
g
n
f
n
h
n
h
⋅
Θ
=
⋅
.
4.
Let
)
(
n
f
and
)
(
n
g
be asymptotically positive functions (i.e.
0
)
(
>
n
f
and
0
)
(
>
n
g
for all sufficiently
large
n
), and suppose that
))
(
(
)
(
n
g
n
f
Θ
=
.
Does it necessarily follow that
Θ
=
)
(
1
)
(
1
n
g
n
f
?
Either
prove this statement, or give a counterexample.
5.
Let
)
(
n
g
be an asymptotically nonnegative function.
Prove that
∅
=
Ω
∩
))
(
(
))
(
((
n
g
n
g
o
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Tantalo,P
 Recurrence relation, Asymptotic analysis, Master Theorem, recurrence formula, asymptotic growth rates

Click to edit the document details