Absolutely Dominated
Let , we say that
g
is absolutely dominated by
f
iff for all
n
∈
N
0
,
g
(
n
)
≤
f
(
n
)
Absolutely Dominated up to a Constant Factor
Let
f
,
g
:
N
0
→
R
≥
0
, we say that
g
is
absolutely dominated up to a constant factor by
f
iff there exists
c
∈
R
+
such that
g
is
absolutely dominated by
c
f
Eventually Dominated
Let
f
,
g
:
N
0
→
R
≥
0
, we say that
g
is eventually dominated by
f
iff there exists some
n
0
∈
R
+
such that
∀
n
∈
N
+
,
n
≥
n
0
then we know
g
(
n
)
≤
f
(
n
)
Lower Degree Polynomial plus Constant is Eventually Dominated by Higher Degree
Let
f
(
n
)
=
n
2
and
g
(
n
)
=
n
+
90
Eventually Dominated up to a Constant Factor
Let
f
,
g
:
N
0
→
R
≥
0
, we say that
g
is
eventually dominated by
f
iff there exists some
c
∈
R
+
such that
g
is
eventually dominated by
c
f
Big-O of
Let
f
,
g
:
N
0
→
R
≥
0
, we say that
g
is
big-o of
f
iff
g
∈
O
(
f
)
Note that if
g
is big-o of
f
, then we know that
f
is an asymptotic upper bound for
g
Basic Polynomial
Suppose
f
(
n
)
=
n
3
and
g
(
n
)
=
n
3
+
100
n
+
5000
, show that
g
∈
O
(
f
)
TODO
Omega
Suppose we have
f
:
N
0
→
R
≥
0
, then we define the set
Ω
(
f
)
as the set of all functions
g
:
N
0
→
R
≥
0
such that
∃
c
,
n
0
∈
R
+
s.t.
∀
n
∈
N
0
,
n
≥
n
0
⟹
g
(
n
)
≥
c
⋅
f
(
n
)
Omega of
Let
f
,
g
:
N
0
→
R
≥
0
, we say that
g
is
omega of
f
iff
g
∈
Ω
(
f
)
Here we can think of
Ω
as being
O
's counter part, so that when
g
is omega of
f
, then
f
is a lower bound on
g
's eventual growth rate.
Theta
Suppose we have
f
:
N
0
→
R
≥
0
, then we define the set
Θ
(
f
)
as the set of all functions
g
:
N
0
→
R
≥
0
such that
g
∈
O
(
f
)
∩
Ω
(
f
)
where we are using
big-o and
theta
The point of theta is to say that
g
is both eventually upper and lower bounded by
f
which means that
g
should grow asymptotically the same as
f