The University of Texas at Austin
Lecture 2
Department of Computer Sciences
Professor Vijaya Ramachandran
Growth of functions; asymptotic analysis; summations
CS357: ALGORITHMS, Spring 2006
Growth of Functions
A function
f
(
n
)is
asymptotically nonnegative
if
f
(
n
) is nonnegative for all values suﬃciently large
values of
n
(i.e., for all values of
n
above a certain value).
‘BigOh’
.
O
(
g
(
n
)) =
{
f
(
n
) : there exist constants
c>
0
,n
0
>
0 such that 0
≤
f
(
n
)
≤
cg
(
n
)
∀
n
≥
n
0
}
Note.
O
(
g
(
n
)) refers to a
set of functions
, yet (by convention) we will use the equality symbol
(‘=’) when dealing with bigOh (the same holds for bigOmega and Theta), i.e., we will say
f
(
n
)=
O
(
g
(
n
)) when we should actually say that
f
(
n
)
∈
O
(
g
(
n
)). It is important to keep this distinction
in mind.
‘BigOmega’
.
Ω(
g
(
n
)) =
{
f
(
n
) : there exist constants
c>
0
,n
0
>
0 such that 0
≤
cg
(
n
)
≤
f
(
n
)
∀
n
≥
n
0
}
‘Theta’
.
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 Spring '06
 Ramachandran
 Calculus, Algorithms, Mathematical Series, Professor Vijaya Ramachandran

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