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Asymptotic Notations Explained

Asymptotic notation is used to analyze the efficiency of algorithms as the input size grows. There are three main notations: O notation provides an asymptotic upper bound and models worst-case analysis; Ω notation provides an asymptotic lower bound and models best-case analysis; Θ notation provides both asymptotic upper and lower bounds and models average-case analysis. Examples are given to demonstrate how functions fit into each notation based on growth rate comparisons as n approaches infinity.

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112 views4 pages

Asymptotic Notations Explained

Asymptotic notation is used to analyze the efficiency of algorithms as the input size grows. There are three main notations: O notation provides an asymptotic upper bound and models worst-case analysis; Ω notation provides an asymptotic lower bound and models best-case analysis; Θ notation provides both asymptotic upper and lower bounds and models average-case analysis. Examples are given to demonstrate how functions fit into each notation based on growth rate comparisons as n approaches infinity.

Uploaded by

Nivi V
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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ASYMPTOTIC NOTATIONS AND ITS PROPERTIES

Asymptotic notation is a notation, which is used to take meaningful statement about


the efficiency of a program.
The efficiency analysis framework concentrates on the order of growth of an algorithm’s
basic operation count as the principal indicator of the algorithm’s efficiency.
To compare and rank such orders of growth, computer scientists use three notations, they
are:
 O - Big oh notation
 Ω - Big omega notation
 Θ - Big theta notation
Let t(n) and g(n) can be any nonnegative functions defined on the set of natural numbers.
The algorithm’s running time t(n) usually indicated by its basic operation count C(n), and
g(n),
some simple function to compare with the count.

Example 1:

where g(n) = n2.


(i) O - Big oh notation
A function t(n) is said to be in O(g(n)), denoted if t (n) t(n) ∈ O(g(n)),t(n)
is bounded above by some constant multiple of g(n) for all large n, i.e., if there
exist some positive constant c and some nonnegative integer n0 such that
t (n) ≤ cg(n) for all n ≥ n0.

Where t(n) and g(n) are nonnegative functions defined on the set of natural
numbers. O = Asymptotic upper bound = Useful for worst case analysis =
Loose bound

FIGURE 1.5 Big-oh notation:

Example 2: Prove the assertions


Proof: 100n + 5 ≤ 100n + n (for all n ≥ 5)
= 101n
≤ 101n2
Since, the definition gives us a lot of freedom in choosing specific values for
constants c
and n0. We have c=101 and n0=5
Example 3: Prove the assertions
Proof: 100n + 5 ≤ 100n + 5n (for
all n ≥ 1)
=
105n i.e.,
100n + 5 ≤
105n i.e.,
t(n) ≤ cg(n)

(ii) Ω - Big omega notation


A function t(n) is said to be in Ω(g(n)), denoted t(n) ∈ Ω(g(n)), if t(n) is bounded
below by some positive constant multiple of g(n) for all large n, i.e., if there exist some
positive constant c and some nonnegative integer n0 such that
t (n) ≥ cg(n) for all n ≥ n0.
Where t(n) and g(n) are nonnegative functions defined on the set of natural numbers.
Ω = Asymptotic lower bound = Useful for best case analysis = Loose bound

FIGURE 1.6 Big-omega notation: t (n) ∈ Ω (g(n)).

Example 4: Prove the assertions n3+10n2+4n+2 ∈ Ω (n2).


Proof: n3+10n2+4n+2 ≥ n2 (for all n ≥ 0)
i.e., by definition t(n) ≥ cg(n), where c=1 and n0=0

(iii) Θ - Big theta notation


A function t(n) is said to be in Θ(g(n)), denoted t(n) ∈ Θ(g(n)), if t(n) is bounded
both above and below by some positive constant multiples of g(n) for all large n,
i.e., if there exist some positive constants c1 and c2 and some nonnegative integer n0 such
that
c2g(n) ≤ t (n) ≤ c1g(n) for all n ≥ n0.
Where t(n) and g(n) are nonnegative functions defined on the set of natural numbers.
Θ = Asymptotic tight bound = Useful for average case analysis

FIGURE 1.7 Big-theta notation: t (n) ∈ Θ(g(n)).


Note: asymptotic notation can be thought of as "relational operators" for functions similar
to the corresponding relational operators for values.
= ⇒ Θ(), ≤ ⇒ O(), ≥ ⇒ Ω(), < ⇒ o(), >
⇒ ω()

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