DATA STRUCTURE
UNIT-1 LECTURE.NO:5
Asymptotic Notations:
The asymptotic behavior of a function is the study of how the value of a function
varies for larger values of “n” where “n” is the size of the input.
Using asymptotic behavior we can easily find the time efficiency
of an algorithm.
Different asymptotic notations are:
1.Big Ο (big oh) it is the formal method of expressing the upper bound of an
algorithms running time.
Its a measure of longest amount of time it could possibly take for the algorithm
to complete. Let f(n) is the time efficiency of an algorithm.
The function f(n) is said to be Big-oh of g(n), denoted by
f(n) € O(g(n)) OR f(n) ≈ O(g(n))
Such that there exist a +ve constant “c” and +ve integer n0 satisfying the constraint.
f(n) ≤ c * g(n) for all n≥ n0
E.g.: Consider the following assertion:
Let f(n)=100n + 5 Express f(n) using big-oh
Solution:
It is given that f(n) =100n + 5. Replacing 5 with n (so that next higher order term
is obtained), we get 100n + n and call it c* g(n).
i.e. c* g(n) = 100n + n for n = 5
= 101n for n = 5
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DATA STRUCTURE
UNIT-1 LECTURE.NO:5
Now, the following constraint is satisfied
f(n) ≤ c * g(n) for all n≥ n0
i.e. 100n + 5 <= 101* n
for n>= 5 it is clear from the above relation that c=101,
g(n)=n and n0 = 5, so by definition f(n) € O(g(n)) i.e. f(n) € O(n)
2. Ω notation (Big-Omega)
Let f(n) be the time complexity of an algorithm. The function f(n) is said to be Big-Omega of g(n)
which is denoted by
f(n) € Ω(g(n)) OR f(n) ≈ Ω(g(n))
such that there exist a +ve constant “c” and non negative integer n0 satisfying the constraint
f(n) ≥ c * g(n) for all n ≥ n0
f(n) ≥ c * g(n) for all n≥ n0
for n>= 0
i.e.
100n + 5 ≥ 100 * n
it is clear from the above relation that c = 100, g(n) = n and n0 ,So by definition f(n)
€ Ω(g(n)) i.e. f(n) ≈ Ω(n)
3
Ex: let f(n) = 10n3
+5
10n + 5 ≥ 10 *
3
n for n ≥ 0
3
f(n) € Ω(n )
3. Θ notation (Theta)
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UNIT-1 LECTURE.NO:5
Let f(n) be the time complexity of an algorithm. The function f(n) is said to be Big-Theta of g(n)
which is denoted by f(n) € θ(g(n)) OR f(n) ≈ θ (g(n))
such that there exist a +ve constant“c1, c2” and non negative integer n0 satisfying the constraint
c2g(n) ≤ f(n) ≥ c1g(n) for all n ≥ n0
Ex: Let f(n) = 100n + 5 express f(n) using big-theta
c2g(n) ≤ f(n) ≥ c1g(n) for all n ≥ n0
100 * n ≤ 100n + 5 ≥ 100n+n
for n ≥ 5
It is clear from the above relation that c1=100, c2=105, n0= 5, g(n)= n So by definition
f(n) € θ(g(n)) f(n) € θ(n)
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