This document discusses asymptotic notations which are mathematical tools used to analyze the time and space complexity of algorithms. It introduces Big O, Big Omega, and Big Theta notations. Big O notation represents the upper bound and worst case time complexity. Big Omega notation represents the lower bound and best case time complexity. Big Theta notation defines the average time complexity of an algorithm. Examples are provided for how to determine the asymptotic notation of polynomial functions.

1. ASYMPTOTIC NOTATIONS
NADAR SARAWATHI COLLEGE OF ARTS
AND SCIENCE.

2. Asymptotic notations are mathematical tools to
represent time complexity of algorithms for
asymptotic analysis.
Asymptotic notations are the mathematical
representations of space and time complexity of an
algorithm.
It’s used to measure speed and efficiency.

3. Big Oh(0)
Big Omega(W)
BigTheta(q)

4. Big Oh notation is a mathematical notation that
describes the limiting behaviour of a function when
the argument tends towards a particular value or
infinity.
This notation is known as the upper bound of the
algorithm, or a worst case of an algorithm.
It tells us that a certain function will never exceed a
specified time for any value of input n.

5. If f(x)=anxn+an-1*xn-1+…..+a1x+a0 Where
aieR,i=0,….,n; then f(x) is 0(xn).
If x>1 we have
|f(x)|=|anxn+an-1*xn-1+…..+a1x+a0|
<|an|xn+|an-1|xn-1+…..+|a1|x+|a0|
=xn(|an|+|an-1|/x+……+|a1|/xn-1+|a0|/xn)
<xn(|an|+|an-1|+…..+|a1|+|a0|)
So, |f(x)|<cxn where c=|an|+|an-1|+…..+|a1|+|a0|

7. Big Omega notation is used to define the lower
bound of any algorithm or we can say the best case
of any algorithm.
It’s always indicates the minimum time required for
any algorithm for all input values.
It is used to find out the least time for an algorithm
to takes an completion.

8. FeW(g) if f lim infn-> |f(n)/g(n)|>0.
If lim inf |f(n)/g(n)|=c>0, then we have for
each e>0 at most finitely many positive
integers satisfying |f(n)/g(n)|<C-e.Thus, there
exists an n0 such that.
|f(n)|>(C-e)|g(n)|
holds for all n>n0, proving that feW(g).

10. TheTheta notation bounds a functions from above
and below, so it defines exact asymptotic behaviour.
It’s used to average time for an algorithm it takes
place an complete an or execution.

11. Let f(x)=anxn+an-1xn-1+…..+a1x+a0
Where a0,a1,…..,an are real numbers with
an=0.Then f(x) is of order xn(or Q(xn))
The polynomial f(x)=8x5+5x2+10 is order of
x5(or Q(x5)).
The polynomial f(x)=8x199+7x100+x99+5x2+25 is
order of x199(or Q(x199)).