The document discusses asymptotic analysis and asymptotic notation. It defines Big-Oh, Big-Omega, and Theta notation and provides examples. Some key points: - Asymptotic analysis examines an algorithm's behavior for large inputs by analyzing its growth rate as the input size n approaches infinity. - Common notations like O(n^2), Ω(n log n), θ(n) describe an algorithm's asymptotic growth in relation to standard functions. - O notation describes asymptotic upper bounds, Ω describes lower bounds, and θ describes tight bounds between two functions. - Theorems describe relationships between the notations and how to combine functions when using the notations.

2. Asymptotic Analysis-1

3. Asymptotic Analysis-1
Topics
 Objectives
 Asymptotic Notation
 Definitions ( Big-Oh, Big-Omega and Theta)
 Asymptotic Notation Theorems

4. Asymptotic Analysis
Objectives
 The purpose of asymptotic analysis is to examine the behavior of an algorithm for large
input size. More specifically, if T(n) is the running time for an input of size n , we would
want to know the behavior or growth rate of T(n) for very large values of n. An analysis of
algorithm for large input is referred to as asymptotic analysis.
 The asymptotic behavior of an algorithm is often compared to some standard
mathematical function, such as n2, n lg n etc The relationship or similarity of behavior is
often expressed by a special notation which is called asymptotic notation.
 The standard asymptotic notations commonly used in the analysis of algorithms are known
as O (Big Oh), Ω (Big Omega), and θ(Theta).
 Sometimes, additional notations o( small-oh) and ω( small-omega) are also used to show
the growth rates of algorithms

5. Asymptotic Notation

6. O-Notation
Definition
If f(n) is running time of an algorithm, and g(n) is some standard growth function such that
for some positive constants c and integer n0 ,
f(n) ≤ c.g(n) for all n ≥ n0
then f(n) = O(g(n)) (Read f(n) is Big-Oh of g(n) )
 The behavior of f(n) and g(n) is portrayed in the diagram. It follows that for n<n0, ,f(n) may
lie above or below g(n), but for all n ≥ n0, f(n) falls consistently below g(n)..
Trend of running time

7. O-Notation
Asymptotic Upper Bound
Asymptotic Upper Bound
If f(n) = O(g(n)), then the function g(n) is called asymptotic upper bound of f(n)
Since the worst-case running time of an algorithm is the maximum running time for any
input, it would follow that g(n) provides an upper bound on the worst running time
The notation O( g(n) ) does not imply that g(n) is the worse running time; it simply means
that worst running time would never exceed upper limit determined by g(n).
Asymptotic Upper Bound of f(n)

8. O-Notation

9. cg(n)=4n2
f(n)=3n2 + 10 n
cg(n)= 13n2
O-Notation Example
Comparison of Growth Rates
The results of analysis in the preceding examples are plotted in the diagram below.
It can be seen that the function cg(n) =4n2 ( c= 4) overshoots the function f(n)= 3n2 + 10n
for n0=10. Also, the function cg(n) =13n2 ( c =13 ) grows faster than f(n) = 3n2 + 10n for
n0=1
n0=1 n0=10
Growth of functions 13n2 and 4n2 versus the function 3n2 + 10 n

10. Set Builder O-Notation
Definition
 Consider the functions , say f1(n), f2(n), f3(n), ..fk(n) for which g(n) is the asymptotic
upper bound. By definition,
f1(n) ≤ c1 g(n) for n ≥ n1
f2(n) ≤ c2 g(n) for n ≥ n2
f3(n) ≤ c3 g(n) for n ≥ n3
……………………
fk(n) ≤ ck g(n) for ≥ nk
where c1, c2, c3 ,.. ck are constants and n1 ,n2 ,n3,.. nk are positive integers. The functions f1(n),
f2(n), f3(n), .. fk(n) are said to belong to the class O( g(n)). In set notation, the relation is
denoted by
O( g(n) )= { f1(n) , f2(n), f3(n)…,fk(n)}
 Alternatively, using set-builder notation, if
O(g(n)) ={ f(n): there exist positive constants c and n0, such that f(n) ≤ c.g(n), for all n ≥ n0 }
then, f(n) 

O(g(n))

11. Ω-Notation
Definition
If f(n) is running time of an algorithm, and g(n) is some standard growth function such that
for some positive constants c, positive integer n0 ,
c.g(n) ≤ f(n) for all n ≥ n0
then f(n) = Ω(g(n)) (Read f(n) is Big-Omega of g(n) )
 The behavior of f(n) and g(n) is portrayed in the graph. It follows that for n < n0, f(n) may
lie above or below g(n), but for all n ≥ n0, f(n) falls consistently above g(n). It also implies that
g(n) grows slower than f(n)
Trend of running time

12. Ω-Notation
Asymptotic Lower Bound
Asymptotic Lower Bound
If f(n) = Ω(g(n)), then the function g(n) is called asymptotic lower bound of f(n)
Since the best-case running time of an algorithm is the minimum running time for any input,
it would follow that g(n) provides a lower bound on best running time
As before, the notation Ω( g(n) ) does not imply that g(n) is the best running time; it simply
means that best running time would never be lower than g(n).
Asymptotic lower bound of f(n)

13. Ω-Notation

14. Set Builder Ω-Notation
 hich g(n) is the asymptotic lower bound. By definition,
f1(n) ≥ c1 g(n) for n ≥ n1
f2(n) ≥ c2 g(n) for n ≥ n2
f3(n) ≥ c3 g(n) for n ≥ n3
……………………
fk(n) ≥ ck g(n) for ≥ nk
where c1, c2, c3 ,.. ck are constants and n1 ,n2 ,n3,.. nk are positive integers. The functions f1(n),
f2(n), f3(n), .. fk(n) are said to belong to the class Ω(g(n)). In set notation, the relation is
denoted by
Ω( g(n) )= { f1(n) , f2(n), f3(n)…,fk(n)}
 Alternatively, using set-builder notation , if
Ω(g(n)) ={ f(n): there exist positive constants c and n0, such that f(n) ≥ c.g(n), for all n ≥ n0 }
then f(n) 

Ω(g(n))

15. θ-Notation
Definition
If f(n) is running time of an algorithm, and g(n) is some standard growth function
such that for some positive constants c1 , c2 and positive integer n0 ,
0 < c2.g(n) ≤ f(n) ≤ c1.g(n) for all n ≥ n0
then f(n) = θ(g(n)) (Read f(n) is theta of g(n) )
 The behavior of f(n) and g(n) is portrayed in the graph. It follows that for n < n0, f(n) may
be above or below g(n), but for all n ≥ n0, f(n) falls consistently between c1.g(n) and c2.g(n).
It also implies that g(n) grows as fast as f(n) The function g(n) is said to be the asymptotic
tight bound for f(n)
Asymptotic tight bound of f(n)
Asymptotic Upper Bound
Asymptotic Lower Bound

16. Set Builder θ-Notation
Definition
 There can be a several functions for which the g(n) is asymptotic tight bound. All such
functions are said to belong to the class Ө(g(n))
 Using set builder notation,, if
Ө(g(n)) = { f(n): there exit positive constants c1,c2 and positive integer n0 such that
c1 g(n) ≤ f(n) ≤ c2 g(n) }
then f(n)  θ(g(n))

17. Asymptotic Notation
Constant Running Time
 If running time T(n)=c is a constant, i. e independent of input size, then by convention,
the asymptotic behavior is denoted by the notation
O(c) = O(1) , θ (c) = θ(1), Ω(c) = Ω(1)
 The convention implies that the running time of an algorithm ,which does not depend
on the size of input, can be expressed in any of the above ways.
 If c is constant then using basic definition it can be shown that
O( c.f(n) ) = O( f(n) )
θ( c.f(n) ) = θ( f(n) )
Ω( c.f(n) ) = Ω( f(n ))
Example: (i) O(1000n) = O(n),
(ii) θ(7lgn ) = θ (lg n),
(ii) Ω(100 n!) = Ω(n!)
 The above results imply that in asymptotic notation the multiplier constants in an
expression for the running time can be dropped

18. Asymptotic Notations
Theorem: If f(n) = θ( g(n) ) then
Relationships
f(n) = Ω( g(n) ), and f(n)= O( g(n) )
Conversely, if f(n) = Ω( g(n) ) and f(n) = O( g(n) )
then f(n) = θ( g(n) )
 The above relations can be established by using basic definitions
Example(1): Since, n(n-1)/2 = θ(n2 ), therefore it follows that
n(n-1)/2 = Ω(n2 )
n(n-1)/2 = O(n2 )
Example(2): It can be shown that
5n2+1 = Ω(n2 )
and 5n2+1 = O(n2 )
Therefore, 5n2 + 1 = θ(n2 )

19. Asymptotic Set Notations
O(g(n)) Ө(g(n)) Ω(g(n))
Relationship
We have seen that if f(n) = O( g(n) ) and f(n) = Ω(g(n)) then f(n) = Ө( g(n) )
Using set notation we can express the relationship as follows:
if f(n)O( g(n) ),
and f(n)Ω (g(n)),
then f(n) Ө( g(n) ) ,
where Ө(g(n)) = O( g(n) ) ∩ Ω (g(n) )
 The above property is illustrated by the following Venn diagram

20. Asymptotic Notation
Order Theorem
Theorem: If f1(n) = O( g1(n) ) and f2(n) = O( g2(n) ) then
f1(n) + f2(n)= O( max( g1(n) , g2(n) )
Proof: By definition,
f1(n) ≤ c1. g1(n) for n ≥ n1
f2(n) ≤ c2. g2(n) for n ≥ n2
Let n0 = max(n1, n2) c3=max(c1, c2)
f1(n) ≤ c3. g1(n) for n ≥ n0
f2(n) ≤ c3 g2(n) for n ≥ n0
f1(n) + f2(n) ≤ c3.g1(n) + c3. g2(n) )
Let h(n) = max( g1(n) , g2(n) )
for n ≥ n0
f1(n) + f2(n) ≤ 2c3.h(n) = c. h(n) where c=2c3
f1(n) + f2(n) ≤ c. h(n) = c. max( g1(n) , g2(n) )
Therefore, f1(n) + f2(n) = O( max( g1(n) , g2(n) )
 The theorem also applies to θ and Ω notations
for n ≥ n0
for n ≥ n0

21. General Theorem
Theorem: If f1(n)= O(g1(n)) , f2(n) =O(g2(n)), f3(n)=O(g3(n))…., fk(n)=O(gk(n)) then
f1(n) + f2(n) + f3(n) ….+ fk(n) = O( max (g1 (n), g2 (n), g3 (n)…, gk(n) ) )
where max means fastest growing function
The theorem can be proved by using basic definition of Big-Oh
 It follows from the theorem that in an expression consisting of sum of several functions,
the comparatively slower growing functions can be discarded in favor of the fastest
growing function to obtain the Big oh notation for the whole expression This also true for
the Ө and Ω notations