Asymptotic Notation

 What is the goal of analysis of algorithms?
― To compare algorithms mainly in terms of running time but
also in terms of other factors (e.g., memory requirements,
programmer's effort etc.)
 What do we mean by running time analysis?
― Determine how running time increases as the size of the
problem increases.
An algorithm is a finite set of precise instructions for
performing a computation or for solving a problem.

Worst case
⍟Provides an upper bound on running time
⍟An absolute guarantee that the algorithm would not run longer, no
matter what the inputs are
Best case
⍟Provides a lower bound on running time
⍟Input is the one for which the algorithm runs the fastest
Average case
⍟Provides a prediction about the running time
⍟Assumes that the input is random

• To compare two algorithms with running times f(n) and g(n),
we need a rough measure that characterizes how fast each function grows.
• Express running time as a function of the input size n (i.e., f(n)).
• Compare different functions corresponding to running times.
• Such an analysis is independent of machine time, programming style,etc.
• Compare functions in the limit, that is, asymptotically!
(i.e., for large values of n)

Asymptotic Notation
A way to describe the behavior of functions in
the limit or without bounds.
The notations are defined in terms of functions whose domains
are the set of natural numbers N={0,1,2,...}.
Such notations are convenient for describing the worst-case
running time function T(n).
It can also be extended to the domain of real numbers.

Example : -
x is asymptotic with x + 1
limit - lim
𝑥→∞
𝑓 𝑥 = 𝑘
Roughly translated might read as:
x approaches ∞, f(x) approaches k for
x close to ∞, f(x) is close to k
Two limits often used in analysis are:
lim
𝑥→∞
𝑓 𝑥 = lim
𝑥→∞
𝑓 𝑥 1/x = 0
lim
𝑥→∞
𝑓 𝑥 = lim
𝑥→∞
𝑓 𝑥 cx = ∞ for c>0
Asymptotic Notation

possibly asymptotically tight upper
bound for f(n) - Cannot do worse, can
do better
n is the problem size.
f(n) ∈ O(g(n)) where:
Meaning for all values of n ≥ 𝑛0f(n) is
on or below g(n).
O(g(n)) is a set of all the functions f(n)
that are less than or equal to cg(n), ∀ n
≥𝑛0.
Big-O Notation (Omicron)
O(g(n)) = { f(n): ∃ positive constants c,
𝑛0such that 0 ≤ f(n) ≤ cg(n), ∀ n ≥𝑛0}
If f(n) ≤ cg(n), c > 0, ∀ n ≥𝑛0then f(n) ∈ O(g(n))

Example of Big O notation
𝐒𝐡𝐨𝐰 𝟐𝐧 𝟐 = 𝐎 𝐧 𝟑 𝟎 ≤ 𝐟 𝐧 ≤ 𝐜𝐠 𝐧 𝐃𝐞𝐟𝐢𝐧𝐚𝐭𝐢𝐨𝐧 𝐨𝐟 𝐎(𝐠 𝐧 )
Solution :-
0 ≤ 2n2
≤ cn3
0/n3≤ 2n2 /n3≤ cn3/n3
Determine C
0 ≤ 2/n ≤ c
0 ≤ 2/1 ≤ c = 2
Substitute
Divide by n3
lim
𝑛→∞
2/𝑛 = 0
2/n maximum when n=1
Satisfied by c=2
1000𝒏 𝟐 + 50n =
O(𝒏 𝟐
)
with c=1050 and 𝐧 𝟎=1
Satisfied by 𝑛0=1
∀ n ≥ 𝑛0=1
If f(n) ≤ cg(n), c > 0, ∀ n ≥ 𝒏 𝟎then f(n) ∈ O(g(n))
Determine 𝒏 𝟎
0 ≤ 2/𝑛0 ≤ 2
0 ≤ 2/2 ≤ 𝑛0
0 ≤ 1 ≤ 𝑛0= 1
0 ≤ 2n2
≤ 2n3

Big Omega Notation (Ω)
possibly asymptotically tight lower bound
for f(n) - Cannot do better, can do worse
f(n) ∈ Ω(g(n)) where:
Meaning for all values of n ≥ 𝑛0f(n) is on or
above g(n).
Ω(g(n)) is a set of all the functions f(n) that are
greater than or equal cg(n), ∀ n ≥𝑛0.
Ω(g(n)) = {f(n): ∃ positive constants c >
0, 𝑛0 such that 0 ≤ cg(n) ≤ f(n), ∀ n ≥𝑛0}
If cg(n) ≤ f(n), c > 0 and ∀ n ≥𝑛0, then f(n) ∈ Ω(g(n))

Example of Ω notation
Solution :-
0 ≤ cg(n) ≤ f(n)
0 ≤ c n2 ≤ 3 n2 + n
0/ n2 ≤ c n2 / n2 ≤ 3 n2 / n2 + n/ n2
0 ≤ c ≤ 3 + 1/n
0 ≤ c ≤ 3
0 ≤ 3 ≤ 3 + 1/ 𝑛0
-3 ≤ 3-3 ≤ 3-3 + 1/ 𝑛0
-3 ≤ 0 ≤ 1/ 𝑛0
𝑛0 =1 satisfies
log 𝑛→∞ 1/𝑛
log 𝑛→∞ 3 + 1 /𝑛 = 3
c = 3
𝐒𝐡𝐨𝐰that 3 𝒏 𝟐
+ n = Ω(𝒏 𝟐
)
3𝒏 𝟐 + n = Ω(𝒏 𝟐 ) with
c=1 and 𝒏 𝟎 =1

Big Theta Notation()
asymptotically tight bound for f(n)
f(n) ∈ (g(n)) where :
Positive means greater than 0.
(g(n)) is a set of all the functions f(n)
that are between 𝑐1 g(n) and 𝑐2g(n),
∀ n ≥ 𝑛0.
If f(n) is between 𝑐1 g(n) and 𝑐2g(n), ∀ n
≥ 𝑛0, then f(n) ∈ (g(n))
(g(n)) =
{f(n): ∃ positive constantsc1,c2, n0such that
0 ≤ c1 g(n) ≤ f(n) ≤ c2g(n), ∀ n ≥ n0}

Example of  notation
Θ: ½n2
- 3n ∈ Θ(n2
) when c1= 1/14, c2= ½ and𝑛0 = 7
Divide by n2
c1n2 ≤ ½n2
- 3n ≤ c2n2
c1≤ ½ - 3/n ≤ c2
O: Determine c2= ½
½ - 3/n ≤ c2
as n → ∞, 3/n → 0
lim
𝑛→∞
1/𝑛 − 3/𝑛 = ½
therefore ½ ≤ c2 o rc2 = ½ maximum for as n → ∞ Ω: Determine c1= 1/14
0 <c1≤ ½ - 3/n
- 3/n > 0 minimum for 𝑛0 = 7
0 <c1= ½ - 3/7 = 7/14 - 6/14 = 1/14
𝑛0: Determine 𝑛0 = 7
c1≤ ½ - 3/ 𝑛0 ≤ c2
1/14 ≤ ½ - 3/𝑛0 ≤ ½
-6/14 ≤ -3/𝑛0 ≤ 0
-𝑛0 ≤ -3*14/6 ≤ 0
𝑛0 ≥ 42/6 ≥ 0
𝑛0 ≥ 7
Solution :-
Show that ½𝒏 𝟐
- 3n ∈ Θ(𝒏 𝟐
) Determine ∃ 𝒄 𝟏, 𝒄 𝟐, 𝒏 𝟎positive constants such that:

Relations Between , W, O
Theorem : For any two functions g(n) and f(n),
f(n) = (g(n)) iff
f(n) = O(g(n)) and f(n) = W(g(n)).
 I.e., (g(n)) = O(g(n))  WW(g(n))
 In practice, asymptotically tight bounds are obtained from asymptotic
upper and lower bounds.

Little-o Notation (omicron)
non-asymptotically tight upper bound for f(n)
f(n) ∈ o(g(n)) where :
lim
𝑛→∞
f(n) / g(n) = 0
for any 0 < c < ∞
o(g(n)) = {f(n): for any constant c >
0, ∃ a constant n0 >0, such that: 0
≤ f(n) <cg(n), ∀ n ≥n0}
o(g(n)) = {f(n): for any constant c > 0, ∃ a constant
n0 >0,
such that: 0 ≤ f(n) <cg(n), ∀ n ≥n0}
o(f(n))
f(n)
n0

Little-ω Notation (omega)
non-asymptotically tight lower bound for f(n)
f(n) ∈ ω(g(n)) where :
for any 0 < c ≤ ∞
lim
𝑛→∞
f(n) / g(n) = c
for some 0 < c ≤ ∞
Ω possibly asymptotically tight lower bound
ω non-asymptotically tight lower bound
ω(g(n)) = {f(n): for any constant c > 0, ∃ a constant n0 >0,
such that 0 ≤ cg(n) <f(n), ∀ n ≥n0}
Ω(g(n)) = {f(n): for some constant c > 0, ∃ a
constant n0 >0, such that 0 ≤ cg(n) ≤ f(n),
∀ n ≥n0}
lim
𝑛→∞
f(n) / g(n) = ∞
(f(n))
f(n)
n0

Comparison of Functions
 An imprecise analogy between the asymptotic comparison of
two function f and g and the relation between their values:
f(n) = O(g(n)) ≈ f(n) ≤ g(n)
f(n) = Ω(g(n)) ≈ f(n) ≥ g(n)
f(n) = Θ(g(n)) ≈ f(n) = g(n)
f(n) = o(g(n)) ≈ f(n) < g(n)
f(n) = ω(g(n)) ≈ f(n) > g(n)

lim [f(n) / g(n)] = 0  f(n)  o(g(n))
n
lim [f(n) / g(n)] <   f(n)  O(g(n))
n
0 < lim [f(n) / g(n)] <   f(n)  (g(n))
n
0 < lim [f(n) / g(n)]  f(n) W(g(n))
n
Lim [f(n) / g(n)] =   f(n)  (g(n))
n
lim [f(n) / g(n)] undefined can’t say
n
Limits

Properties
 Transitivity
f(n) = (g(n)) & g(n) = (h(n))  f(n) = (h(n))
f(n) = O(g(n)) & g(n) = O(h(n))  f(n) = O(h(n))
f(n) = W(g(n)) & g(n) = W(h(n))  f(n) = W(h(n))
f(n) = o (g(n)) & g(n) = o (h(n))  f(n) = o (h(n))
f(n) = (g(n)) & g(n) = (h(n))  f(n) = (h(n))
 Reflexivity
f(n) = (f(n))
f(n) = O(f(n))
f(n) = W(f(n))
 Symmetry
f(n) = (g(n)) iff g(n) = (f(n))
 Complementarity
f(n) = O(g(n)) iff g(n) = W(f(n))
f(n) = o(g(n)) iff g(n) = ((f(n))

Asymptotic notation in
equations and inequalities
 On the right-hand side :-
(n2) stands for some anonymous function in (n2)
2n2 + 3n + 1 = 2n2 + (n) means:
There exists a function f(n)  (n) such that
2n2 + 3n + 1 = 2n2 + f(n)
 On the left-hand side :-
2n2 + (n) = (n2) means :
No matter how the anonymous function is chosen on the left-
hand side, there is a way to choose the anonymous function
on the right-hand side to make the equation valid.

Class Name Comment
1 constant – Instructions are executed once or a few times
log n logarithmic – A big problem is solved by cutting the original problem in smaller
sizes, by a constant fraction at each step
n linear – A small amount of processing is done on each input element
n log n linearithmic – A problem is solved by dividing it into smaller problems, solving
them independently and combining the solution
𝒏 𝟐 quadratic – Typical for algorithms that process all pairs of data items (double
nested loops)
𝒏 𝟑 cubic – Processing of triples of data (triple nested loops)
𝟐 𝒏 exponential – Few exponential algorithms are appropriate for practical use
n! factorial – Typical for algorithms that generate all permutations of an n-element
set.

Acknowledgement
Firstly , I would like to thanks Mr. Abhishek
Mukhopadhyay for giving us such an interesting
topic to work with and Of course last but not the
least I express my esteemed gratitude to my group
members Meraj,Papiya,Nazaan,Protap who put up
with me during the whole period and provided me
valuable guidance in times of need and gave
wonderful cooperation and support to complete
the project.

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