Asymptotic analysis

UCSE010 - DESIGN AND ANALYSIS OF ALGORITHMS
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Dr.Nisha Soms/SRIT
Semester-06/2020-2021

• There are three types of analysis that we
perform on a particular algorithm.
• Best Case indicates the minimum time required for
program execution.
• For example, the best case for a sorting algorithm would be
data that's already sorted.
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Dr.Nisha Soms/SRIT

• Average Case indicates the average time required for
program execution.
• Finding average case can be very difficult.
• Worst Case indicates the maximum time required for
program execution.
• For example, the worst case for a sorting algorithm might
be data that's sorted in reverse order (but it depends on the
particular algorithm).
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Dr.Nisha Soms/SRIT

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Dr.Nisha Soms/SRIT
Ref: https://www.cse.iitd.ac.in/~mausam/courses/col106/autumn2017/lectures/02-
asymptotic.pdf

 The standard notations used to define the time
required and the space required by the algorithm.
 The word Asymptotic means approaching a value or
curve arbitrarily closely (i.e., as some sort of limit is
taken).
 There are three types of asymptotic notations to
represent the growth of any algorithm, as input
increases:
▪ Big Theta (Θ)
▪ Big Oh(O)
▪ Big Omega (Ω)
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Dr.Nisha Soms/SRIT

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Semester-06/2020-2021 Dr.Nisha Soms/SRIT
 f(n) is O(g(n)), if there exists constants c and
n0 , s.t. f(n) ≤ c g(n) for all n ≥ n0
 f(n) and g(n) are functions over non-negative
integers

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 The notation Ο(n) is the formal way to express
the upper bound of an algorithm's running
time.
 It measures the worst case time complexity
or the longest amount of time an algorithm can
possibly take to complete.

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 f(n) is Ω(g(n)) if there exists constants c and
n0, such that c g(n) ≤ f(n) for n ≥ n 0

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 The notation Ω(n) is the formal way to express
the lower bound of an algorithm's running
time.
 It measures the best case time complexity or
the best amount of time an algorithm can

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 f(n) is Ɵ(g(n)) if there exists constants c1 , c2 ,
and n0, such that c1 g(n) ≤ f(n) ≤ c2 g(n) for n
≥ n0

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 The notation θ(n) is the formal way to express
both the lower bound and the upper bound of
an algorithm's running time.
 It measures the average case time complexity
average amount of time an algorithm can

 Given f(n) and g(n), which grows faster?
 If Lim n→∞ f(n)/g(n) = 0, then g(n) is faster
 If Lim n→∞ f(n)/g(n) = ∞, then f(n) is faster
 If Lim n→∞ f(n)/g(n) = non-zero constant,
then both grow at the same rate
 Solved Problems are available in
https://www.cse.wustl.edu/~sg/CS241_FL99/h
w1-practice.html
Semester-06/2020-2021 Dr.Nisha Soms/SRIT 12

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 General Property:
If f(n) is O(g(n)) then a*f(n) is also O(g(n)) ; where a is a
constant.
 Example:
f(n) = 2n²+5 is O(n²) then 7*f(n) = 7(2n²+5)
= 14n²+35 is also O(n²)
 Similarly, this property satisfies both Θ and Ω notation.
 We can say
 If f(n) is Θ(g(n)) then a*f(n) is also Θ(g(n)); where a is a constant.
 If f(n) is Ω (g(n)) then a*f(n) is also Ω (g(n)); where a is a
constant.

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 Reflexive Property:
If f(n) is given then f(n) = O(f(n))
 Example:
If f(n) = n3 ⇒ O(n3)
 Similarly,
 f(n) = Ω(f(n))
 f(n) = Θ(f(n))

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 Transitive Property:
If f(n) is O(g(n)) and g(n) is O(h(n)) then f(n) =
O(h(n)) .
 Example: if f(n) = n , g(n) = n² and h(n)=n³
n is O(n²) and n² is O(n³) then n is O(n³)
 Similarly this property satisfies for both Θ and Ω
notation.
 We can say
 If f(n) is Θ(g(n)) and g(n) is Θ(h(n)) then f(n) = Θ(h(n))
 If f(n) is Ω (g(n)) and g(n) is Ω (h(n)) then f(n) = Ω (h(n))

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 Symmetric Property:
If f(n) is Θ(g(n)) then g(n) is Θ(f(n)) .
 Example: f(n) = n² and g(n) = n² then f(n) =
Θ(n²) and g(n) = Θ(n²)
This property only satisfies for Θ notation.

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 Transpose Symmetric Property:
If f(n) is O(g(n)) then g(n) is Ω (f(n)).
 Example: f(n) = n , g(n) = n² then n is O(n²)
and n² is Ω (n)
This property only satisfies for O and Ω
notations.

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 Transpose Symmetric Property:
If f(n) is O(g(n)) then g(n) is Ω (f(n)).
 Example: f(n) = n , g(n) = n² then n is O(n²)
and n² is Ω (n)
This property only satisfies for O and Ω
notations.

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 If f(n) = O(g(n)) and f(n) = Ω(g(n)) then f(n) = Θ(g(n))
 If f(n) = O(g(n)) and d(n)=O(e(n))
then f(n) + d(n) = O( max( g(n), e(n) ))
 Example: f(n) = n i.e O(n), d(n) = n² i.e O(n²)
then f(n) + d(n) = n + n² i.e O(n²)
 If f(n)=O(g(n)) and d(n)=O(e(n)) then
f(n) * d(n) = O( g(n) * e(n) )
 Example: f(n) = n i.e O(n), d(n) = n² i.e O(n²)
then f(n) * d(n) = n * n² = n³ i.e O(n³)

 Solved Problems are available in
https://www.cse.wustl.edu/~sg/CS241_FL99/h
w1-practice.html

 https://dotnettutorials.net/lesson/properties-of-
asymptotic-notations/
 https://unacademy.com/lesson/properties-of-
asymptotic-notation-part-1/1F2FTIU0
 https://www.cse.iitd.ac.in/~mausam/courses/col1
06/autumn2017/lectures/02-asymptotic.pdf
 https://www.cse.iitd.ac.in/~pkalra/col100/slides/1
0-Efficiency.pdf
 https://www.cse.wustl.edu/~sg/CS241_FL99/hw1
-practice.html

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