This document discusses data structures asymptotic analysis and notations. It introduces best case, worst case, and average case analysis of algorithms. There are three main types of asymptotic notations: Big-O notation represents upper bounds (worst case), Omega notation represents lower bounds (best case), and Theta notation represents tight bounds (average case). Common examples of asymptotic notations include constant, logarithmic, linear, quadratic, and exponential time complexities.

1. Data Structures Asymptotic Analysis
V.G. Jyothi Main
Computer Application
V.G.Jyothi Mani
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2. Data Structures Asymptotic Analysis
• The algorithm takes less time or space
Best Case
• The algorithm takes long time or space.
Worst Case
• The algorithm takes time or space that lies between
best and worst case.
Average Case
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Asymptotic Notations are the expressions that are used to represent the
complexity of an algorithm. There are three types of analysis that we perform
on a particular algorithm.

3. Cont....
Ex: The running time of one operation is computed as f(n)
where n = 1,2,3,4,……linear
Another operation it is computed as g(n2).
where n =1,2,3,4 i.e g(n2 )= 1,4,9,16…… exponentially
Similarly, the running time of both operations will be nearly the
same if n is significantly.
• Best Case − Minimum time required for program execution.
• Average Case − Average time required for program execution.
• Worst Case − Maximum time required for program execution.
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4. Types of Data Structure Asymptotic Notation
1. Big-O Notation (Ο) – Big O notation
specifically worst case.
2. Omega Notation (Ω) – Omega(Ω)
notation specifically best case.
3. Theta Notation (θ) – This notation
represents the average complexity of
an algorithm.
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5. Big O Notation (Ο)
The notation Ο(n) is the formal
way to express the upper bound
of an algorithm's running time.
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t
N

6. Omega Notation (Ω
The notation Ω(n) is the formal way
to express the lower bound of an
algorithm's running time.
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N

7. Theta Notation, (θ)
The notation θ(n) is the formal
way to express both the lower
bound and the upper bound of
an algorithm's running time.
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N

8. Common Asymptotic Notations
constant − Ο(1)
logarithmic − Ο(log n)
linear − Ο(n)
n log n − Ο(n log n)
quadratic − Ο(n
2
)
cubic − Ο(n
3
)
polynomial − n
Ο(1)
exponential − 2
Ο(n)
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