Asymptotic Analysis in Data Structures and Analysis

Asymptotic Analysis
Rachana Mehta

Algorithmic Analysis
Why the algorithmic analysis is necessary ?

Efficient :
Less Running Time
Less Memory Used
Efficiency of a Algorithm/Function based on the amount of input

Efficient :
Less Running Time : Time Complexity
Less Memory Used : Space Complexity
Efficiency of a Algorithm/Function based on the amount of input

How can we measure the running time of a
algorithm ?
Execution ?
Is it a good choice ?

Measuring the Running Time with a Pseudo-
Code
Generalize language/flow/stepwise instructions for implementing a
algorithm/function
Primitive Operation : Low level Operations :
Data Assignment
Control Statements (break,continue, function call,return)
Arithmetic or Logical Operation

Measuring the Running Time with a Pseudo-
Code
Generalize language/flow/stepwise instructions for implementing a
algorithm/function
Programming Constructs:
for loop, while loop, do-while loop
if-else statement

Asymptotic Notation
Asymptotic notations are the mathematical notations used to describe the
running time of an algorithm
Three Main Asymptotic Notation :
1. Big-O Notation
2. Omega Notation
3. Theta Notation

Asymptotic
Notation
1. Big-O Notation : Upper
Bound : Worst Case Analysis
A Function f(n) belongs to the
set O(g(n)), if there exist a
positive constant c such that
f(n) lies between 0 and
c*g(n), for sufficiently large n.
O(g(n)) = { f(n): there exist
positive constants c and n0 such
that 0 ≤ f(n) ≤ cg(n) for all n ≥
n0 }

Asymptotic
Notation
2. Omega Notation : Lower
Bound : Best Case Analysis
set Ω(g(n)), if there exist a
positive constant c such that
f(n) lies above c*g(n), for
sufficiently large n.
Ω(g(n)) = { f(n): there exist
positive constants c and n0 such
that 0 ≤ cg(n) ≤ f(n) for all n ≥
n0 }

Asymptotic
Notation
3. Theta Notation : Average
Case Analysis
set ϴ(g(n)), if there exist a
positive constant c1 and c2
such that f(n) lies between c1
*g(n) and c2 *g(n), for
sufficiently large n.
Θ (g(n)) = {f(n): there exist
positive constants c1, c2 and n0
such that 0 ≤ c1 * g(n) ≤ f(n) ≤
c2 * g(n) for all n ≥ n0}

Rules for Order Arithmetic:
Multiplicative Constants O(k * f(n)) = O(f(n))
Addition Rule O(f(n) + g(n)) = max(f(n), g(n))
Multiplication Rule O(f(n) * g(n)) = O(f(n)) * O(g(n))
Examples: O(1000n) = O(n)
O(n2
+ 3n + 2) = O(n2
)
O(3n3
+ 6n2
- 4n + 2) = O(3n3
) = O(n3
)
If f(x) = n2
* log n = O(n2
logn)

Examples
for (i = 0; i < n; i++)
cin>>A[i];

Examples
for (i = 0; i < n; i++)
for (j = 0; j < n; j++)
cin >> A[i][j];

Examples
for (i = 0; i < n; i++)
for (j = 0; j < i; j++)
cin >> A[i][j];

Examples
for (i = 0; i < n; i++)
cin>>B[i];
for (i = 0; i < n; i++)
for (j = 0; j < i; j++)
cin >> A[i][j];

Examples
for (i = 0; i < n; i++)
for (k = 0;k < n; k++)
cin>>B[k];
for (j = 0; j < 2*n; j++)
cin >> “Hello”;
cin >> A[j];

Examples
for (i = 0; i < n; i++)
for (k = 0;k < n; k++)
cin>>B[k];
for (m = 0; m < n; m++)
for (j = 0; j < 2*n; j++)
cin >> “Hello”;
cin >> A[j];

Examples
int i = 0; int A[100]; int n =100;
Take a number for “rollno” from user
while(( A[i] != rollno) && (i<n))
i++;
if (i > n)
cout << "not found";
else cout << "found";

Examples
int i = n;
while( i>=1)
x = x+1 ;
i = i/2;

Asymptotic Analysis in Data Structures and Analysis

More Related Content

Similar to Asymptotic Analysis in Data Structures and Analysis (20)

Recently uploaded (20)

Asymptotic Analysis in Data Structures and Analysis