3. Analysis of Algorithms
• The present day algorithms are based on the
RAM (Random Access Machine) model.
• In RAM model, instructions execute one after
another with no, concurrent operations.
4. Analysis of Algorithms
Computing
best case,
worst case and
average case
efficiency
Measuring
space
complexity
Measuring
input size
Analysis of
Algorithms
Measuring
time
complexity
Measuring
running
time
Computing
order of
growth of
algorithms
6. Worst Case Complexity
• The Worst Case Complexity of an algorithm is
the function defined by the maximum number
of steps taken on any instance size n.
7. Best Case Complexity
• The best case complexity of the algorithm is
the function defined by the minimum number
of steps taken on any instance of size n.
8. Average Case Complexity
• The average case complexity of the algorithm
is the function defined by an average number
of steps taken on an instance of size n.
10. Performance Evaluation of Algorithms
• Performance evaluation can be divided into
two major phases.
– 1) Priori estimates (Performance analysis)
– 2) Posteriori testing (Performance measurement)
11. Performance Analysis
• The efficiency of an algorithm can be decided
by measuring the performance of an
algorithm.
• The performance of an algorithm depends
upon two factors.
– 1) Amount of time required by an algorithm to
execute (known as time complexity).
– 2) Amount of storage required by an algorithm
(known as Space complexity).
13. Computing Space Complexity
The space requirement S(P) of any
algorithm
P
may
be
written
as
S(P)=c+Sp, where c is a constant and
Sp is a instance characteristics.
14. Two factors of Space Complexity
• Two factors are involved in Space complexity
computation
(constant
and
instance
characteristics).
• Constant characteristic (c) is a constant, it
denotes the space of input and outputs. This
space is an amount of space taken by
instructions, variables and identifiers.
• Instance characteristic (Sp) is a space dependent
upon instance (particular problem instance).
15. Addition of three number-Space
complexity
Algorithm Add(a,b,c)
{
//Problem Description: This algorithm computes
//the addition of three elements
//Input: a,b and c are of floating type
//Output: The addition is returned
return a+b+c;
}
16. • The space requirement for addition of three
numbers algorithm is
• S(P)=C+ Sp
• The problem instance is characterized by
specific values of a, b and c. By assuming a, b
and c occupies one word then total size comes
to 3. Space needed by a, b and c is
independent of instance characteristics.
Consequently Sp (instance characteristics)=0.
17. Sum of ‘n’ numbers
Algorithm Sum(a,n)
{
S<-0.0;
for i<-1 to n do
S<-S+a[i];
return s;
}
The Space requirement
for sum of n numbers
algorithm is
S(P)>=(n+3)
The ‘n’ space required
for a[], one unit space
for n, one unit for i and
one unit for S.
18. Sum of ‘n’ numbers using Recursion
Algorithm Rsum(a,n)
{
if(n<=0) then return 0.0;
else return Rsum(a, n-1)+a[n]; }
The Space requirement is S(P)>=3(n+1)
The internal stack used for recursion includes space for
formal parameters, local variables and return address. The
space required by each call to function Rsum requires
atleast three words (space for n values + space for return
address + pointer to a[]). The depth of recursion is n+1 ( n
times call to function and one return call). The recursion
stack space will be >=3(n+1).
19. Time Complexity
• The time complexity of an algorithm is the
amount of computer time required by an
algorithm to run to completion.
• The time T(P) by a program P is the sum of the
compile time and the run (or execution) time.
• The compile time does not depend on the
instance characteristics and the compiled
program runs several times without
recompilation.
20. Run time complexity
• Run time complexity of a program will be
determined by tp ( instance characteristics).
• Run time complexity depends upon so many
factors.
21. Issues in Time Complexity
• It is difficult to compute the time complexity
in terms of physically clocked time or instance
in multiuser system, executing time depends
on may factors such as:
• System load
• Number of other programs running
• Instruction set used
• Speed of underlying hardware
22. Frequency count
• The time complexity is therefore given in
terms of frequency count.
• Frequency count is a count denoting number
of times of execution of statement.
• Time efficiency is analyzed by determining the
number of repetitions of the basic operation
as a function of input size.
23. Basic operation
• Basic operation is nothing but core operation,
generally basic operation resides in inner loop.
Example in Sorting algorithm the basic
operation is comparing the elements and
placing them in appropriate position.
24. Input Size
• One of the instance characteristics for run
time complexity of an algorithm is input size.
• Usually longer input size make the algorithm
to run longer time.
• The input size for the problem of summing an
array with ‘n’ elements is n+1 (n for listing the
‘n’ elements and 1 for ‘n’ value)
25. Input size and basic operation examples
Problem
Input size measure
Basic operation
Searching for key in Number of list’s items,
Key comparison
a list of n items
i.e. n
Multiplication of
two matrices
Matrix dimensions or
total number of
elements
Multiplication of
two numbers
Checking primality
of a given integer n
n’size = number of
digits (in binary
representation)
Division
#vertices and/or edges
Visiting a vertex
or traversing an
edge
Typical graph
problem
26. Measuring Running Time
T(n)=cop C(n)
Running time of
basic operation
Time taken by the
basic operation to
execute
Number of
times the
operation
needs to be
executed
27. sum of ‘n’ numbers -Time complexity
Statement
Algorithm Sum(a,n)
{
S<-0.0;
for i<-1 to n do
S<-S+a[i];
return s;
}
Total
Steps per Frequency
execution
0
0
1
1
1
n+1
1
n
1
1
0
--
Total
steps
0
0
1
n+1
n
1
0
2n+3
28. Sum of ‘n’ using Recursion-Time Complexity
Statement
Steps per
execution
Frequency
n=0 n>0
Total steps
n=0
n>0
Algorithm RSum(a,n)
{
if(n<=0) then
return 0.0;
else return
Rsum(a,n-1)+a[n];
}
Total
0
-
-
-
-
1
1
1
1
1
0
1
1
1
0
1+x
0
1
0
1+x
2
X=tRsum(n-1)
2+x
29. Order of Growth
• Measuring the performance of an algorithm in
relation with the input size ‘n’ is called order of
growth.
Order of growth for varying input size of ‘n’
n
Log n
n log n
n2
2n
1
0
0
1
1
2
1
2
4
4
4
2
8
16
16
8
3
24
64
256
16
4
64
256
65,536
32
5
160
1024
4,294,967,296
32. Asymptotic Notations
• Asymptotic running time of an algorithm is
defined in terms of functions.
• Asymptotic notation is useful describe the
running time of the algorithm.
• Asymptotic notations give time complexity as
“fastest possible”, “slowest possible” or “average
time”.
• Bigh Oh (Ο) , Omega (Ω) and Theta (Θ) notations
are useful to represent the asymptotic complexity
of algorithms.
