Digital Analysis and Algorithms

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UNIT – I
Analysis of Algorithm:
INTRODUCTION – ANALYZING CONTROL STRUCTURES-AVERAGE CASE ANALYSIS-SOLVING RECURRENCES.

ALGORITHM

Informal Definition:
An Algorithm is any well-defined computational procedure that takes some value or set of values as Input and produces a set of values or some value as output. Thus algorithm is a sequence of computational steps that transforms the i/p into the o/p.

Formal Definition:
An Algorithm is a finite set of instructions that, if followed, accomplishes a particular task. In addition, all algorithms should satisfy the following criteria.

1.INPUT  Zero or more quantities are externally supplied. 2.OUTPUT  At least one quantity is produced.
3.DEFINITENESS  Each instruction is clear and unambiguous. 4.FINITENESS  If we trace out the instructions of an algorithm, then for all cases, the algorithm terminates after a finite number of steps. 5.EFFECTIVENESS  Every instruction must very basic so that it can be carried out, in principle, by a person using only pencil & paper.

Issues or study of Algorithm:

How to device or design an algorithm  creating and algorithm. •How to express an algorithm  definiteness.
How to analysis an algorithm  time and space complexity. •How to validate an algorithm  fitness.
Testing the algorithm  checking for error.

Algorithm Specification:

Algorithm can be described in three ways.

1.Natural language like English:
When this way is choused care should be taken, we should ensure that each & every statement is definite.

2. Graphic representation called flowchart:
This method will work well when the algorithm is small& simple.

3.Pseudo-code Method:
In this method, we should typically describe algorithms as program, which resembles language like Pascal & algol.

Pseudo-Code Conventions:

1.Comments begin with // and continue until the end of line.

2.Blocks are indicated with matching braces {and}.

3.An identifier begins with a letter. The data types of variables are not explicitly declared.

4.Compound data types can be formed with records. Here is an example, Node. Record
{
data type – 1 data-1;
.
.
.
data type – n data – n;
node * link;
}

Here link is a pointer to the record type node. Individual data items of a record can be accessed with  and period.

5.Assignment of values to variables is done using the assignment statement. := ;

6.There are two Boolean values TRUE and FALSE.

 Logical Operators AND, OR, NOT
Relational Operators =, =, !=

7.The following looping statements are employed.

For, while and repeat-until
While Loop:
While < condition > do
{

.
.
.

}

For Loop:
For variable: = value-1 to value-2 step step do

{

.
.
.

}
repeat-until:

repeat

.
.
.

until

8.A conditional statement has the following forms.

 If then
 If then
Else

Case statement:

Case
{
: :
.
.
.
: :
: else :
}

9.Input and output are done using the instructions read & write.

10.There is only one type of procedure:
Algorithm, the heading takes the form,

Algorithm Name (Parameter lists)

 As an example, the following algorithm fields & returns the maximum of ‘n’ given numbers:

1.algorithm Max(A,n)
2.// A is an array of size n
3.{
4.Result := A[1];
5.for I:= 2 to n do
6. if A[I] > Result then
7. Result :=A[I];
8. return Result;
9.}

In this algorithm (named Max), A & n are procedure parameters. Result & I are Local variables.

 Next we present 2 examples to illustrate the process of translation problem into an algorithm....
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