What is Algorithm?

In this course, we will learn about algorithms through examples.

An algorithm is a series of well-defined instructions for solving a specific problem in computer programming. It accepts a collection of inputs and outputs the desired result. As an example,

An algorithm for multiplying two numbers:

• Consider two numerical inputs.

• Using the + operator, add two numbers together.

• Show the outcome

Qualities of a good Algorithm

• Input and output must be carefully defined.

• Each stage of the algorithm should be distinct and unequivocal.

• Algorithms should be the most effective solution among many possible solutions to a problem.

• Computer code should not be included in an algorithm. Instead, the method should be constructed in a form that allows it to be utilized in a variety of programming languages.

Characteristics of Algorithm

1. Input- 0 or more

2. Output- At least 1 output

3. Definiteness- Should have definite results or solutions

4. Finite- It must be stop or there should be an end state

5. Effectiveness- No unnecessary statement or redundant statements in algorithm

How to write an Algorithm?

Algorithm Sawap(a, b){

temp:= a ;

a:= b ;

b:= temp ;

}

How to analyze an Algorithm?

1. Time - how much time the algorithm is taken?

2. Space- how much space/memory it is consumed?

3. Network consumption- Reduction of data quality for transferring a data.

4. Power consumption- how much power the algorithm will consume?

5. CPU consumption- how many register it is consuming?

Example:

Algorithm Sawap(a, b){

temp:= a ; // require 1 unit of time

a:= b ; // require 1 unit of time

b:= temp ; // require 1 unit of time

}

Time analysis: f(n)= 1+1+1 = 3, that means this algorithm require order of 1 or, O(1)

Space analysis: how many variables are require to solve this algorithm?

Here, we require 3 variables to solve swap problem. temp, a, and b. So, S(n)=3 or order of 1 or, O(1)

Frequency Count Method

Example 1:

Algorithm Sum(A, n){

s=0; \\require 1 unit of time

for(i=0; i<n; i++){ \\ require n+1 unit of time

s=s+A[i] \\ require n unit of time

}

return s; \\require 1 unit of time

}

Time analysis: f(n)= 1+n+1+n+1=2n+3, that means this algorithm require order of n or, O(n) [the highest power of n in the time function]

Space analysis:

Here we have one array of length n, 'A', variables n, s, and i.

array, A = requires n unit of space

variable, n= requires 1 unit of space

variable, s= requires 1 unit of space

variable, i= requires 1 unit of space

Total S(n) = n+1+1+1= n+3 that means this algorithm require order of n or, O(n) [the highest power of n in the space function]

Example 2:

Algorithm Add_Array(A, B, n){

for(i=0; i<n; i++){ \\ require n+1 unit of time

for(j=0; j<n; j++){ \\ require n(n+1) unit of time

C[i,j]= A[i,j]+B[i,j] ; \\ require n * n unit of time

}

}

return C; \\require 1 unit of time

}

Time analysis: f(n)= n+1+n(n+1)+n*n+1=2n^2+2n+1, that means this algorithm require order of n^2 or, O(n^2) [the highest power of n in the time function]

Space analysis:

Here we have one array of length n, 'A, B, C', variables n, j, and i.

array, A = requires n^2 unit of space [As we are using 2d array]

array, B = requires n^2 unit of space [As we are using 2d array]

array, C = requires n^2 unit of space [As we are using 2d array]

variable, n= requires 1 unit of space

variable, i= requires 1 unit of space

variable, j= requires 1 unit of space

Total S(n) = 3n^2+3 that means this algorithm require order of n^2 or, O(n^2) [the highest power of n in the space function]

Example 3: [Do it now!]

Algorithm Multiply_Array(A, B, n){

for(i=0; i<n; i++){

for(j=0; j<n; j++){

C[i,j]= 0 ;

for(k=0; k<n; k++){

C[i,j]= C[i,j]+ A[i,k]*B[k,j];

}

}

}

return C;

}