UNIT-1
Algorithm Specification- Introduction.
Recursive algorithms.
Data Abstraction.
Performance analysis- time complexity and space complexity.
Asymptotic Notation-Big O.
Omega and Theta notations.
Arrays: Arrays – ADT.
Polynomials.
Sparse matrices.

WHAT IS AN ALGORITHM?
It is a finite sequence of well-defined, computer-implementable instructions, typically
to solve a class of specific problems or to perform a computation.
Advantages of Algorithms:
1.It is easy to understand.
2.Algorithm is a step-wise representation of a solution to a given problem.
3.In Algorithm the problem is broken down into smaller pieces or steps hence,
it is easier for the programmer to convert it into an actual program.

Disadvantages of Algorithms:
1.Writing an algorithm takes a long time so it is time-consuming.
2.Branching and Looping statements are difficult to show in Algorithms.
How to Design an Algorithm?
In order to write an algorithm, following things are needed as a pre-requisite:
1.The problem that is to be solved by this algorithm.
2.The constraints of the problem that must be considered while solving the problem.
3.The input to be taken to solve the problem.
4.The output to be expected when the problem the is solved.
5.The solution to this problem, in the given constraints.

I
TERATIVE ALGORITHM
Iterative algorithms are loop based repetitions of a process
Iterative algorithm uses repetition structure
An infinite loop occurs with iteration if the loop-condition test never becomes false
Iteration terminates when the loop-condition fails
Iteration does not use stack so it's faster than recursion
Iterative algorithm consume less memory
Infinite looping uses CPU cycles repeatedly
Iteration makes code longer
RECURSIVE ALGORITHM
Recursive algorithm uses a function that is partially defined by itself
Recursive algorithm uses selection structure
Infinite recursion occurs if the recursion step does not reduce the problem in a manner that
converges on some condition.(base case)

Recursion terminates when a base case is recognized
Recursion is usually slower then iteration due to overhead of maintaining stack
Recursive algorithm uses more memory than iteration
Infinite recursion can crash the system
Recursion makes code smaller
WHAT IS DATA ABSTRACTION?
Data abstraction is the programming process of creating a data type, usually a class,
that hides the details of the data representation in order to make the data type easier to
work with. Data abstraction involves creating a representation for data that separates
the interface from the implementation so a programmer or user only has to understand
the interface, the commands to use, and not how the internal structure of the data is
represented and/or implemented.

Smartphone Example
An example in the real-world is a mobile or smartphone. The typical smartphone user
does not know how calls are made, or how the phone accesses the Internet, or even
how data is stored in the phone. Instead, the mobile or smartphone user is provided
with an interface that hides these details from them. Making a call simply involves
selecting a contact's phone number. Accessing the Internet involves selecting an icon
on the screen.
The user does not need to understand the details of how a phone call is placed or
how the Internet works. These details are abstracted away from the phone's interface
in order to make the phone easier and more effective to use. The design of the mobile
or smartphone clearly separates the interface from the implementation. The user can
learn the functions of the phone without ever having to know anything about how the
phone works internally.

PERFORMANCE ANALYSIS
Analyzing the performance of an algorithm is an important part of its design. One of
the ways to estimate the performance of an algorithm is to analyze its complexity.
There are two possible types of analysis to quantify the complexity of an algorithm:
• Space complexity analysis: Estimates the runtime memory requirements needed to
execute the algorithm.
int square(int a)
{
return a*a;
}
In the above piece of code, it requires 2 bytes of memory to store variable 'a' and
another 2 bytes of memory is used for return value.
That means, totally it requires 4 bytes of memory to complete its execution. And this 4
bytes of memory is fixed for any input value of 'a'. This space complexity is said to
be Constant Space Complexity.

Calculate the sum of “n” numbers given in the array.
int calculate_sum(int arr[], int length)
{
int total_sum = 0;
int i = 0;
for( i= 0; i< len; i++)
{
total_sum = total_sum + arr[i];
}
}
From the above code, we can see that:
A variable “total_sum” that takes constant sum of 1 unit
A variable “length” that takes constant sum of 1 unit
A variable “i” that takes constant sum of 1 unit

But the variable arr[] is an array, it’s space consumption increases with the increase of
input size. i.e n.
Hence the space consumption is [3 + n] units.
the total space complexity of the above algorithm is O(n).
Calculate space complexity for matrix multiplication.
int add_matrix( int a[m] [n], int b [m] [n], int m, int n)
{
for (i = 0; i< n; i++)
{
for(j = 0; j<n; j++)
{
c[i][j] = a[i][j] * b[i][j]
}
}
}

From the above code, we can see that:
A variable “m” that takes constant sum of 1 unit
A variable “n” that takes constant sum of 1 unit
A variable “i” that takes constant sum of 1 unit
A variable “j” that takes constant sum of 1 unit
But we have 3 matrix variables of size arr[m][n] space consumption increases with the
increase of input size. For one matrix will consume “n^2” space, for 3 it will consume
“3*n^2” space.
Hence total space consumed is “4 + 3n^2”.
For larger size of input we ignore the constants, hence the space consumed is “n^2”.

WHAT IS ASYMPTOTIC ANALYSIS?
Asymptotic analysis of an algorithm refers to defining the mathematical
boundation/framing of its run-time performance. Using asymptotic analysis, we can
very well conclude the best case, average case, and worst case scenario of an
algorithm.Usually, the time required by an algorithm falls under three types .
Best Case − Minimum time required for program execution.
Average Case − Average time required for program execution.
Worst Case − Maximum time required for program execution.
Asymptotic Notations
Following are the commonly used asymptotic notations to calculate the running time
complexity of an algorithm.
Ο Notation
Ω Notation
θ Notation

Big-O Notation (O-notation)
Big-O notation represents the upper bound of the running time of an algorithm.
Thus, it gives the worst-case complexity of an algorithm.
O(g(n)) = { f(n): there exist positive constants c and n0
such that 0 ≤ f(n) ≤ cg(n) for all n ≥ n0 }
Omega Notation (Ω-notation)
Omega notation represents the lower bound of the running time of an algorithm.
Thus, it provides the best case complexity of an algorithm.
Ω(g(n)) = { f(n): there exist positive constants c and n0
such that 0 ≤ cg(n) ≤ f(n) for all n ≥ n0 }

ARRAYS
An array is a variable that can store multiple values. For example, if you want to store
100 integers, you can create an array for it.
int data[100];
APPLICATIONS OF ARRAYS
To store list of values
To perform matrix operations
To implement search algorithms
To implement sort algorithms
To implement stack
To implement queue
To implement CPU scheduling algorithms
To implement graph traversal technique depth first search
To implement hash tables
To implement heaps etc.

TYPES OF ARRAYS
1D (One dimensional)
2D (Two dimensional)
3D (Three dimensional)
ND (Multi dimensional)
To access array elements we need loops those are equal to number of dimensions
How to initialize an array?
It is possible to initialize an array during declaration.
For example,int mark[5] = {19, 10, 8, 17, 9};
You can also initialize an array like this.
int mark[] = {19, 10, 8, 17, 9};

Runtime Array initialization
An array can also be initialized at runtime using scanf() function.
TWO DIMENSIONAL ARRAYS
C language supports multidimensional arrays also. The simplest form of a
multidimensional array is the two-dimensional array. Both the row's and column's
index begins from 0.
Two-dimensional arrays are declared as follows,
data-type array-name[row-size][column-size]
/* Example */
int a[3][4];

An array can also be declared and initialized together. For example,
int arr[][3] = {
{0,0,0},
{1,1,1}
};
you can declare a three-dimensional (3d) array. For example,
float y[2][4][3];
Here, the array y can hold 24 elements.
POLYNOMIALS
A polynomial is composed of different terms where each of them holds a coefficient
and an exponent.
A polynomial p(x) is the expression in variable x which is in the form
(axn + bxn-1 + …. + jx+ k), where a, b, c …., k fall in the category of real numbers
and 'n' is non negative integer, which is called the degree of polynomial.

The simple way is to represent a polynomial with degree 'n' and store the coefficient of
n+1 terms of the polynomial in the array. So every array element will consist of two
values:
Coefficient and Exponent
In the Polynomial linked list, the coefficients and exponents of the polynomial are
defined as the data node of the list.
For adding two polynomials that are stored as a linked list. We need to add the
coefficients of variables with the same power. In a linked list node contains 3
members, coefficient value link to the next node.
a linked list that is used to store Polynomial looks like Polynomial : 4x7 + 12x2 + 45

Input :
p1= 13x8 + 7x5 + 32x2 + 54
p2= 3x12 + 17x5 + 3x3 + 98
Output : 3x12 + 13x8 + 24x5 + 3x3 + 32x2 + 152
Polynomials Addition
Let us illustrate the way the two polynomials are added, let us consider P and Q be
two polynomials having these two polynomials three terms each.
p= 30x2 + 20x + 100x —————(1)
Q= 60x3 + 50x2 + 60x——————(2)

we can represent these two polynomials as:-
Step 1: Compare the exponent of P and the corresponding exponent of q.
here expo(p) < expo(q) so, added the terms pointer to by q to the resultant list and
now advanced the q pointer.
Step 2:

first of all, here we compare the exponents of the current items between given P and Q.
expo(p)=expo(q)
therefore, add the coefficients of these two terms and link this to the resultant list and
advance the pointers p and q to their next nodes.
Step3:

furthermore, we compare the exponents of the current terms again
expo(p)=expo(q)
therefore, we add the coefficients of these two terms and link this to the resultant
linked list and advance the pointers to their next.
you will notice that nodes Q reaches the NULL and P points the last node.
Step 4:

In the above figure, you will notice that there is no node in the second polynomial to
compare with. you can understand in a better way after seeing the figure, so the last
node in the first polynomial is added to the end of the resultant linked list. the next
below figure is the output which comes Polynomials Addition of two polynomials.
Step 5:
Therefore, the display the resultant linked list, the resultant linked list is the pointed
to by the pointer.
i.e. R= 60x3 + 80x2 + 80x + 100

Degree of a polynomial is defined as the highest exponent in the given equation
SPARSE MATRIX
Sparse Matrix is a matrix that contains a few non-zero elements. Almost all the
places are filled with zero. Matrix of m*n dimension refers to a 2-D array with m
number of rows and n number of columns. And if the non-zero elements in the
matrix are more than zero elements in the matrix then it is called a sparse matrix.

to limit the processing time and space usage instead of storing a lesser number of
non-zero elements in a matrix, we use the below 2 representation:
1. Array representation
The 2D array is converted to 1 D array with 3 columns representing:
a. Row – Row index of non-zero element
b. Column – Column index of non-zero element
c. Value – value at the same row, column index in 2D matrix

2. Linked List representation:
In linked list representation, each node has four fields as given below:
Row: Row index of the non-zero elements in the matrix.
Column: Column index of the non-zero elements in the matrix.
Value: Value of the non zero elements at (row, column) position in the matrix
Next node: Reference to the next node.