Data structure and algorithm notes

Prerequisites:
 Knowledge of C
Note:- The complete C programs involved in this note are not study in detailed for the
point of exam view these are only for practical purpose.
 For point of exam view writing algorithms and functions are important
Unit 1:
Concept and definition of data structures
• Information and its meaning
• Array in C
• The array as an ADT
• One dimensional array
• Two dimensional array
• Multidimensional array
• Structure
• Union
• Pointer
What is data structure?
- Data structure is a way of organizing all data items and establishing relationship
among those data items.
- Data structures are the building blocks of a program.
Data structure mainly specifies the following four things:
• Organization of data.
• Accessing methods
• Degree of associativity
• Processing alternatives for information
To develop a program of an algorithm, we should select an appropriate data structure for
that algorithm. Therefore algorithm and its associated data structures form a program.
Algorithm + Data structure = Program
A static data structure is one whose capacity is fixed at creation. For example, array. A
dynamic data structure is one whose capacity is variable, so it can expand or contract at any
time. For example, linked list, binary tree etc.
Abstract Data Types (ADTs)
An abstract data type is a data type whose representation is hidden from, and of no concern to
the application code. For example, when writing application code, we don’t care how strings
are represented: we just declare variables of type String, and manipulate them by using string
operations.
Once an abstract data type has been designed, the programmer responsible for implementing
that type is concerned only with choosing a suitable data structure and coding up the methods.
On the other hand, application programmers are concerned only with using that type and
calling its methods without worrying much about how the type is implemented.
DSA By Bhupendra Saud1
csitnepal
Source: www.csitnepal.com

Classification of data structure:
Fig:- Classification of data structure
Array
• Array is a group of same type of variables that have common name
• Each item in the group is called an element of the array
• Each element is distinguished from another by an index
• All elements are stored contiguously in memory
• The elements of the array can be of any valid type- integers, characters, floating-
point types or user-defined types
Types of Array:
1). One dimensional array:
The elements of the array can be represented either as a single column or as a
single row.
Declaring one-dimensional array:
data_type array_name[size];
Following are some valid array declarations:
int age[15];
float weight[50];
int marks[100];
DSA By Bhupendra Saud
Data structure
Primitive
Data structure
Non-
Primitive
Data structure
Integer Float Characte
r
Pointer Array List File
Linear list Non-linear list
Stack
Queue
Trees
Graphs
2
csitnepal

char section[12];
char name[10];
Following are some invalid array declarations in c:
int value[0];
int marks[0.5];
int number[-5];
Array Initialization (1-D):
The general format of array initialization is:
data_type array_name[size]={element1,element2,…………..,element n};
for example:
int age[5]={22,33,43,24,55};
int weight[]={55,6,77,5,45,88,96,11,44,32};
float a[]={2,3.5,7.9,-5.9,-8};
char section[4]={‘A’,’B’,’C’,’D’};
char name[10]=”Bhupendra”;
Example 1: A program to read n numbers and to find the sum and average of those
numbers.
#include<stdio.h>
void main()
{
int a[100], i, n, sum=0;
float avg;
printf(“Enter number of elements”);
scanf(“%d”,&n);
printf(“Enter %d numbers”,n);
for(i=0;i<n;i++)
{
scanf(“%d”,&a[i]);
sum=sum+a[i];
//sum+=a[i];
}
avg=sum/n;
printf(“sum=%dn Average=%f”, sum, avg);
}
Some common operations performed in one-dimensional array are:
• Creating of an array
• Inserting new element at required position
• Deletion of any element
• Modification of any element
• Traversing of an array
• Merging of arrays
Insertion of new element at required position:
Let we have an array
a[6]=(1,5,7,6,22,90};
Suppose we want to insert 20 in array a, at location with index 4, it means the elements
22 and 90 must shift 1 position downwards as follows.
csitnepal

Original array Elements shifted downwards Array after insertion
C- code for above problem:
#include<stdio.h>
#include<conio.h>
void main()
{
int a[100], pos, nel, i;
clrscr();
printf(“Enter no of elements to be inserted”);
scanf(“%d”, &n);
printf(“Enter %d elements”, n);
for(i=0;i<n;i++)
{
scanf(“%d”, &a[i]);
}
printf(“Enter position at which you want to insert new element”);
scanf(“%d”, &pos);
printf(“Enter new element”);
scanf(“%d”, &nel);
for(i=n-1; i>=pos; i--)
{
a[i+1] = a[i];
}
a[pos]=nel;
n++;
printf(”New array is:n”);
for(i=0; i<n; i++)
{
printf(“%dt”, a[i]);
}
getch();
}
a[0] 1
a[1] 5
a[2] 7
a[3] 6
a[4] 22
a[5] 90
a[6] 0
a[7] 0
a[8] 0
a[9] 0
a[0] 1
a[1] 5
a[2] 7
a[3] 6
a[4] ………..
a[5] 22
a[6] 9 0
a[7] 0
a[8] 0
a[9] 0
a[0] 1
a[1] 5
a[2] 7
a[3] 6
a[4] 20
a[5] 22
a[6] 90
a[7] 0
a[8] 0
a[9] 0
4
csitnepal

Deletion of any element from an array:
Suppose we want to delete the element a[5]=90, then the elements following it were
moved upward by one location as shown in fig. below:
Fig: - Deleting an element from one-dimensional array
#include<stdio.h>
#include<conio.h>
void main()
{
int a[100], pos, i;
clrscr();
for(i=0;i<n;i++)
{
}
printf(“Enter position at which you want to delete an element”);
for(i=pos; i<n; i++)
{
a[i] = a[i+1];
}
n--;
for(i=0; i<n; i++)
{
}
}
a[0] 1
a[1] 5
a[2] 7
a[3] 6
a[4] 22
a[5] 90
a[6] 30
a[7] 10
a[8] 50
a[9] 8
a[0] 1
a[1] 5
a[2] 7
a[3] 6
a[4] 22
a[5] ………
a[6] 3 0
a[7] 10
a[8] 50
a[9] 8
a[0] 1
a[1] 5
a[2] 7
a[3] 6
a[4] 22
a[5] 30
a[6] 10
a[7] 50
a[8] 8
a[9]
5
csitnepal

Modification of any element:
#include<stdio.h>
#include<conio.h>
void main()
{
int a[100], pos, nel, i;
clrscr();
for(i=0;i<n;i++)
{
}
printf(“Enter position at which you want to modify an element”);
printf(“Enter new element”);
scanf(“%d”, &nel);
a[pos]=nel;
for(i=0; i<n; i++)
{
}
getch();
}
Traversing of an array:
Traversing means to access all the elements of the array, starting from first
element upto the last element in the array one-by-one.
#include<stdio.h>
#include<conio.h>
void main()
{
int a[100], pos, i;
clrscr();
for(i=0;i<n;i++)
{
}
printf(“Traversing of the array:n”);
for(i=0; i<n; i++)
{
}
getch();
}
csitnepal

Merging of two arrays:
Merging means combining elements of two arrays to form a new array. Simplest
way of merging two arrays is the first copy all elements of one array into a third empty
array, and then copy all the elements of other array into third array.
Suppose we want to merge two arrays a[6] and b[4]. The new array says c will be having
(6+4) =10 elements as shown in figure below.
Fig: - Merging of two arrays
#include<stdio.h>
#include<conio.h>
void main()
{
int a[6], b[4], c[10], i, j;
clrscr();
printf(“Enter elements of first arrayn”);
for(i=0;i<6;i++)
printf(“Enter elements of second arrayn”);
for(i=0;i<4;i++)
{
}
for(i=0; i<6; i++)
{
c[i]=a[i];
}
j=i; // here i=j=6
for(i=0; i<4; i++)
{
c[j]=a[i];
j++;
}
printf(“The resulting array is:n”);
for(i=0; i<10; i++)
{
printf(”%dt”, c[i]);
}
a[0] 1
a[1] 5
a[2] 7
a[3] 6
a[4] 12
a[5] 15
b[0] 20
b[1] 25
b[2] 30
b[3] 35
c[0] 1
c[1] 5
c[2] 7
c[3] 6
c[4] 12
c[5] 15
c[6] 20
c[7] 25
c[8] 30
c[9] 35
7
csitnepal

getch();
}
Two-Dimensional array:
When we declare two dimensional array, the first subscript written is for the number of
rows and the second one is for the column.
Declaration of 2- D array:
Return_type array_name[row_size][column_size];
Example;
int a[3][4];
float b[10][10];
int this first example, 3 represents number of rows and 4 represents number of columns.
• Think, two-dimensional arrays as tables/matrices arranged in rows and columns
• Use first subscript to specify row no and the second subscript to specify column no.
Array Initialization (2-D):
The general format of array initialization is:
data_type array_name[row_size][col_size]={element1,element2,…………..,element n};
for example:
int a[2][3]={33,44,23,56,77,87};
or
int a[2][3]={{33,44,23},
{56, 77, 87}};
Example 1: A program to find addition of any two matrices by using function
#include<stdio.h>
#include<conio.h>
void display(int [][], int, int); //function prototype
void main()
{
int a[10][10], b[10][10],c[10][10],i, j, r, c;
clrscr();
printf(“Enter size of a matrix”);
scanf(“%d%d”, &r,&c);
printf(“Enter elements of first matrixn”);
for(i=0;i<r;i++)
{
Row 0
Row 1
Row 2
Column 0 Column 1 Column 2 Column 3
a[ 0 ][ 0 ]
a[ 1 ][ 0 ]
a[ 2 ][ 0 ]
a[ 0 ][ 1 ]
a[ 1 ][ 1 ]
a[ 2 ][ 1 ]
a[ 0 ][ 2 ]
a[ 1 ][ 2 ]
a[ 2 ][ 2 ]
a[ 0 ][ 3 ]
a[ 1 ][ 3 ]
a[ 2 ][ 3 ]
Row subscript
Array name
Column subscript
8
csitnepal

for(j=0;j<c;j++)
{
scanf(“%d”,&a[i][j]);
}
}
printf(“Enter elements of second matrixn”);
for(i=0;i<r;i++)
{
for(j=0;j<c;j++)
{
scanf(“%d”,&b[i][j]);
}
}
// finding sum
for(i=0;i<r;i++)
{
for(j=0;j<c;j++)
{
c[i][j]=a[i][j]+b[i][j];
}
}
printf(“The first matrix isn”);
display(a, r, c); //function call
printf(“The second matrix isn”);
display(b, r, c); //function call
printf(“The resulting matrix isn”);
display(c, r, c); //function call
getch();
}
void display(int d[10][10], int r, int c) //function definition
{
int i, j;
for(i=0;i<r;i++)
{
for(j=0;j<c;j++)
{
printf(“%dt”, d[i][j]);
}
printf(“n”);
}
}
Example 2: A program to find transposition of a matrix by using function
#include<stdio.h>
void display(int [][], int, int); //function prototype
void main()
{
int a[10][10], t[10][10],i, j, r, c;
printf(“Enter no of rows and no of columns”);
scnaf(“%d%d”,&r,&c);
printf(“Enter elements of a matrixn”);
for(i=0;i<r;i++)
{
for(j=0;j<c;j++)
{
scanf(“%d”,&a[i][j]);
csitnepal

}
}
//finding transpose of a matrix
for(i=0;i<r;i++)
{
for(j=0;j<c;j++)
{
t[i][j]=a[j][i];
}
}
printf(“the original matrix isn”);
display(a, r, c); //function call
printf(“the transposed matrix isn”);
display(t, r, c); //function call
}
void display(int x[][], int r, int c) //function definition
{
int i, j;
for(i=0;i<r;i++)
{
for(j=0;j<c;j++)
{
printf(“%dt”,x [i][j]);
}
printf(“n”);
}
}
Implementation of a two-dimensional array:
A two dimensional array can be implemented in a programming language in two ways:
• Row-major implementation
• Column-major implementation
Row-major implementation:
Row-major implementation is a linearization technique in which elements of array are
reader from the keyboard row-wise i.e. the complete first row is stored, and then the
complete second row is stored and so on. For example, an array a[3][3] is stored in the
memory as shown in fig below:
Row 1 Row 2 Row 3
Column-major implementation:
In column major implementation memory allocation is done column by column
i.e. at first the elements of the complete first column is stored, and then elements of
a[0][0] a[0][1] a[0][2] a[1][0] a[1][1] a[1][2] a[2][0] a[2][1] a[2][2]
10
csitnepal

complete second column is stored, and so on. For example an array a [3] [3] is stored in
the memory as shown in the fig below:
Column 1 column 2 column 3
Multi-Dimensional array
C also allows arrays with more than two dimensions. For example, a three
dimensional array may be declared by
int a[3][2][4];
Here, the first subscript specifies a plane number, the second subscript a row number
and the third a column number.
However C does allow an arbitrary number of dimensions. For example, a six-
dimensional array may be declared by
int b[3][4][6][8][9][2];
Column 0 Column 1 Column 2 Column 3
Show that an array is an ADT:
Let A be an array of type T and has n elements then it satisfied the following operations:
• CREATE(A): Create an array A
• INSERT(A,X): Insert an element X into an array A in any location
• DELETE(A,X): Delete an element X from an array A
• MODIFY(A,X,Y): modify element X by Y of an array A
• TRAVELS(A): Access all elements of an array A
• MERGE(A,B): Merging elements of A and B into a third array C
a[0][0] a[1][0] a[2][0] a[0][1] a[1][1] a[2][1] a[0][2] a[1][2] a[2][2]
Row 0
Row 1
Plane 0
Plane 1
Plane 2
11
csitnepal

Thus by using a one-dimensional array we can perform above operations thus an array
acts as an ADT.
Structure:
A structure is a collection of one or more variables, possibly of different types,
grouped together under a single name.
An array is a data structure in which all the members are of the same data type. Structure
is another data structure in which the individual elements can differ in type. Thus, a
single structure might contain integer elements, floating-point elements and character
elements. The individual structure elements are referred to as members.
Defining a structure: A structure is defined as
struct structure_name
{
member 1;
member 2;
………..
member n;
};
We can define a structure to hold the information of a student as follows:
struct Student
{
char name[2];
int roll;
char sec;
float marks;
};
Structure variable declaration:
struct Student s1, s2, s3;
We can combine both template declaration and structure variable declaration in one
statement.
Eg,
struct Student
{
char name[2];
int roll;
char sec;
float marks;
} s1, s2, s3;
csitnepal

Accessing members of a structure:
There are two types of operators to access members of a structure. Which are:
• Member operator (dot operator or period operator (.))
• Structure pointer operator (->).
Structure initialization:
Like any data type, a structure variable can be initialized as follows:
struct Student
{
char name[20];
int roll;
char sec;
float marks;
};
struct Student s1={“Raju”, 22, ‘A’, 55.5};
The s1 is a structure variable of type Student, whose members are assigned initial
values. The first member (name[20[) is assigned the string “Raju”, the second member
(roll) is assigned the integer value 22, the third member (sec) is assigned the character
‘A’, and the fourth member (marks) is assigned the float value 55.5.
Example: program illustrates the structure in which read member elements of
structure and display them.
#include<stdio.h>
void main()
{
struct Student
{
char name[20];
int roll;
char sec;
float marks;
};
struct Student s1;
clrscr();
printf(“Enter the name of a student”);
gets(s1.name);
printf(“Enter the roll number of a student”);
scanf(“%d”,&s1.roll);
printf(“Enter the section of a student”);
scanf(“%c”,&s1.sec);
printf(“Enter the marks obtained by the student”);
scanf(“%f”,&s1.marks);
//displaying the records
printf(“Name=%sn Roll number =%dn Section=%cn Obtained marks=%f”,s1.name,
s1.roll, s1.sec, s1.marks);
}
Structures within Structures:
Structures within structures mean nesting of structures. Study the following
example and understand the concepts.
csitnepal

Example: the following example shows a structure definition having another
structure as a member. In this example, person and students are two structures.
Person is used as a member of student.(person within Student)
#include<stdio.h>
struct Psrson
{
char name[20];
int age;
};
struct Student
{
int roll;
char sec;
struct Person p;
};
void main()
{
struct Student s;
gets(s.p.name);
printf(“Enter age”);
scanf(“%d”,&s.p.age);
scanf(“%d”,&s.roll);
scanf(“%c”,&s.sec);
printf(“Name=%sn Roll number =%dn Age=%dn Section=%cn”,s.p.name, s.roll,
s.p.age, s.sec);
}
Passing entire structures to functions:
#include<stdio.h>
void display(struct student);
struct student
{
char name[20];
int age;
int roll;
char sec;
};
void main()
{
struct student s;
int i;
gets(s.name);
printf(“Enter age”);
scanf(“%d”,&s.age);
scanf(“%d”,&s.roll);
scanf(“%c”,&s.sec);
display(s); //function call
Equivalent form of nested structure is:
struct Student
{
int roll;
char sec;
struct Person
{
char name[20];
int age;
}p;
};
14
csitnepal

}
void sisplay(struct student st)
{
printf(“Name=%sn Roll number =%dn Age=%dn Section=%cn”,st.name, st.roll,
st.age, st.sec);
}
Unions:
Both structure and unions are used to group a number of different variables
together. Syntactically both structure and unions are exactly same. The main difference
between them is in storage. In structures, each member has its own memory location but
all members of union use the same memory location which is equal to the greatest
member’s size.
Declaration of union:
The general syntax for declaring a union is:
union union_name
{
data_type member1;
data_type member2;
data_type member3;
…………………………
…………………………
data_type memberN;
};
We can define a union to hold the information of a student as follows:
union Student
{
char name[2];
int roll;
char sec;
float marks;
};
union variable declaration:
union Student s1, s2, s3;
we can combine both template declaration and union variable declaration in one
statement.
Eg,
union Student
{
char name[2];
int roll;
char sec;
float marks;
} s1, s2, s3;
csitnepal

Differences between structure and unions
1. The amount of memory required to store a structure variable is the sum of sizes of all
the members. On the other hand, in case of a union, the amount of memory required is
the same as member that occupies largest memory.
#include<stdio.h>
#include<conio.h>
struct student
{
int roll_no;
char name[20];
};
union employee
{
int ID;
char name[20];
};
void main()
{
struct student s;
union employee e;
printf(“nsize of s = %d bytes”,sizeof(s)); // prints 22 bytes
printf(“nSize of e = %d bytes”,sizeof(e)); // prints 20 bytes
getch();
}
Pointers
A pointer is a variable that holds address (memory location) of another variable rather
than actual value. Also, a pointer is a variable that points to or references a memory
location in which data is stored. Each memory cell in the computer has an address that
can be used to access that location. So, a pointer variable points to a memory location
and we can access and change the contents of this memory location via the pointer.
Pointers are used frequently in C, as they have a number of useful applications. In
particular, pointers provide a way to return multiple data items from a function via
function arguments.
Pointer Declaration
Pointer variables, like all other variables, must be declared before they may be used in a
C program. We use asterisk (*) to do so. Its general form is:
ata-type *ptrvar;
For example,
int* ptr;
float *q;
char *r;
csitnepal

This statement declares the variable ptr as a pointer to int, that is, ptr can hold address of
an integer variable.
Reasons for using pointer:
• A pointer enables us to access a variable that is defined outside the function.
• Pointers are used in dynamic memory allocation.
• They are used to pass array to functions.
• They produce compact, efficient and powerful code with high execution speed.
• The pointers are more efficient in handling the data table.
• They use array of pointers in character strings result in saving of data storage
space in memory. Sorting stings using pointer is very efficient.
• With the help of pointer, variables can be swapped without physically moving
them.
• Pointers are closely associated with arrays and therefore provides an alternate way
to access individual array elements.
Pointer initialization:
Once a pointer variable has been declared, it can be made to point to a variable
using an assignment statement as follows:
int marks;
int *marks_pointer;
Marks_pointer=&marks;
Passing (call) by Value and Passing (call) by Reference
Arguments can generally be passed to functions in one of the two ways:
o Sending the values of the arguments (pass by value)
o Sending the addresses of the arguments (pass by reference)
Pass by value: In this method, the value of each of the actual arguments in the calling
function is copied into corresponding formal arguments of the called function. With this
method the changes made to the formal arguments in the called function have no effect
on the values of actual arguments in the calling function. The following program
illustrates ‘call by value’.
#include<stdio.h>
#include<conio.h>
void main()
{
int a,b;
void swap(int, int );
clrscr();
a = 10;
b = 20;
swap(a,b);
printf("a = %dtb = %d",a,b);
getch();
}
void swap(int x, int y)
{
int t;
csitnepal

t = x;
x = y;
y = t;
printf("x = %dty = %dn",x,y);
}
The output of the above program would be
x = 20 y = 10
a = 10 b = 20
Note that values of a and b are unchanged even after exchanging the values of x and y.
Pass by reference: In this method, the addresses of actual arguments in the calling
function are copied into formal arguments of the called function. This means that using
these addresses we would have an access to the actual arguments and hence we would
be able to manipulate them. The following program illustrates this fact.
#include<stdio.h>
#include<conio.h>
void main()
{
int a,b;
void swap(int*, int*);
clrscr();
a = 10;
b = 20;
swap(&a,&b);
printf("a = %dtb = %d",a,b);
getch();
}
void swap(int *x, int *y)
{
int t;
t = *x;
*x = *y;
*y = t;
printf("x = %dty = %dn",*x,*y);
}
The output of the above program would be
x = 20 y = 10
a = 20 b = 10
Note: We can use call by reference to return multiple values from the function.
Pointers and Arrays
An array name by itself is an address, or pointer. A pointer variable can take different
addresses as values. In contrast, an array name is an address, or pointer, that is fixed.
Pointers and One-dimensional Arrays
In case of one dimensional array, an array name is really a pointer to the first element in
the array. Therefore, if x is a one-dimensional array, then the address of the first array
element can be expressed as either &x[0] or simply x. Moreover, the address of the
csitnepal

second array element can be expressed as either &x[1] or as (x+1), and so on. In general,
the address of array element (x+i) can be expressed as either &x[i] or as (x+i). Thus we
have two different ways to write the address of any array element: we can write the
actual array element, preceded by an ampersand; or we can write an expression in which
the subscript is added to the array name.
Since, &x[i] and (x+i) both represent the address of the ith element of x, it would seem
reasonable that x[i] and *(x+i) both represent the contents of that address, i.e., the value
of the ith element of x. The two terms are interchangeable. Hence, either term can be
used in any particular situation. The choice depends upon your individual preferences.
For example,
/* Program to read n numbers in an array and display their sum and average */
#include<stdio.h>
#include<conio.h>
#define SIZE 100
void main()
{
float a[SIZE],sum=0,avg;
int n,i;
clrscr();
printf("How many numbers?");
scanf("%d",&n);
printf("Enter numbers:n");
for(i=0;i<n;i++)
{
scanf("%f",(a+i)); // scanf("%f",&a[i]);
sum=sum+*(a+i); //sum=sum+a[i];
}
avg=sum/n;
printf("Sum=%fn",sum);
printf("Average=%f",avg);
getch();
}
/* using pointer write a program to add two 3 × 2 matrices and print the result in
matrix form */
#include<stdio.h>
#include<conio.h>
#define ROW 3
#define COL 2
void main()
{
int a[ROW][COL],b[ROW][COL],i,j,sum;
clrscr();
printf("Enter elements of first matrix:n");
for(i=0;i<ROW;i++)
{
for(j=0;j<COL;j++)
scanf("%d", (*(a+i)+j));
printf("n");
}
csitnepal

printf("Enter elements of second matrix:n");
for(i=0;i<ROW;i++)
{
for(j=0;j<COL;j++)
scanf("%d", (*(b+i)+j));
printf("n");
}
printf("Addition matrix is:n");
for(i=0;i<ROW;i++)
{
for(j=0;j<COL;j++)
{
sum = *(*(a+i)+j)+*(*(b+i)+j);
printf("%dt",sum);
}
printf("n");
}
getch();
}
/*Sum of two matrix using dynamic memory allocation*/
#include<stdio.h>
#include<conio.h>
#include<stdlib.h>
void read(int**, int, int);
void write(int**, int, int);
void main()
{
int **a;
int **b;
int **s;
int r,c,i,j;
clrscr();
printf("Enter no of row and columns of a matrixn");
scanf("%d%d",&r,&c);
for(i=0;i<r;i++)
{
*(a+i)=(int*)malloc(sizeof(int)*c);
*(b+i)=(int*)malloc(sizeof(int)*c);
*(s+i)=(int*)malloc(sizeof(int)*c);
}
printf(“Enter elements of first matrix”);
read(a,,r,c);
printf(“Enter elements of Second matrix”);
read(b,,r,c);
for(i=0;i<r;i++)
{
for(j=0;j<c;j++)
{
*(*(s+i)+j)=*(*(a+i)+j)+*(*(b+i)+j);
}
}
printf("Matrix A is:nn");
write(a,r,c);
printf("Matrix B is:nn");
csitnepal

write(b,r,c);
printf("Sum ofmatrix A and B is:nn");
write(s,r,c);
getch();
}
void read(int **x,,int r,int c)
{
int i,j;
for(i=0;i<r;i++)
{
for(j=0;j<c;j++)
{
scanf("%d",*(x+i)+j);
}
}
}
void write(int**y,int r,int c)
{
int i,j;
for(i=0;i<r;i++)
{
for(j=0;j<c;j++)
{
printf("%dt",*(*(y+i)+j));
}
printf("n");
}
}
Lab work No:1
#Write a program using user defined functions to sum two 2-dimensional arrays and
store the sum of the corresponding elements into third array and print all the three
arrays with its values at the corresponding places (not in a single row), get the
transpose of the third matrix and print
#include<stdio.h>
#include<conio.h>
void read(int[10][10],char,int,int);
void write(int[10][10],int,int);
void main()
{
int a[10][10];
int b[10][10];
int s[10][10];
int t[10][10];
int r,c,i,j;
clrscr();
printf("Enter no of row and columns of a matrixn");
scanf("%d%d",&r,&c);
read(a,'a',r,c);
read(b,'b',r,c);
for(i=0;i<r;i++)
{
for(j=0;j<c;j++)
{
s[i][j]=a[i][j]+b[i][j];
csitnepal

}
}
write(a,r,c);
write(b,r,c);
printf("Sum of matrix A and B is:nn");
write(s,r,c);
for(i=0;i<r;i++)
{
for(j=0;j<c;j++)
{
t[i][j]=s[j][i];
}
}
printf("Transpose of tesultant matrix is:n");
write(t,r,c);
getch();
}
void read(int x[10][10],char ch,int r,int c)
{
int i,j;
printf("Enter elements of a matrix[%c]n",ch);
for(i=0;i<r;i++)
{
for(j=0;j<c;j++)
{
scanf("%d",&x[i][j]);
}
}
}
void write(int y[10][10],int r,int c)
{
int i,j;
for(i=0;i<r;i++)
{
for(j=0;j<c;j++)
{
printf("%dt",y[i][j]);
}
printf("n");
}
}
Lab work No:2
Write a program using an array to perform the following tasks:(use switch case for
menu)
a. Insert element into an array at specified position
b. Delete element from an array
c. Traversing
d. Searching a particular element in the array
#include<stdio.h>
csitnepal

#include<conio.h>
void insert(int [100], int*);
void delet(int [100], int*);
void traverse(int [100], int*);
void searching(int [100], int*);
void main()
{
int a[100],nel,pos,i;
int n;
int choice;
clrscr();
printf("Enter no of elements to be inserted");
scanf("%d", &n);
printf("Enter %d elements", n);
for(i=0;i<n;i++)
{
scanf("%d", &a[i]);
}
do
{
printf("nmanu for program:n");
printf("1:insertn2:deleten3:Traversen4:searchingn5:exitn");
printf("Enter your choicen");
scanf("%d",&choice);
switch(choice)
{
case 1:
insert(a,&n);
break;
case 2:
delet(a,&n);
break;
case 3:
traverse(a,&n);
break;
case 4:
searching(a,&n);
break;
case 5:
exit(1);
break;
default:
printf("Invalied choice");
}
}while(choice<6);
}
void insert(int a[100], int *n)
{
int pos, nel, i;
printf("Enter position at which you want to insert new element");
scanf("%d", &pos);
printf("Enter new element");
scanf("%d", &nel);
for(i=*n-1; i>=pos; i--)
{
csitnepal

a[i+1] = a[i];
}
a[pos]=nel;
*n=*n+1;
printf("New array is:n");
for(i=0; i<*n; i++)
{
printf("%dt", a[i]);
}
}
void delet(int a[100], int *n)
{
int pos, i;
printf("Enter position at which you want to delete an element");
scanf("%d", &pos);
for(i=pos; i<*n; i++)
{
a[i] = a[i+1];
}
*n=*n-1;
printf("New array is:n");
for(i=0; i<*n; i++)
{
printf("%dt", a[i]);
}
}
void traverse(int a[100], int *n)
{
int i;
printf("Elements of array are:n");
for(i=0;i<*n;i++)
{
printf("%dt",a[i]);
}
}
void searching(int a[100], int *n)
{
int k,i;
printf("Enter searched item");
scanf("%d",&k);
for(i=0;i<*n;i++)
{
if(k==a[i])
{
printf("******successful search******");
break;
}
}
if(i==*n)
printf("*********unsuccessful search*********");
}
csitnepal

Lab work No:3
Write a program to multiply two m*n matrices using dynamic memory allocation.
Hint c[i][j]+=a[i][k]*b[[k][j];
#include<stdio.h>
#include<conio.h>
#include<stdlib.h>
void read(int**,int,int);
void write(int**,int,int);
void main()
{
int **a;
int **b;
int **m;
int r1,c1,r2,c2,i,j,k;
clrscr();
printf("Enter no of row and columns of first matrixn");
scanf("%d%d",&r1,&c1);
printf("Enter no of row and columns of second matrixn");
scanf("%d%d",&r2,&c2);
for(i=0;i<r1;i++)
{
*(a+i)=(int*)malloc(sizeof(int)*c1);
*(m+i)=(int*)malloc(sizeof(int)*c2);
}
for(i=0;i<r2;i++)
{
*(b+i)=(int*)malloc(sizeof(int)*c2);
}
printf("Enter elements of first matrix”);
read(a,r1,c1);
printf("Enter elements of Second matrixn”);
read(b,r2,c2);
for(i=0;i<r1;i++)
{
for(j=0;j<c2;j++)
{
*(*(m+i)+j)=0;
for(k=0;k<c1;k++)
{
(*(*(m+i)+j))+=*(*(a+i)+k)*(*(*(b+k)+j));
}
}
}
write(a,r1,c1);
write(b,r2,c2);
printf("product of matrix A and B is:nn");
write(m,r1,c2);
getch();
}
void read(int **x,int r,int c)
{
csitnepal

int i,j;
for(i=0;i<r;i++)
{
for(j=0;j<c;j++)
{
scanf("%d",*(x+i)+j);
}
}
}
void write(int**y,int r,int c)
{
int i,j;
for(i=0;i<r;i++)
{
for(j=0;j<c;j++)
{
printf("%dt",*(*(y+i)+j));
}
printf("n");
}
}
Unit 2
Algorithm
• Concept and definition
• Design of algorithm
• Characteristic of algorithm
• Big O notation
Algorithm:
An algorithm is a precise specification of a sequence of instructions to be carried out in
order to solve a given problem. Each instruction tells what task is to be done. There
should be a finite number of instructions in an algorithm and each instruction should be
executed in a finite amount of time.
Properties of Algorithms:
• Input: A number of quantities are provided to an algorithm initially before the
algorithm begins. These quantities are inputs which are processed by the
algorithm.
• Definiteness: Each step must be clear and unambiguous.
• Effectiveness: Each step must be carried out in finite time.
• Finiteness: Algorithms must terminate after finite time or step
• Output: An algorithm must have output.
csitnepal

• Correctness: Correct set of output values must be produced from the each set of
inputs.
Write an algorithm to find the greatest number among three numbers:
Step 1: Read three numbers and store them in X, Y and Z
Step 2: Compare X and Y. if X is greater than Y then go to step 5 else step 3
Step 3: Compare Y and Z. if Y is greater than Z then print “Y is greatest” and go to step
7 otherwise go to step 4
Step 4: Print “Z is greatest” and go to step 7
Step 5: Compare X and Z. if X is greater than Z then print “X is greatest” and go to step
7 otherwise go to step 6
Step 6: Print “Z is greatest” and go to step 7
Step 7: Stop
Big Oh (O) notation:
When we have only asymptotic upper bound then we use O notation. A function
f(x)=O(g(x)) (read as f(x) is big oh of g(x) ) iff there exists two positive constants c and
x0 such that
for all x >= x0, f(x) <= c*g(x)
The above relation says that g(x) is an upper bound of f(x)
O(1) is used to denote constants.
Example:
f(x)=5x3+3x2+4 find big oh(O) of f(x)
solution:f(x)= 5x3+3x2+4<= 5x3+3x3+4x3 if x>0
<=12x3
=>f(x)<=c.g(x)
where c=12 and g(x)=x3
Thus by definition of big oh O(f(x))=O(x3)
Big Omega (Ω) notation:
Big omega notation gives asymptotic lower bound. A function f(x) =Ω (g(x)) (read as
g(x) is big omega of g(x) ) iff there exists two positive constants c and x0 such that for all x >=
x0, 0 <= c*g(x) <= f(x).
The above relation says that g(x) is a lower bound of f(x).
csitnepal

Big Theta (Θ) notation:
When we need asymptotically tight bound then we use notation. A function f(x) =
(g(x)) (read as f(x) is big theta of g(x) ) iff there exists three positive constants c1, c2 and x0
such that for all x >= x0, c1*g(x) <= f(x) <= c2*g(x)
The above relation says that f(x) is order of g(x)
Example:
f(n) = 3n2 + 4n + 7
g(n) = n2 , then prove that f(n) = (g(n)).
Proof: let us choose c1, c2 and n0 values as 14, 1 and 1 respectively then we can have,
f(n) <= c1*g(n), n>=n0 as 3n2 + 4n + 7 <= 14*n2 , and
f(n) >= c2*g(n), n>=n0 as 3n2 + 4n + 7 >= 1*n2
for all n >= 1(in both cases).
So c2*g(n) <= f(n) <= c1*g(n) is trivial.
Hence f(n) = Q (g(n)).
Example : Fibonacci Numbers
Input: n
Output: nth
Fibonacci number.
Algorithm: assume a as first(previous) and b as second(current) numbers
fib(n)
{
a = 0, b= 1, f=1 ;
for(i = 2 ; i <=n ; i++)
{
f = a+b ;
a=b ;
b=f ;
}
return f ;
}
Efficiency:
Time Complexity: The algorithm above iterates up to n-2 times, so time complexity is
O(n).
Space Complexity: The space complexity is constant i.e. O(1).
Example : Bubble sort
Algorithm
BubbleSort(A, n)
{
for(i = 0; i <n-1; i++)
{
for(j = 0; j < n-i-1; j++)
{
if(A[j] > A[j+1])
{
csitnepal

temp = A[j];
A[j] = A[j+1];
A[j+1] = temp;
}
}
}
}
Time Complexity:
Inner loop executes for (n-1) times when i=0, (n-2) times when i=1 and so on:
Time complexity = (n-1) + (n-2) + (n-3) + …………………………. +2 +1
= O(n2)
Unit:-3
The stack
a. concept and definition
• primitive operations
• Stack as an ADT
• Implementing PUSH and POP operation
• Testing for overflow and underflow conditions
b. The infix, postfix and prefix
• Evaluating the postfix operation
• Converting from infix to postfix
c. Recursion
• Implementation of:
 Multiplication of natural numbers
 Factorial
 Fibonacci sequences
 The tower of Hanoi
Introduction to Stack
A stack is an ordered collection of items into which new items may be inserted and
from which items may be deleted at one end, called the top of the stack. The deletion
and insertion in a stack is done from top of the stack.
The following fig shows the stack containing items:
Top
(Fig: A stack containing elements or items)
D
C
B
A
29
csitnepal

Intuitively, a stack is like a pile of plates where we can only (conveniently) remove a
plate from the top and can only add a new plate on the top.
In computer science we commonly place numbers on a stack, or perhaps place records
on the stack
Applications of Stack:
Stack is used directly and indirectly in the following fields:
 To evaluate the expressions (postfix, prefix)
 To keep the page-visited history in a Web browser
 To perform the undo sequence in a text editor
 Used in recursion
 To pass the parameters between the functions in a C program
 Can be used as an auxiliary data structure for implementing algorithms
 Can be used as a component of other data structures
Stack Operations:
The following operations can be performed on a stack:
PUSH operation: The push operation is used to add (or push or insert) elements in a
stack
 When we add an item to a stack, we say that we push it onto the stack
 The last item put into the stack is at the top
POP operation: The pop operation is used to remove or delete the top element from
the stack.
 When we remove an item, we say that we pop it from the stack
csitnepal

 When an item popped, it is always the top item which is removed
The PUSH and the POP operations are the basic or primitive operations on a stack.
Some others operations are:
 CreateEmptyStack operation: This operation is used to create an empty stack.
 IsFull operation: The isfull operation is used to check whether the stack is full
or not ( i.e. stack overflow)
 IsEmpty operation: The isempty operation is used to check whether the stack
is empty or not. (i. e. stack underflow)
 Top operations: This operation returns the current item at the top of the stack,
it doesn’t remove it
The Stack ADT:
A stack of elements of type T is a finite sequence of elements of T together with the
operations
 CreateEmptyStack(S): Create or make stack S be an empty stack
 Push(S, x): Insert x at one end of the stack, called its top
 Top(S): If stack S is not empty; then retrieve the element at its top
 Pop(S): If stack S is not empty; then delete the element at its top
 IsFull(S): Determine if S is full or not. Return true if S is full stack; return false
otherwise
 IsEmpty(S): Determine if S is empty or not. Return true if S is an empty stack;
return false otherwise.
Implementation of Stack:
Stack can be implemented in two ways:
1. Array Implementation of stack (or static implementation)
2. Linked list implementation of stack (or dynamic)
Array (static) implementation of a stack:
It is one of two ways to implement a stack that uses a one dimensional array to
store the data. In this implementation top is an integer value (an index of an array) that
indicates the top position of a stack. Each time data is added or removed, top is
incremented or decremented accordingly, to keep track of current top of the stack. By
convention, in C implementation the empty stack is indicated by setting the value of top
to -1(top=-1).
#define MAX 10
csitnepal

sruct stack
{
int items[MAX]; //Declaring an array to store items
int top; //Top of a stack
};
typedef struct stack st;
Creating Empty stack :
The value of top=-1 indicates the empty stack in C implementation.
/*Function to create an empty stack*/
void create_empty_stack(st *s)
{
s->top=-1;
}
Stack Empty or Underflow:
This is the situation when the stack contains no element. At this point the top of stack is
present at the bottom of the stack. In array implementation of stack, conventionally
top=-1 indicates the empty.
The following function return 1 if the stack is empty, 0 otherwise.
int isempty(st *s)
{
if(s->top==-1)
return 1;
else
return 0;
}
Stack Full or Overflow:
This is the situation when the stack becomes full, and no more elements can be pushed
onto the stack. At this point the stack top is present at the highest location (MAXSIZE-
1) of the stack. The following function returns true (1) if stack is full false (0) otherwise.
int isfull(st *s)
{
if(s->top==MAX-1)
return 1;
else
return 0;
}
Algorithm for PUSH and POP operations on Stack
Let Stack[MAXSIZE] be an array to implement the stack. The variable top denotes the
top of the stack.
i) Algorithm for PUSH (inserting an item into the stack) operation:
This algorithm adds or inserts an item at the top of the stack
1. [Check for stack overflow?]
if top=MAXSIZE-1 then
print “Stack Overflow” and Exit
else
Set top=top+1 [Increase top by 1]
csitnepal

Set Stack[top]:= item [Inserts item in new top position]
2. Exit
ii) Algorithm for POP (removing an item from the stack) operation
This algorithm deletes the top element of the stack and assign it to a variable item
1.[Check for the stack Underflow]
If top<0 then
Print “Stack Underflow” and Exit
else
[Remove the top element]
Set item=Stack [top]
[Decrement top by 1]
Set top=top-1
Return the deleted item from the stack
2. Exit
The PUSH and POP functions
The C function for push operation
void push(st *s, int element)
{
if(isfull(s)) /* Checking Overflow condition */
printf("n n The stack is overflow: Stack Full!!n");
else
s->items[++(s->top)]=element; /* First increase top by 1 and store element at top position*/
}
OR
Alternatively we can define the push function as give below:
void push()
{
int item;
if(top == MAXSIZE - 1) //Checking stack overflow
printf("n The Stack Is Full");
else
{
printf("Enter the element to be inserted");
scanf("%d",&item); //reading an item
top= top+1; //increase top by 1
stack[top] = item; //storing the item at the top of
the stack
}
}
The C function for POP operation
void pop(stack *s)
{
if(isempty(s))
printf("nnstack Underflow: Empty Stack!!!");
else
printf("nthe deleted item is %d:t",s->items[s->top--]);/*deletes top element and
decrease top by 1 */
}
OR
Alternatively we can define the push function as give below:
csitnepal

void pop()
{
int item;
if(top <01) //Checking Stack Underflow
printf("The stack is Empty");
else
{
item = stack[top]; //Storing top element to item variable
top = top-1; //Decrease top by 1
printf(“The popped item is=%d”,item); //Displaying the deleted item
}
}
The following complete program illustrates the implementation of stack with
operations:Z
Program: 1
/* Array Implementation of Stack */
#include<stdio.h>
#include<conio.h>
#define MAX 10
struct stack
{
int items[MAX]; //Declaring an array to store items
int top; //Top of a stack
};
void create_empty_stack(st *s); //function prototype
void push(st *s, int);
void pop(st *s);
void display(st *s);
//Main Function
void main()
{
int element, choice;
st *s;
int flag=1;
clrscr();
create_empty_stack(s); /* s->top=-1; indicates empty stack */
do
{
printf("nn Enter your choice");
printf(" nnt 1:Push the elements");
printf(" nnt 2: To display the elements");
printf(" nnt 3: Pop the element");
printf(" nnt 4: Exit");
printf("nnn Enter of your choice:t");
switch(choice)
{
case 1:
printf("n Enter the number:");
scanf("%d", &element); /*Read an element from keyboard*/
push(s,element);
break;
case 2:
display(s);
csitnepal

break;
case 3: clrscr();
pop(s);
break;
case 4:
flag=0;
break;
default:
printf("n Invalid Choice");
}
}while(flag);
getch();
}
/*Function to create an empty stack*/
void create_empty_stack(st *s)
{
s->top=-1;
}
/*Function to check whether the stack is empty or not */
int isempty(st *s)
{
if(s->top==-1)
return 1;
else
return 0;
}
/*function to check whether the stack is full or not*/
int isfull(st *s)
{
if(s->top==MAX-1)
return 1;
else
return 0;
}
/* push() function definition */
void push(st *s, int element)
{
if(isfull(s)) /* Checking Overflow condition */
printf("n nThe stack is overflow: Stack Full!!n");
else
s->items[++(s->top)]=element;
}
/* Function for displaying elements of a stack*/
void display(st *s)
{
int i;
if(isempty(s))
printf("nThe Stack does not contain any Elements");
else
{
printf("nThe elements in the stack is/are:n");
for(i=s->top;i>=0;i--)
printf("%dn",s->items[i]);
}
}
csitnepal

/* the POP function definition*/
void pop(st *s)
{
if(isempty(s))
printf("nnstack Underflow: Empty Stack!!!");
else
printf("nnthe deleted item is %d:t",s->items[s->top--]);
}
Lab work No:4
/*implementation of stack by 2nd
ways*/
#include<stdio.h>
#include<conio.h>
#define MAX 100
struct stack
{
int item[MAX];
int tos;
};
void push(st*, int);
int pop(st*);
void display(st*);
void main()
{
int dta, ch, x;
st *s;
clrscr();
s->tos=-1;
printf("n**************menu for program*************:n");
printf("1:pushn2:popn3:displayn4:exitn");
do
{ printf("nEnter yout choicen");
scanf("%d",&ch);
switch(ch)
{
case 1:
printf("Enter data to be insertedn");
scanf("%d",&dta);
push(s,dta);
break;
case 2:
x=pop(s);
printf("npoped item is:");
printf("%dn",x);
break;
case 3:
display(s);
break;
default:
exit(1);
}
}while(ch<4);
getch();
}
/*******push function**************/
csitnepal

void push(st *s,int d)
{
if(s->tos==MAX-1)
{
printf("Stack is fulln");
}
else
{
++s->tos;
s->item[s->tos]=d;
}
}
/***********pop function**************/
int pop(st *s)
{
int itm;
if(s->tos==-1)
{
printf("Stack is emptyn");
return(0);
}
else
{
itm=s->item[s->tos];
s->tos--;
return(itm);
}
}
/*************display function********************/
void display(st *s)
{
int i;
if(s->tos==-1)
printf("There is no data item to displayn");
else
{
for(i=s->tos; i>=0; i--)
{
printf("%dt", s->item[i]);
}
}
}
Infix, Prefix and Postfix Notation
One of the applications of the stack is to evaluate the expression. We can represent the
expression following three types of notation:
 Infix
 Prefix
 Postfix
csitnepal

 Infix expression: It is an ordinary mathematical notation of expression where
operator is written in between the operands. Example: A+B. Here ‘+’ is an
operator and A and B are called operands
 Prefix notation: In prefix notation the operator precedes the two operands. That is
the operator is written before the operands. It is also called polish notation.
Example: +AB
 Postfix notation: In postfix notation the operators are written after the operands
so it is called the postfix notation (post mean after). In this notation the operator
follows the two operands. Example: AB+
Examples:
A + B (Infix)
+ AB (Prefix)
AB + (Postfix)
 Both prefix and postfix are parenthesis free expressions. For example
(A + B) * C Infix form
* + A B C Prefix form
A B + C * Postfix form
Converting an Infix Expression to Postfix
First convert the sub-expression to postfix that is to be evaluated first and repeat
this process. You substitute intermediate postfix sub-expression by any variable
whenever necessary that makes it easy to convert.
 Remember, to convert an infix expression to its postfix equivalent, we first convert
the innermost parenthesis to postfix, resulting as a new operand
 In this fashion parenthesis can be successively eliminated until the entire
expression is converted
 The last pair of parenthesis to be opened within a group of parenthesis encloses
the first expression within the group to be transformed
 This last in, first-out behavior suggests the use of a stack
Precedence rule:
While converting infix to postfix you have to consider the precedence rule, and the
precedence rules are as follows
csitnepal

1. Exponentiation ( the expression A$B is A raised to the B power, so that 3$2=9)
2. Multiplication/Division
3. Addition/Subtraction
When un-parenthesized operators of the same precedence are scanned, the order is
assumed to be left to right except in the case of exponentiation, where the order is
assumed to be from right to left.
 A+B+C means (A+B)+C
 A$B$C means A$(B$C)
By using parenthesis we can override the default precedence.
Consider an example that illustrate the converting of infix to postfix expression, A + (B* C).
Use the following rule to convert it in postfix:
1. Parenthesis for emphasis
2. Convert the multiplication
3. Convert the addition
4. Post-fix form
Illustration:
A + (B * C). Infix form
A + (B * C) Parenthesis for emphasis
A + (BC*) Convert the multiplication
A (BC*) + Convert the addition
ABC*+ Post-fix form
Consider an example:
(A + B) * ((C - D) + E) / F Infix form
(AB+) * ((C – D) + E) / F
(AB+) * ((CD-) + E) / F
(AB+) * (CD-E+) / F
(AB+CD-E+*) / F
AB+CD-E+*F/ Postfix form
Examples
xercise: Convert the infix expression listed in the above table into postfix notation and
verify yourself.
csitnepal

Algorithm to convert infix to postfix notation
Let here two stacks opstack and poststack are used and otos & ptos represents the
opstack top and poststack top respectively.
1. Scan one character at a time of an infix expression from left to right
2. opstack=the empty stack
3. Repeat till there is data in infix expression
3.1 if scanned character is ‘(‘ then push it to opstack
3.2 if scanned character is operand then push it to poststack
3.3 if scanned character is operator then
if(opstack!=-1)
while(precedence (opstack[otos])>precedence(scan character)) then
pop and push it into poststack
otherwise
push into opstack
3.4 if scanned character is ‘)’ then
pop and push into poststack until ‘(‘ is not found and ignore both symbols
4. pop and push into poststack until opstack is not empty.
5. return
Trace of Conversion Algorithm
The following tracing of the algorithm illustrates the algorithm. Consider an infix
expression
((A-(B+C))*D)$(E+F)
Scan
character
Poststack opstack
( …… (
( …… ((
A A ((
- A (( -
( A (( -(
B AB (( -(
+ AB (( -( +
C ABC (( -( +
) ABC+ (( -
) ABC+- (
* ABC+- (*
D ABC+-D (*
) ABC+-D* …….
$ ABC+-D* $
( ABC+-D* $(
E ABC+-D*E $(
+ ABC+-D*E $(+
F ABC+-D*EF $(+
) ABC+-D*EF+ $
…… ABC+-D*EF+$ (postfix) …………….
csitnepal

Converting an Infix expression to Prefix expression
The precedence rule for converting from an expression from infix to prefix are identical.
Only changes from postfix conversion is that the operator is placed before the operands
rather than after them. The prefix of
A+B-C is -+ABC.
A+B-C (infix)
=(+AB)-C
=-+ABC (prefix)
Example Consider an example:
A $ B * C – D + E / F / (G + H) infix form
= A $ B * C – D + E / F /(+GH)
=$AB* C – D + E / F /(+GH)
=*$ABC-D+E/F/(+GH)
=*$ABC-D+(/EF)/(+GH)
=*$ABC-D+//EF+GH
= (-*$ABCD) + (//EF+GH)
=+-*$ABCD//EF+GH which is in prefix form.
Evaluating the Postfix expression
Each operator in a postfix expression refers to the previous two operands in the
expression.
To evaluate the postfix expression we use the following procedure:
Each time we read an operand we push it onto a stack. When we reach an operator, its
operand s will be the top two elements on the stack. We can then pop these two elements
perform the indicated operation on them and push the result on the stack so that it will
be available for use as an operand of the next operator.
Consider an example
3 4 5 * +
=3 20 +
=23 (answer)
Evaluating the given postfix expression:
6 2 3 + - 3 8 2 / + * 2 $ 3 +
=6 5 - 3 8 2 / + * 2 $ 3 +
=1 3 8 2 / + * 2 $ 3 +
=1 3 4 + * 2 $ 3 +
=1 7 * 2 $ 3 +
=7 2 $ 3 +
=49 3 +
= 52
csitnepal

Algorithm to evaluate the postfix expression
Here we use only one stack called vstack(value stack).
1. Scan one character at a time from left to right of given postfix expression
1.1 if scanned symbol is operand then
read its corresponding value and push it into vstack
1.2 if scanned symbol is operator then
– pop and place into op2
– op and place into op1
– compute result according to given operator and push result into vstack
2. pop and display which is required value of the given postfix expression
3. return
Trace of Evaluation:
Consider an example to evaluate the postfix expression tracing the algorithm
ABC+*CBA-+*
123+*321-+*
Scanned
character
value Op2 Op1 Result vstack
A 1 …… ……. ……. 1
B 2 …… …… ….. 1 2
C 3 …… …… ….. 1 2 3
+ ……. 3 2 5 1 5
* …… 5 1 5 5
C 3 ……. …… …… 5 3
B 2 … ……. 5 3 2
A 1 ……. ……. …… 5 3 2 1
- ……. 1 2 1 5 3 1
+ ……. 1 3 4 5 4
* ……. 4 5 20 20
Its final value is 20.
Evaluating the Prefix Expression
To evaluate the prefix expression we use two stacks and some time it is called two stack
algorithms. One stack is used to store operators and another is used to store the
operands. Consider an example for this
+ 5 *3 2 prefix expression
= +5 6
=11
Illustration: Evaluate the given prefix expression
/ + 5 3 – 4 2 prefix equivalent to (5+3)/(4-2) infix notation
= / 8 – 4 2
= / 8 2
= 4
csitnepal

/*program for evaluating postfix expression*/
#include<stdio.h>
#include<conio.h>
#include<math.h>
#include<string.h>
void push(int);
int pop();
int vstack[100];
int tos=-1;
void main()
{
int i,res,l,op1,op2,value[100];
char postfix[100],ch;
clrscr();
printf("Enter a valid postfixn");
gets(postfix);
l=strlen(postfix);
for(i=0;i<=l-1;i++)
{
if(isalpha(postfix[i]))
{
printf("Enter value of %c",postfix[i]);
scanf("%d",&value[i]);
push(value[i]);
}
else
{
ch=postfix[i];
op2=pop();
op1=pop();
switch(ch)
{
case '+':
push(op1+op2);
break;
case'-':
push(op1-op2);
break;
case'*':
push(op1*op2);
break;
case'/':
push(op1/op2);
break;
case'$':
push(pow(op1,op2));
break;
case'%':
push(op1%op2);
break;
}
}
}
printf("The reault is:");
res=pop();
printf("%d", res);
csitnepal

getch();
}
/***********insertion function*************/
void push(int val)
{
vstack[++tos]=val;
}
/***********deletion function***************/
int pop()
{
int n;
n=vstack[tos--];
return(n);
}
/*program to convert infix to postfix expression*/
#include<stdio.h>
#include<conio.h>
#include<math.h>
#include<string.h>
int precedency(char);
void main()
{
int i,otos=-1,ptos=-1,l, l1;
char infix[100],poststack[100],opstack[100];
printf("Enter a valid infixn");
gets(infix);
l=strlen(infix);
l1=l;
for(i=0;i<=l-1;i++)
{
if(infix[i]=='(')
{
opstack[++otos]=infix[i];
l1++;
}
else if(isalpha(infix[i]))
{
poststack[++ptos]=infix[i];
}
else if (infix[i]==')')
{
l1++;
while(opstack[otos]!='(')
{
poststack[++ptos]=opstack[otos];
otos--;
}
otos--;
}
else //operators
{
if(precedency(opstack[otos])>precedency(infix[i]))
{
poststack[++ptos]=opstack[otos--];
csitnepal

}
}
}
while(otos!=-1)
{
poststack[++ptos]=opstack[otos];
otos--;
}
/********for displaying***************/
for(i=0;i<l1;i++)
{
printf("%c",poststack[i]);
}
getch();
}
/****************precedency function*********************/
int precedency(char ch)
{
switch(ch)
{
case '$':
return(4);
// break;
case'*':
case'/':
return(3);
// break;
case'+':
case'-':
return(2);
// break;
default:
return(1);
}
}
Recursion:
Recursion is a process by which a function calls itself repeatedly, until some
specified condition has been satisfied. The process is used for repetitive computations in
which each action is stated in terms of a previous result.
In order to solve a problem recursively, two conditions must be satisfied. First, the
problem must be written in a recursive form, and second, the problem statement must
include a stopping condition.
csitnepal

Example:
/*calculation of the factorial of an integer number using recursive function*/
#include<stdio.h>
#include<conio.h>
void main()
{
int n;
long int facto;
long int factorial(int n);
printf("Enter value of n:");
scanf("%d",&n);
facto=factorial(n);
printf("%d! = %ld",n,facto);
getch();
}
long int factorial(int n)
{
if(n == 0)
return 1;
else
return n * factorial(n-1);
}
Let’s trace the evaluation of factorial(5):
Factorial(5)=
5*Factorial(4)=
5*(4*Factorial(3))=
5*(4*(3*Factorial(2)))=
5*(4*(3*(2*Factorial(1))))=
5*(4*(3*(2*(1*Factorial(0)))))=
5*(4*(3*(2*(1*1))))=
5*(4*(3*(2*1)))=
5*(4*(3*2))=
5*(4*6)=
5*24=
120
Example:
/*calculation of the factorial of an integer number without using recursive function*/
#include<stdio.h>
#include<conio.h>
void main()
{
int n;
long int facto;
long int factorial(int n);
printf("Enter value of n:");
scanf("%d",&n);
facto=factorial(n);
printf("%d! = %ld",n,facto);
getch();
}
csitnepal

long int factorial(int n)
{
long int facto=1;
int i;
if(n==0)
return 1;
else {
for(i=1;i<=n;i++)
facto=facto*i;
return facto;
}
}
/* Program to generate Fibonacci series up to n terms using recursive function*/
#include<stdio.h>
#include<conio.h>
void main()
{
int n,i;
int fibo(int);
printf("Enter n:");
scanf("%d",&n);
printf("Fibonacci numbers up to %d terms:n",n);
for(i=1;i<=n;i++)
printf("%dn",fibo(i));
getch();
}
int fibo(int k)
{
if(k == 1 || k == 2)
return 1;
else
return fibo(k-1)+fibo(k-2);
}
/* Program to find sum of first n natural numbers using recursion*/
#include<stdio.h>
#include<conio.h>
void main()
{
int n;
int sum_natural(int );
printf("n = ");
scanf("%d",&n);
printf("Sum of first %d natural numbers = %d",n,sum_natural(n));
getch();
}
int sum_natural(int n)
{
if(n == 1)
return 1;
else
return n + sum_natural(n-1);
}
csitnepal

Tower of Hanoi problem:
Initial state:
• There are three poles named as origin, intermediate and destination.
• n number of different-sized disks having hole at the center is stacked around the
origin pole in decreasing order.
• The disks are numbered as 1, 2, 3, 4, ……………….,n.
Objective:
• Transfer all disks from origin pole to destination pole using intermediate pole for
temporary storage.
Conditions:
• Move only one disk at a time.
• Each disk must always be placed around one of the pole.
• Never place larger disk on top of smaller disk.
Algorithm: - To move a tower of n disks from source to dest (where n is positive integer):
1. If n ===1:
1.1. Move a single disk from source to dest.
2. If n > 1:
2.1. Let temp be the remaining pole other than source and dest.
2.2. Move a tower of (n – 1) disks form source to temp.
2.3. Move a single disk from source to dest.
2.4. Move a tower of (n – 1) disks form temp to dest.
3. Terminate.
Example: Recursive solution of tower of Hanoi:
#include <stdio.h>
#include <conio.h>
void TOH(int, char, char, char); //Function prototype
void main()
{
int n;
printf(“Enter number of disks”);
scanf(“%d”,&n);
TOH(n,’O’,’D’,’I’);
getch();
}
void TOH(int n, char A, char B, char C)
{
if(n>0)
{
TOH(n-1, A, C, B);
Printf(“Move disk %d from %c to%cn”, n, A, B);
TOH(n-1, C, B, A);
}
}
csitnepal

/* Program to find multiplication of first n natural numbers using
recursion*/
#include<stdio.h>
#include<conio.h>
void main()
{
int n;
int mul_natural(int );
printf("n = ");
scanf("%d",&n);
printf("Product of first %d natural numbers = %d", n, mul_natural(n));
getch();
}
int mul_natural(int n)
{
if(n == 1)
return 1;
else
return (n * mul_natural(n-1));
}
Advantages of Recursion:
• The code may be much easier to write.
• To solve some problems which are naturally recursive such as tower of Hanoi.
Disadvantages of Recursion:
• Recursive functions are generally slower than non-recursive functions.
• May require a lot of memory to hold intermediate results on the system stack.
• It is difficult to think recursively so one must be very careful when writing
recursive functions.
Unit 4
Queues
a) Concept and definition
b) Queue as ADT
c) Implementation of insert and delete operation of
• Linear queue
• Circular queue
d) Concept of priority queue
What is a queue?
> A Queue is an ordered collection of items from which items may be deleted at
one end (called the front of the queue) and into which items may be inserted at the other
end (the rear of the queue).
csitnepal

> The first element inserted into the queue is the first element to be removed. For
this reason a queue is sometimes called a fifo (first-in first-out) list as opposed to the
stack, which is a lifo (last-in first-out).
Example:
Operations on queue:
• MakeEmpty(q): To make q as an empty queue
• Enqueue(q, x): To insert an item x at the rear of the queue, this is also called by
names add, insert.
• Dequeue(q): To delete an item from the front of the queue q. this is also known as
Delete, Remove.
• IsFull(q): To check whether the queue q is full.
• IsEmpty(q): To check whether the queue q is empty
• Traverse (q): To read entire queue that is display the content of the queue.
Enqueue(A): Enqueue(B,C,D):
Dequeue(A): Dequeue(B):
csitnepal

Initialization of queue:
• The queue is initialized by having the rear set to -1, and front set to 0. Let us assume
that maximum number of the element we have in a queue is MAXQUEUE elements as
shown below.
Applications of queue:
• Task waiting for the printing
• Time sharing system for use of CPU
• For access to disk storage
• Task scheduling in operating system
The Queue as a ADT:
A queue q of type T is a finite sequence of elements with the operations
•MakeEmpty(q): To make q as an empty queue
• IsEmpty(q): To check whether the queue q is empty. Return true if q is empty,
return false otherwise.
• IsFull(q): To check whether the queue q is full. Return true in q is full, return
false otherwise.
• Enqueue(q, x): To insert an item x at the rear of the queue, if and only if q is not full.
• Dequeue(q): To delete an item from the front of the queue q. if and only if q is not empty.
• Traverse (q): To read entire queue that is display the content of the queue.
Implementation of queue:
There are two techniques for implementing the queue:
• Array implementation of queue(static memory allocation)
• Linked list implementation of queue(dynamic memory allocation)
Array implementation of queue:
In array implementation of queue, an array is used to store the data elements. Array
implementation is also further classified into two types
 Linear array implementation:
A linear array with two indices always increasing that is rear and front. Linear
array implementation is also called linear queue
rear=-1
front=0
csitnepal

 Circular array implementation:
This is also called circular queue.
Linear queue:
Algorithm for insertion (or Enqueue ) and deletion (Dequeue) in queue:
Algorithm for insertion an item in queue:
1. Initialize front=0 and rear=-1
if rear>=MAXSIZE-1
print “queue overflow” and return
else
set rear=rear+1
queue[rear]=item
2. end
Algorithm to delete an element from the queue:
1. if rear<front
print “queue is empty” and return
else
item=queue[front++]
2. end
Declaration of a Queue:
# define MAXQUEUE 50 /* size of the queue items*/
struct queue
{
int front;
int rear;
int items[MAXQUEUE];
};
typedef struct queue qt;
Defining the operations of linear queue:
• The MakeEmpty function:
void makeEmpty(qt *q)
{
q->rear=-1;
q->front=0;
}
• The IsEmpty function:
int IsEmpty(qt *q)
{
if(q->rear<q->front)
return 1;
else
return 0;
}
• The Isfull function:
int IsFull(qt *q)
csitnepal

{
if(q->rear==MAXQUEUEZIZE-1)
return 1;
else
return 0;
}
• The Enqueue function:
void Enqueue(qt *q, int newitem)
{
if(IsFull(q))
{
printf(“queue is full”);
exit(1);
}
else
{
q->rear++;
q->items[q->rear]=newitem;
}
}
• The Dequeue function:
int Dequeue(qt *q)
{
if(IsEmpty(q))
{
printf(“queue is Empty”);
exit(1);
}
else
{
return(q->items[q->front]);
q->front++;
}
}
Problems with Linear queue implementation:
• Both rear and front indices are increased but never decreased.
• As items are removed from the queue, the storage space at the beginning of the array
is discarded and never used again. Wastage of the space is the main problem with linear
queue which is illustrated by the following example.
0 1 2 3 4 5 6 front=2, rear=6
f r
This queue is considered full, even though the space at beginning is vacant.
11 22 33 44 55
53
csitnepal

/*Array implementation of linear queue*/
#include<stdio.h>
#include<conio.h>
#define SIZE 20
struct queue
{
int item[SIZE];
int rear;
int front;
};
typedef struct queue qu;
void insert(qu*);
void delet(qu*);
void display(qu*);
void main()
{
int ch;
qu *q;
q->rear=-1;
q->front=0;
clrscr();
printf("Menu for program:n");
printf("1:insertn2:deleten3:displayn4:exitn");
do
{ printf("Enter youer choicen");
scanf("%d",&ch);
switch(ch)
{
case 1:
insert(q);
break;
case 2:
delet(q);
break;
case 3:
display(q);
break;
case 4:
exit(1);
break;
default:
printf("Your choice is wrongn");
}
}while(ch<5);
getch();
}
/**********insert function*************/
void insert(qu *q)
{ int d;
printf("Enter data to be insertedn");
scanf("%d",&d);
csitnepal

if(q->rear==SIZE-1)
{
printf("Queue is fulln");
}
else
{
q->rear++;
q->item[q->rear]=d;
}
}
/***********delete function*****************/
void delet(qu *q)
{
int d;
{
printf("Queue is emptyn");
}
else
{
d=q->item[q->front];
q->front++;
printf("Deleted item is:");
printf("%dn",d);
}
}
/**************display function***********/
void display(qu *q)
{
int i;
{
}
else
{
for(i=q->front;i<=q->rear;i++)
{
printf("%dt",q->item[i]);
}
}
}
Circular queue:
A circular queue is one in which the insertion of a new element is done at very
first location of the queue if the last location of the queue is full.
csitnepal

Fig:- Circular queue
 A circular queue overcomes the problem of unutilized space in linear queue
implementation as array.
 In circular queue we sacrifice one element of the array thus to insert n elements in a
circular queue we need an array of size n+1.(or we can insert one less than the size
of the array in circular queue).
Initialization of Circular queue:
rear=front=MAXSIZE-1
Algorithms for inserting an element in a circular queue:
This algorithm is assume that rear and front are initially set to MAZSIZE-1.
1. if (front==(rear+1)%MAXSIZE)
print Queue is full and exit
else
rear=(rear+1)%MAXSIZE; [increment rear by 1]
2. cqueue[rear]=item;
3. end
Algorithms for deleting an element from a circular queue:
This algorithm is assume that rear and front are initially set to MAZSIZE-1.
1. if (rear==front) [checking empty condition]
print Queue is empty and exit
2. front=(front+1)%MAXSIZE; [increment front by 1]
3. item=cqueue[front];
4. return item;
5. end.
Declaration of a Circular Queue:
# define MAXSIZE 50 /* size of the circular queue items*/
struct cqueue
{
int front;
int rear;
int items[MAXSIZE];
};
typedef struct cqueue cq;
csitnepal

Operations of a circular queue:
 The MakeEmpty function:
void makeEmpty(cq *q)
{
q->rear=MAXSIZE-1;
q->front=MAXSIZE-1;
}
 The IsEmpty function:
int IsEmpty(cq *q)
{
return 1;
else
return 0;
}
 The Isfull function:
int IsFull(cq *q)
{
if(q->front==(q->rear+1)%MAXDIZE)
return 1;
else
return 0;
}
 The Enqueue function:
void Enqueue(cq *q, int newitem)
{
if(IsFull(q))
{
printf(“queue is full”);
exit(1);
}
else
{
q->rear=(q->rear+1)%MAXDIZE;
q->items[q->rear]=newitem;
}
}
 The Dequeue function:
int Dequeue(cq *q)
{
if(IsEmpty(q))
{
printf(“queue is Empty”);
exit(1);
}
else
{
q->front=(q->front+1)%MAXSIZE;
return(q->items[q->front]);
csitnepal

}
}
/*implementation of circular queue with secrifying one cell */
#include<stdio.h>
#include<conio.h>
#define SIZE 20
struct cqueue
{
int item[SIZE];
int rear;
int front;
};
typedef struct cqueue qu;
void insert(qu*);
void delet(qu*);
void display(qu*);
void main()
{
int ch;
qu *q;
q->rear=SIZE-1;
q->front=SIZE-1;
clrscr();
do
{
printf("Enter youer choicen");
scanf("%d",&ch);
switch(ch)
{
case 1:
insert(q);
break;
case 2:
delet(q);
break;
case 3:
display(q);
break;
case 4:
exit(1);
break;
default:
break;
}
}while(ch<5);
getch();
}
/**********insert function*************/
void insert(qu *q)
{
int d;
if((q->rear+1)%SIZE==q->front)
csitnepal

else
{
q->rear=(q->rear+1)%SIZE;
printf ("Enter data to be insertedn");
scanf("%d",&d);
q->item[q->rear]=d;
}
}
/**********delete function*****************/
void delet(qu *q)
{
if(q->rear==q->front)
else
{
q->front=(q->front+1)%SIZE;
printf("%dn",q->item[q->front]);
}
}
void display(qu *q)
{
int i;
else
{
printf("Items of queue are:n");
for(i=(q->front+1)%SIZE;i!=q->rear;i=(i+1)%SIZE)
{
}
printf("%dt",q->item[q->rear]);
}
}
/*implementation of circular queue without secrifying one cell by using a count
variable */
#include<stdio.h>
#include<conio.h>
#define SIZE 20
struct cqueue
{
int item[SIZE];
int rear;
int front;
};
int count=0;
typedef struct cqueue qu;
void insert(qu*);
void delet(qu*);
void display(qu*);
void main()
{
csitnepal

int ch;
qu *q;
q->rear=SIZE-1;
q->front=SIZE-1;
clrscr();
do
{
scanf("%d",&ch);
switch(ch)
{
case 1:
insert(q);
break;
case 2:
delet(q);
break;
case 3:
display(q);
break;
case 4:
exit(1);
break;
default:
break;
}
}while(ch<5);
getch();
}
/**********insert function*************/
void insert(qu *q)
{
int d;
if(count==SIZE)
else
{
q->rear=(q->rear+1)%SIZE;
scanf("%d",&d);
q->item[q->rear]=d;
count++;
}
}
/**********delete function*****************/
void delet(qu *q)
{
if(count==0)
else
{
csitnepal

q->front=(q->front+1)%SIZE;
printf("%dn",q->item[q->front]);
count--;
}
}
void display(qu *q)
{
int i;
else
{
for(i=(q->front+1)%SIZE; i!=q->rear; i=(i +1)%SIZE)
{
}
printf("%dt",q->item[q->rear]);
}
}
Priority queue:
A priority queue is a collection of elements such that each element has been
assigned a priority and the order in which elements are deleted and processed comes
from the following rules:.
 An element of higher priority is processed before any element of lower priority.
 If two elements has same priority then they are processed according to the order in
which they were added to the queue.
The best application of priority queue is observed in CPU scheduling.
✔ The jobs which have higher priority are processed first.
✔ If the priority of two jobs is same this jobs are processed according to their
position in queue.
✔ A short job is given higher priority over the longer one.
Types of priority queues:
 Ascending priority queue(min priority queue):
An ascending priority queue is a collection of items into which items can be inserted
arbitrarily but from which only the smallest item can be removed.
 Descending priority queue(max priority queue):
csitnepal

An descending priority queue is a collection of items into which items can be inserted
arbitrarily but from which only the largest item can be removed.
Priority QUEUE Operations:
 Insertion :
The insertion in Priority queues is the same as in non-priority queues.
 Deletion :
Deletion requires a search for the element of highest priority and deletes the
element with highest priority. The following methods can be used for deletion/removal
from a given Priority Queue:
✔ An empty indicator replaces deleted elements.
✔ After each deletion elements can be moved up in the array decrementing the rear.
✔ The array in the queue can be maintained as an ordered circular array
Priority Queue Declaration:
Queue data type of Priority Queue is the same as the Non-priority Queue.
#define MAXQUEUE 10 /* size of the queue items*/
struct pqueue
{
int front;
int rear;
int items[MAXQUEUE];
};
struct pqueue *pq;
The priority queue ADT:
A ascending priority queue of elements of type T is a finite sequence of elements
of T together with the operations:
 MakeEmpty(p): Create an empty priority queue p
 Empty(p): Determine if the priority queue p is empty or not
 Insert(p,x): Add element x on the priority queue p
 DeleteMin(p): If the priority queue p is not empty, remove the minimum element
of the quque and return it.
 FindMin(p): Retrieve the minimum element of the priority queue p.
Array implementation of priority queue:
 Unordered array implementation:
✔ To insert an item, insert it at the rear end of the queue.
✔ To delete an item, find the position of the minimum element and
✗ Either mark it as deleted (lazy deletion) or
✗ shift all elements past the deleted element by on position and then decrement rear.
csitnepal

Fig Illustration of unordered array implementation
 Ordered array implementation:
✔ Set the front as the position of the smallest element and the rear as the position
of the largest element.
✔ To insert an element, locate the proper position of the new element and shift
preceding or succeeding elements by one position.
✔ To delete the minimum element, increment the front position.
Fig Illustration of ordered array implementation
csitnepal

Application of Priority queue:
In a time-sharing computer system, a large number of tasks may be waiting for the
CPU, some of these tasks have higher priority than others. The set of tasks waiting for
the CPU forms a priority queue.
/*implementation of ascending priority queue */
#include<stdio.h>
#include<conio.h>
#define SIZE 20
struct cqueue
{
int item[SIZE];
int rear;
int front;
};
typedef struct queue pq;
void insert(pq*);
void delet(pq*);
void display(pq*);
void main()
{
int ch;
pq *q;
q->rear=-1;
q->front=0;
clrscr();
do
{
scanf("%d",&ch);
switch(ch)
{
case 1:
insert(q);
break;
case 2:
delet(q);
break;
case 3:
display(q);
break;
case 4:
exit(1);
break;
default:
break;
}
}while(ch<5);
getch();
}
/**********insert function*************/
csitnepal

void insert(pq *q)
{
int d;
if(q->rear==SIZE-1)
else
{
scanf("%d",&d);
q->rear++;
q->item[q->rear]=d;
}
}
/**********delete function*****************/
void delet(pq *q)
{
int i, temp=0, x;
x=q->item[q->front];
{
return 0;
}
else
{
for(i=q->front+1; i<q->rear; i++)
{
if(x>q->item[i])
{
temp=i;
x=q->item[i];
}
}
for(i=temp;i< q->rear-1;i++)
{
q->item[i]=q->item[i+1];
}
q->rear--;
return x;
}
}
/************display function***********/
void display(pq *q)
{
int i;
if(q->rear < q->front)
else
{
for(i=(q->front i<=q->rear;i++)
{
}
csitnepal

}
}
Unit 4
Linked List:
b) Inserting and deleting nodes
c) Linked implementation of a stack (PUSH / POP)
d) Linked implementation of a queue (insert / delete)
e) Circular linked list
 Stack as a circular list (PUSH / POP)
 Queue as a circular list (Insert / delete)
f) Doubly linked list (insert / delete)
Self referential structure:
It is sometimes desirable to include within a structure one member that is a pointer
to the parent structure type. Hence, a structure which contains a reference to itself is
called self-referential structure. In general terms, this can be expressed as:
struct node
{
member 1;
member 2;
…….
struct node *name;
};
For example,
struct node
{
int info;
struct node *next;
};
This is a structure of type node. The structure contains two members: a info integer
member, and a pointer to a structure of the same type (i.e., a pointer to a structure of
type node), called next. Therefore this is a self-referential structure.
Linked List:
A linked list is a collection of nodes, where each node consists of two parts:
 info: the actual element to be stored in the list. It is also called data field.
 link: one or two links that points to next and previous node in the list. It is also
called next or pointer field.
Illustration:
fig:- Singly linked list of integer values
info next
list
info next info next
3 8 null
csitnepal

 The nodes in a linked list are not stored contiguously in the memory
 You don't have to shift any element in the list.
 Memory for each node can be allocated dynamically whenever the need arises.
 The size of a linked list can grow or shrink dynamically
Operations on linked list:
The basic operations to be performed on the linked list are as follows:
 Creation: This operation is used to create a linked list
 Insertion: This operation is used to insert a new nose in a kinked list in a
specified position. A new node may be inserted
✔At the beginning of the linked list
✔ At the end of the linked list
✔At he specified position in a linked list
 Deletion: The deletion operation is used to delete a node from the linked list. A
node may be deleted from
✔The beginning of the linked list
✔the end of the linked list
✔ the specified position in the linked list.
 Traversing: The list traversing is a process of going through all the nodes of the
linked list from on end to the other end. The traversing may be either forward or
backward.
 Searching or find: This operation is used to find an element in a linked list. In the
desired element is found then we say operation is successful otherwise
unsuccessful.
 Concatenation: It is the process of appending second list to the end of the first list.
Types of Linked List:
basically we can put linked list into the following four types:
 Singly linked list
 doubly linked list
 circular linked list
 circular doubly linked list
Singly linked list:
A singly linked list is a dynamic data structure which may grow or shrink, and
growing and shrinking depends on the operation made. In this type of linked list each
node contains two fields one is info field which is used to store the data items and
another is link field that is used to point the next node in the list. The last node has a
NULL pointer.
The following example is a singly linked list that contains three elements 5, 3, 8.
Representation of singly linked list:
5
info next
list
info next info next
3 8 null
csitnepal

We can create a structure for the singly linked list the each node has two members,
one is info that is used to store the data items and another is next field that store the
address of next node in the list.
We can define a node as follows:
struct Node
{
int info;
struct Node *next;
};
typedef struct Node NodeType;
NodeType *head; //head is a pointer type structure variable
This type of structure is called self-referential structure.
 The NULL value of the next field of the linked list indicates the last node and we define
macro for NULL and set it to 0 as below:
#define NULL 0
Creating a Node:
 To create a new node, we use the malloc function to dynamically allocate memory
for the new node.
 After creating the node, we can store the new item in the node using a pointer to that
nose.
The following steps clearly shows the steps required to create a node and storing an
item.
Note that p is not a node; instead it is a pointer to a node.
The getNode function:
we can define a function getNode() to allocate the memory for a node
dynamically. It is user-defined function that return a pointer to the newly created node.
Nodetype *getNode()
{
NodeType *p;
p==(NodeType*)malloc(sizeof(NodeType));
return(p);
}
Creating the empty list:
csitnepal

void createEmptyList(NodeType *head)
{
head=NULL;
}
Inserting Nodes:
To insert an element or a node in a linked list, the following three things to be done:
 Allocating a node
 Assigning a data to info field of the node
 Adjusting a pointer and a new node may be inserted
 At the beginning of the linked list
 At the end of the linked list
 At the specified position in a linked list
Insertion requires obtaining a new node ans changing two links
fig:- Inserting the new node with 44 between 33 and 55.
An algorithm to insert a node at the beginning of the singly linked list:
let *head be the pointer to first node in the current list
1. Create a new node using malloc function
NewNode=(NodeType*)malloc(sizeof(NodeType));
2. Assign data to the info field of new node
NewNode->info=newItem;
3. Set next of new node to head
NewNode->next=head;
4. Set the head pointer to the new node
head=NewNode;
5. End
The C function to insert a node at the beginning of the singly linked list:
void InsertAtBeg(int newItem)
{
NodeType *NewNode;
NewNode=getNode();
NewNode->next=head;
head=NewNode;
}
csitnepal

An algorithm to insert a node at the end of the singly linked list:
3. Set next of new node to NULL
NewNode->next=NULL;
4. if (head ==NULL)then
Set head =NewNode.and exit.
5. Set temp=head;
6 while(temp->next!=NULL)
temp=temp->next; //increment temp
7. Set temp->next=NewNode;
8. End
The C function to insert a node at the end of the linked list:
void InsertAtEnd(int newItem)
{
NodeType *NewNode;
NewNode=getNode();
NewNode->next=NULL;
if(head==NULL)
{
head=NewNode;
}
else
{
temp=head;
while(temp->next!=NULL)
{
temp=temp->next;
}
temp->next=NewNode;
}
}
An algorithm to insert a node after the given node in singly linked list:
let *head be the pointer to first node in the current list and *p be the pointer to the node
after which we want to insert a new node.
3. Set next of new node to next of p
csitnepal

NewNode->next=p->next;
4. Set next of p to NewNode
p->next =NewNode..
5. End
The C function to insert a node after the given node in singly linked list:
void InsertAfterNode(NodeType *p int newItem)
{
NodeType *NewNode;
NewNode=getNode();
if(p==NULL)
{
printf(“Void insertion”);
exit(1);
}
else
{
NewNode->next=p->next;
p->next =NewNode..
}
}
An algorithm to insert a node at the specified position in a singly linked list:
3. Enter position of a node at which you want to insert a new node. Let this position is pos.
4. Set temp=head;
printf(“void insertion”); and exit(1).
6. for(i=1; i<pos-1; i++)
temp=temp->next;
7. Set NewNode->next=temp->next;
set temp->next =NewNode..
8. End
The C function to insert a node at the specified position in a singly linked list:
void InsertAtPos(int newItem)
{
NodeType *NewNode;
int pos , i ;
printf(“ Enter position of a node at which you want to insert a new node”);
scanf(“%d”,&pos);
if(head==NULL)
{
printf(“void insertion”);
exit(1).
}
else
{
temp=head;
csitnepal

for(i=1; i<pos-1; i++)
{
temp=temp->next;
}
NewNode=getNode();
NewNode->next=temp->next;
temp->next =NewNode;
}
}
Deleting Nodes:
A node may be deleted:
 From the beginning of the linked list
 from the end of the linked list
 from the specified position in a linked list
Deleting first node of the linked list:
An algorithm to deleting the first node of the singly linked list:
1. If(head==NULL) then
print “Void deletion” and exit
2. Store the address of first node in a temporary variable temp.
temp=head;
3. Set head to next of head.
head=head->next;
4. Free the memory reserved by temp variable.
free(temp);
5. End
The C function to deleting the first node of the singly linked list:
void deleteBeg()
{
NodeType *temp;
if(head==NULL)
{
printf(“Empty list”);
exit(1).
}
else
{
temp=head;
printf(“Deleted item is %d” , head->info);
head=head->next;
free(temp);
}
}
Deleting the last node of the linked list:
An algorithm to deleting the last node of the singly linked list:
csitnepal

1. If(head==NULL) then //if list is empty
print “Void deletion” and exit
2. else if(head->next==NULL) then //if list has only one node
Set temp=head;
print deleted item as,
printf(“%d” ,head->info);
head=NULL;
free(temp);
3. else
set temp=head;
while(temp->next->next!=NULL)
set temp=temp->next;
End of while
free(temp->next);
Set temp->next=NULL;
4. End
The C function to deleting the last node of the singly linked list:
void deleteEnd()
{
NodeType *temp;
if(head==NULL)
{
return;
}
else if(head->next==NULL)
{
temp=head;
head=NULL;
printf(“Deleted item is %d”, temp->info);
free(temp);
}
else
{
temp=head;
{
temp=temp->next;
}
printf(“deleted item is %d'” , temp->next->info):
free(temp->next);
temp->next=NULL;
}
}
An algorithm to delete a node after the given node in singly linked list:
let *head be the pointer to first node in the current list and *p be the pointer to the node
after which we want to delete a new node.
1. if(p==NULL or p->next==NULL) then
print “deletion not possible and exit
2. set q=p->next
3. Set p->next=q->next;
4. free(q)
csitnepal

5. End
The C function to delete a node after the given node in singly linked list:
let *p be the pointer to the node after which we want to delete a new node.
void deleteAfterNode(NodeType *p)
{
NodeType *q;
if(p==NULL || p->next==NULL )
{
printf(“Void insertion”);
exit(1);
}
else
{
q=p->next;
p->next=q->next;
free(q);
}
}
An algorithm to delete a node at the specified position in a singly linked list:
1. Read position of a node which to be deleted, let it be pos.
2. if head==NULL
print “void deletion” and exit
3. Enter position of a node at which you want to delete a new node. Let this position is pos.
4. Set temp=head
declare a pointer of a structure let it be *p
print “void ideletion” and exit
otherwise;.
6. for(i=1; i<pos-1; i++)
temp=temp->next;
7. print deleted item is temp->next->info
8. Set p=temp->next;
9. Set temp->next =temp->next->next;
10. free(p);
11. End
The C function to delete a node at the specified position in a singly linked list
void deleteAtSpecificPos()
{
NodeType *temp *p;
int pos, i;
if(head==NULL)
{
return;
}
else
{
printf(“Enter position of a node which you wand to delete”);
scanf(“%d” , &pos);
csitnepal

temp=head;
for(i=1; i<pos-1; i++)
{
temp=temp->next;
}
p=temp->next;
` printf(“Deleted item is %d”, p->info);
temp->next =p->next;
free(p);
}
}
Searching an item in a linked list:
To search an item from a given linked list we need to find the node that contain this data item.
If we find such a node then searching is successful otherwise searching unsuccessful.
void searchItem()
{
NodeType *temp;
int key;
if(head==NULL)
{
printf(“empty list”);
exit(1);
}
else
{
printf(“Enter searched item”);
scanf('%d” ,&key);
temp=head;
while(temp!=NULL)
{
if(temp->info==key)
{
printf(“Search successful”);
break;
}
temp=temp->next;
}
if(temp==NULL)
printf(“Unsuccessful search”);
}
}
Complete program:
/******Various operations on singly linked list**************/
#include<stdio.h>
#include<conio.h>
#include<malloc.h> //for malloc function
#include<process.h> //fpr exit function
struct node
{
csitnepal

int info;
struct node *next;
};
typedef struct node NodeType;
NodeType *head;
head=NULL;
void insert_atfirst(int);
void insert_givenposition(int);
void insert_atend(int);
void delet_first();
void delet_last();
void delet_nthnode();
void info_sum();
void count_nodes();
void main()
{
int choice;
int item;
clrscr();
do
{
printf("n manu for program:n");
printf("1. insert first n2.insert at given position n3 insert at last n 4:Delete first
noden 5:delete last noden6:delete nth noden7:count nodesn8Display itemsn10:exitn");
printf("enter your choicen");
switch(choice)
{
case 1:
printf(“Enter item to be inserted”);
scanf(“%d”, &item)
insert_atfirst(item);
break;
case 2:
insert_givenposition(item);
break;
case 3:
insert_atend();
break;
case 4:
delet_first();
break;
case 5:
delet_last();
break;
case 6:
delet_nthnode();
break;
case 7:
info_sum();
break;
case 8:
csitnepal

count_nodes();
break;
case 9:
exit(1);
break;
default:
printf("invalid choicen");
break;
}
}while(choice<10);
getch();
}
/************function definitions**************/
void insert_atfirst(int item)
{
NodeType *nnode;
nnode=(NodeType*)malloc(sizeof(NodeType));
nnode->info=item;
nnode->next=head;
head=nnode;
}
void insert_givenposition(int item)
{
NodeType *nnode;
NodeType *temp;
temp=head;
int p,i;
nnode=( NodeType *)malloc(sizeof(NodeType));
nnode->info=item;
if (head==NULL)
{
nnode->next=NULL;
head=nnode;
}
else
{
printf("Enter Position of a node at which you want to insert an new noden");
scanf("%d",&p);
for(i=1;i<p-1;i++)
{
temp=temp->next;
}
nnode->next=temp->next;
temp->next=nnode;
}
}
void insert_atend(int item)
{
NodeType *nnode;
NodeType *temp;
temp=head;
nnode->info=item;
csitnepal

if(head==NULL)
{
nnode->next=NULL;
head=nnode;
}
else
{
{
temp=temp->next;
}
nnode->next=NULL;
temp->next=nnode;
}
}
void delet_first()
{
NodeType *temp;
if(head==NULL)
{
printf("Void deletion|n");
return;
}
else
{
temp=head;
head=head->next;
free(temp);
}
}
void delet_last()
{
NodeType *hold,*temp;
if(head==NULL)
{
return;
}
else if(head->next==NULL)
{
hold=head;
head=NULL;
free(hold);
}
else
{
temp=head;
{
temp=temp->next;
}
hold=temp->next;
temp->next=NULL;
free(hold);
}
csitnepal

}
void delet_nthnode()
{
NodeType *hold,*temp;
int pos, i;
if(head==NULL)
{
return;
}
else
{
temp=head;
printf("Enter position of node which node is to be deletedn");
scanf("%d",&pos);
for(i=1;i<pos-1;i++)
{
temp=temp->next;
}
hold=temp->next;
temp->next=hold->next;
free(hold);
}
}
void info_sum()
{
NodeType *temp;
temp=head;
while(temp!=NULL)
{
printf("%dt",temp->info);
temp=temp->next;
}
}
void count_nodes()
{
int cnt=0;
NodeType *temp;
temp=head;
while(temp!=NULL)
{
cnt++;
temp=temp->next;
}
printf("total nodes=%d",cnt);
}
}
Linked list implementation of Stack:
Push function:
let *top be the top of the stack or pointer to the first node of the list.
void push(item)
{
NodeType *nnode;
int data;
csitnepal

if(top==0)
{
nnode->info=item;
nnode->next=NULL;
top=nnode;
}
else
{
nnode->info=item;
nnode->next=top;
top=nnode;
}
}
Pop function:
let *top be the top of the stack or pointer to the first node of the list.
void pop()
{
NodeType *temp;
if(top==0)
{
printf("Stack contain no elements:n");
return;
}
else
{
temp=top;
top=top->next;
printf("ndeleted item is %dt",temp->info);
free(temp);
}
}
A Complete C program for linked list implementation of stack:
/*************Linked list implementation of stack*************/
#include<stdio.h>
#include<conio.h>
#include<malloc.h>
#include<process.h>
struct node
{
int info;
struct node *next;
};
NodeType *top;
top=0;
void push(int);
void pop();
void display();
void main()
{
int choice, item;
clrscr();
csitnepal

do
{
printf("n1.Push n2.Pop n3.Displayn4:Exitn");
printf("enter ur choicen");
switch(choice)
{
case 1:
printf("nEnter the data:n");
scanf("%d",&item);
push(item);
break;
case 2:
pop();
break;
case 3:
display();
break;
case 4:
exit(1);
break;
default:
break;
}
}while(choice<5);
getch();
}
/**************push function*******************/
void push(int item)
{
NodeType *nnode;
int data;
if(top==0)
{
nnode->info=item;
nnode->next=NULL;
top=nnode;
}
else
{
nnode->info=item;
nnode->next=top;
top=nnode;
}
}
/******************pop function********************/
void pop()
{
NodeType *temp;
if(top==0)
{
printf("Stack contain no elements:n");
return;
csitnepal

}
else
{
temp=top;
top=top->next;
printf("ndeleted item is %dt",temp->info);
free(temp);
}
}
/**************display function***********************/
void display()
{
NodeType *temp;
if(top==0)
{
printf("Stack is emptyn");
return;
}
else
{
temp=top;
printf("Stack items are:n");
while(temp!=0)
{
temp=temp->next;
}
}
}
Linked list implementation of queue:
Insert function:
let *rear and *front are pointers to the first node of the list initially and insertion of node
in linked list done at the rear part and deletion of node from the linked list done from
front part.
rear
void insert(int item)
{
NodeType *nnode;
if(rear==0)
{
nnode->info=item;
nnode->next=NULL;
csitnepal

rear=front=nnode;
}
else
{
nnode->info=item;
nnode->next=NULL;
rear->next=nnode;
rear=nnode;
}
}
Delete function:
let *rear and *front are pointers to the first node of the list initially and insertion of
node in linked list done at the rear part and deletion of node from the linked list done
from front part.
void delet()
{
NodeType *temp;
if(front==0)
{
printf("Queue contain no elements:n");
return;
}
else if(front->next==NULL)
{
temp=front;
rear=front=NULL;
printf("nDeleted item is %dn",temp->info);
free(temp);
}
else
{
temp=front;
front=front->next;
free(temp);
}
}
A Complete C program for linked list implementation of queue:
/**************Linked list implementation of queue*****************/
#include<stdio.h>
#include<conio.h>
#include<malloc.h>
#include<process.h>
struct node
{
int info;
struct node *next;
};
NodeType *rear,*front;
rear=front=0;
void insert(int);
csitnepal

void delet();
void display();
void main()
{
int choice, item;
clrscr();
do
{
printf("n1.Insert n2.Delet n3.Displayn4:Exitn");
printf("enter ur choicen");
switch(choice)
{
case 1:
printf("nEnter the data:n");
scanf("%d",&item);
insert(item);
break;
case 2:
delet();
break;
case 3:
display();
break;
case 4:
exit(1);
break;
default:
break;
}
}while(choice<5);
getch();
}
/**************insert function*******************/
{
NodeType *nnode;
if(rear==0)
{
nnode->info=item;
nnode->next=NULL;
rear=front=nnode;
}
else
{
nnode->info=item;
nnode->next=NULL;
rear->next=nnode;
rear=nnode;
}
}
/******************delet function********************/
csitnepal

void delet()
{
NodeType *temp;
if(front==0)
{
printf("Queue contain no elements:n");
return;
}
else if(front->next==NULL)
{
temp=front;
rear=front=NULL;
free(temp);
}
else
{
temp=front;
front=front->next;
free(temp);
}
}
/**************display function***********************/
void display()
{
NodeType *temp;
temp=front;
printf("nqueue items are:t");
while(temp!=NULL)
{
temp=temp->next;
}
}
Circular Linked list:
A circular linked list is a list where the link field of last node points to the very
first node of the list .
Circular linked lists can be used to help the traverse the same list again and again if
needed. A circular list is very similar to the linear list where in the circular list the
pointer of the last node points not NULL but the first node.
In a circular linked list there are two methods to know if a node is the first node or not.
csitnepal

 Either a external pointer, list, points the first node or
 A header node is placed as the first node of the circular list.
The header node can be separated from the others by either heaving a sentinel value as
the info part or having a dedicated flag variable to specify if the node is a header node
or not.
CIRCULAR LIST with header node
C representation of circular linked list:
we declare the structure for the circular linked list in the same way as declared it
for the linear linked list.
struct node
{
int info;
struct node *next;
};
NodeType *start=NULL:
NodeType *last=NULL:
Algorithms to insert a node in a circular linked list:
Algorithm to insert a node at the beginning of a circular linked list:
1. Create a new node as
newnode=(NodeType*)malloc(sizeof(NodeType));
2. if start==NULL then
set newnode->info=item
set newnode->next=newnode
set start=newnode
set last newnode
end if
3. else
set newnode->next=start
set start=newnode
set last->next=newnode
csitnepal

end else
4. End
Algorithm to insert a node at the end of a circular linked list:
1. Create a new node as
set newnode->next=newnode
set start=newnode
set last newnode
end if
3. else
set last->next=newnode
set last=newnode
set last->next=start
end else
4. End
C function to insert a node at the beginning of a circular linked list:
void InsertAtBeg(int Item)
{
NodeType *newnode;
if(start==NULL)
{
newnode->info=item;
newnode->next=newnode;
start=newnode;
last newnode;
}
else
{
newnode->info=item;
last->next=newnode;
last=newnode;
last->next=start;
}
}
C function to insert a node at the end of a circular linked list:
void InsertAtEnd(int Item)
{
NodeType *newnode;
if(start==NULL)
{
newnode->info=item;
newnode->next=newnode;
start=newnode;
last newnode;
}
else
{
csitnepal

newnode->info=item;
last->next=newnode;
last=newnode;
last->next=start;
}
}
Algorithms to delete a node from a circular linked list:
Algorithm to delete a node from the beginning of a circular linked list:
“empty list” and exit
2. else
set temp=start
set start=start->next
print the deleted element=temp->info
set last->next=start;
free(temp)
end else
3. End
Algorithm to delete a node from the end of a circular linked list:
“empty list” and exit
2. else if start==last
set temp=start
print deleted element=temp->info
free(temp)
start=last=NULL
3. else
set temp=start
while( temp->next!=last)
set temp=temp->next
end while
set hold=temp->next
set last=temp
set last->next=start
print the deleted element=hold->info
free(hold)
end else
4. End
C function to delete a node from the beginning of a circular linked list:
void DeleteFirst()
{
if(start==NULL)
{
exit(1);
}
else
{
temp=start;
start=start->next;
printf(“ the deleted element=%d”, temp->info);
last->next=start;
free(temp)
csitnepal

}
}
C function to delete a node from the end of a circular linked list:
void DeleteLast()
{
if(start==NULL)
{
exit(1);
}
else if(start==last) //for only one node
{
temp=start;
printf(“deleted element=%d”, temp->info);
free(temp);
start=last=NULL;
}
else
{
temp=start;
while( temp->next!=last)
temp=temp->next;
hold=temp->next;
last=temp;
last->next=start;
printf(“the deleted element=%d”, hold->info);
free(hold);
}
}
Stack as a circular List:
To implement a stack in a circular linked list, let pstack be a pointer to the last
node of a circular list. Actually there is no any end of a list but for convention let us
assume that the first node(rightmost node of a list) is the top of the stack.
An empty stack is represented by a null list.
The structure for the circular linked list implementation of stack is:
struct node
{
int info;
struct node *next;
};
NodeType *pstack=NULL;
C function to check whether the list is empty or not as follows:
int IsEmpty()
{
if(pstack==NULL)
return(1);
else
return(0);
}
csitnepal

PUSH function:
void PUSH(int item)
{
NodeType newnode;
newnode->info=item;
if(pstack==NULL)
{
pstack=newnode;
pstack->next=pstack;
}
else
{
newnode->next=pstack->next;
pstack->next=newnode;
}
}
fig: circular linked list
POP function:
void POP()
{
NodeType *temp;
if(pstack==NULL)
{
printf(“Stack underflown');
exit(1);
}
else if(pstack->next==pstack) //for only one node
{
printf(“poped item=%d”, pstack->info);
pstack=NULL;
}
else
{
temp=pstack->next;
pstack->next=temp->next;
printf(“poped item=%d”, temp->info);
free(temp);
}
}
csitnepal

Queue as a circular List:
It is easier to represent a queue as a circular list than as a linear list. As a linear list
a queue is specified by two pointers, one to the front of the list and the other to its
rear. However, by using a circular list, a queue may be specified by a single pointer q to
that list. node(q) is the rear of the queue and the following node is its front.
Insertion function:
{
NodeType *nnode;
nnode->info=item;
if(pq==NULL)
pq=nnode;
else
{
nnode->next=pq->next;
pq->next=nnode;
pq=nnode;
}
}
Deletion function:
void delet(int item)
{
NodeType *temp;
if(pq==NULL)
{
printf(“void deletionn”);
exit(1);
}
else if(pq->next==pq) //for only one node
{
printf(“poped item=%d”, pq->info);
pq=NULL;
}
csitnepal

else
{
temp=pq->next;
pq->next=temp->next;
printf(“poped item=%d”, temp->info);
free(temp);
}
}
Doubly Linked List:
A linked list in which all nodes are linked together by multiple number of links ie
each node contains three fields (two pointer fields and one data field) rather than two
fields is called doubly linked list.
It provides bidirectional traversal.
Fig: A node in doubly linked list
fig: A doubly linked list with three nodes
C representation of doubly linked list:
struct node
{
int info;
struct node *prev;
struct node *next;
};
NodeType *head=NULL:
Algorithms to insert a node in a doubly linked list:
Algorithm to insert a node at the beginning of a doubly linked list:
1.Allocate memory for the new node as,
newnode=(NodeType*)malloc(sizeof(NodeType))
2. Assign value to info field of a new node
3. set newnode->prev=newnode->next=NULL
4. set newnode->next=head
5. set head->prev=newnode
6. set head=newnode
7. End
C function to insert a node at the beginning of a doubly linked list:
void InsertAtBeg(int Item)
{
NodeType *newnode;
newnode->info=item;
newnode->prev=newnode->next=NULL;
csitnepal

newnode->next=head;
head->prev=newnode;
head=newnode;
}
Algorithm to insert a node at the end of a doubly linked list:
1. Allocate memory for the new node as,
3. set newnode->next=NULL
4. if head==NULL
set newnode->prev=NULL;
set head=newnode;
5. if head!=NULL
set temp=head
temp=temp->next;
end while
set temp->next=newnode;
set newnode->prev=temp
6. End
Algorithm to delete a node from beginning of a doubly linked list:
1. if head==NULL then
print “empty list” and exit
2. else
set hold=head
set head=head->next
set head->prev=NULL;
free(hold)
3. End
Algorithm to delete a node from end of a doubly linked list:
1. if head==NULL then
2. else if(head->next==NULL) then
set hold=head
set head=NULL
free(hold)
3. else
set temp=head;
while(temp->next->next !=NULL)
temp=temp->next
end while
set hold=temp->next
set temp->next=NULL
free(hold)
4. End
Circular Doubly Linked List:
A circular doubly linked list is one which has the successor and predecessor
pointer in circular manner.
csitnepal

It is a doubly linked list where the next link of last node points to the first node and
previous link of first node points to last node of the list.
The main objective of considering circular doubly linked list is to simplify the
insertion and deletion operations performed on doubly linked list.
C representation of doubly circular linked list:
struct node
{
int info;
struct node *prev;
struct node *next;
};
NodeType *head=NULL:
Algorithm to insert a node at the beginning of a circular doubly linked list:
3. set temp=head->next
4. set head->next=newnode
5. set newnode->prev=head
6. set newnode->next=temp
7. set temp->prev=newnode
8. End
Algorithm to insert a node at the end of a circular doubly linked list:
3. set temp=head->prev
4. set temp->next=newnode
5. set newnode->prev=temp
6. set newnode->next=head
csitnepal

7. set head->prev=newnode
8. End
Algorithm to delete a node from the beginning of a circular doubly linked list:
1. if head->next==NULL then
2. else
set temp=head->next;
set head->next=temp->next
set temp->next=head
free(temp)
3. End
Algorithm to delete a node from the end of a circular doubly linked list:
1. if head->next==NULL then
2. else
set temp=head->prev;
set head->left=temp->left
free(temp)
3. End
Unit 6
Tree data structure:
b) Binary tree
c) Introduction and application
d) operations
e) Types of binary tree
✔ Complete binary tree
✔ Strictly binary tree
✔ Almost complete binary tree
f) Huffman algorithm
g) Binary search tree
✔ insertion
✔ deletion
✔ searching
h) Tree traversal
✔ Pre-order traversal
✔ In-order traversal
✔ post-order traversal
csitnepal

Tree:
A tree is an abstract model of a hierarchical structure that consists of nodes with a
parent-child relationship.
• Tree is a sequence of nodes.
• There is a starting node known as root node.
• Every node other than the root has a parent node.
• Nodes may have any number of children.
A has 3 children, B, C, D
A is parent of B
Recursive definition of tree:
A tree t of order n is either empty or consists of a distinguished node r, called the
root of T, together with at most n trees, T1, T2, …........,Tn called the sub trees of T.
Characteristics of trees:
✔ Non-linear data structure
✔ combines advantages of an ordered array
✔ searching as fast as in ordered array
✔ insertion and deletion as fast as in linked list
Application:
✔ Directory structure of a file store
✔ Structure of arithmetic expressions
✔ Hierarchy of an organization
Some key terms:
Degree of a node:
The degree of a node is the number of children of that node.
In above tree the degree of node A is 3.
Degree of a Tree:
The degree of a tree is the maximum degree of nodes in a given tree.
In the above tree the node A has maximum degree, thus the degree of the tree is 3.
csitnepal

Path:
It is the sequence of consecutive edges from source node to destination node.
There is a single unique path from the root to any node.
Height of a node:
The height of a node is the maximum path length from that node to a leaf node. A leaf
node has a height of 0.
Height of a tree:
The height of a tree is the height of the root.
Depth of a node:
Depth of a node is the path length from the root to that node. The root node has a depth of 0.
Depth of a tree:
Depth of a tree is the maximum level of any leaf in the tree.
This is equal to the longest path from the root to any leaf.
Level of a node:
the level of a node is 0, if it is root; otherwise it is one more then its parent.
Illustration:
✔ A is the root node
✔ B is the parent of E and F
✔ D is the sibling of B and C
✔ E and F are children of B
✔ E, F, G, D are external nodes or leaves
✔ A, B, C are internal nodes
✔ Depth of F is 2
✔ the height of tree is 2
✔ the degree of node A is 3
✔ The degree of tree is 3
Binary Trees:
A binary tree is a finite set of elements that are either empty or is partitioned into three
disjoint subsets. The first subset contains a single element called the root of the tree. The other
csitnepal

two subsets are themselves binary trees called the left and right sub-trees of the original tree. A
left or right sub tree can be empty.
Each element of a binary tree is called a node of the tree. The following figure shows a
binary tree with 9 nodes where A is the root.
•A binary tree consists of a header, plus a number of nodes connected by links in a
hierarchical data structure:
Binary tree properties:
✔ If a binary tree contains m nodes at level l, it contains at most 2m nodes at level l+1.
✔ Since a binary tree can contain at most 1 node at level 0 (the rot), it contains at most 2l
nodes at level l.
Types of binary tree
✔ Complete binary tree
✔ Strictly binary tree
✔ Almost complete binary tree
csitnepal

Strictly binary tree:
If every non-leaf node in a binary tree has nonempty left and right sub-trees, then such
a tree is called a strictly binary tree.
Complete binary tree:
A complete binary tree of depth d is called strictly binary tree if all of whose leaves are
at level d. A complete binary tree with depth d has 2d leaves and 2d -1 non-leaf nodes(internal)
Almost complete binary tree:
A binary tree of depth d is an almost complete binary tree if:
✔ Any node nd at level less than d-1 has two sons.
✔ For any nose nd in the tree with a right descendant at level d, nd must have a left son
and every left descendant of nd is either a leaf at level d or has two sons.
Fig Almost complete binary tree.
csitnepal

Fig Almost complete binary tree but not strictly binary tree.
Since node E has a left son but not a right son.
Operations on Binary tree:
✔ father(n,T):Return the parent node of the node n in tree T. If n is the root, NULL is
returned.
✔ LeftChild(n,T):Return the left child of node n in tree T. Return NULL if n does not
have a left child.
✔ RightChild(n,T):Return the right child of node n in tree T. Return NULL if n does not
have a right child.
✔ Info(n,T): Return information stored in node n of tree T (ie. Content of a node).
✔ Sibling(n,T): return the sibling node of node n in tree T. Return NULL if n has no
sibling.
✔ Root(T): Return root node of a tree if and only if the tree is nonempty.
✔ Size(T): Return the number of nodes in tree T
✔ MakeEmpty(T): Create an empty tree T
✔ SetLeft(S,T): Attach the tree S as the left sub-tree of tree T
✔ SetRight(S,T): Attach the tree S as the right sub-tree of tree T.
✔ Preorder(T): Traverses all the nodes of tree T in preorder.
✔ postorder(T): Traverses all the nodes of tree T in postorder
✔ Inorder(T): Traverses all the nodes of tree T in inorder.
C representation for Binary tree:
struct bnode
{
int info;
struct bnode *left;
struct bnode *right;
};
struct bnode *root=NULL;
csitnepal

Fig: Structure of Binary tree
Tree traversal:
The tree traversal is a way in which each node in the tree is visited exactly once in a
symmetric manner.
There are three popular methods of traversal
✔ Pre-order traversal
✔ In-order traversal
✔ Post-order traversal
Pre-order traversal:
The preorder traversal of a nonempty binary tree is defined as follows:
✔ Visit the root node
✔ Traverse the left sub-tree in preorder
✔ Traverse the right sub-tree in preorder
fig Binary tree
The preorder traversal output of the given tree is: A B D H I E C F G
The preorder is also known as depth first order.
C function for preorder traversing:
void preorder(struct bnode *root)
{
if(root!=NULL)
{
printf(“%c”, root->info);
preorder(root->left);
preorder(root->right);
}
}
csitnepal

In-order traversal:
The inorder traversal of a nonempty binary tree is defined as follows:
✔ Traverse the left sub-tree in inorder
✔ Traverse the right sub-tree in inorder
The inorder traversal output of the given tree is: H D I B E A F C G
C function for inorder traversing:
void inorder(struct bnode *root)
{
if(root!=NULL)
{
inorder(root->left);
inorder(root->right);
}
}
Post-order traversal:
The post-order traversal of a nonempty binary tree is defined as follows:
✔ Traverse the left sub-tree in post-order
✔ Traverse the right sub-tree in post-order
The post-order traversal output of the given tree is: H I D E B F G C A
C function for post-order traversing:
void post-order(struct bnode *root)
{
if(root!=NULL)
{
post-order(root->left);
post-order(root->right);
}
}
Binary search tree(BST):
A binary search tree (BST) is a binary tree that is either empty or in which every node
contains a key (value) ans satisfies the following conditions:
✔ All keys in the left sub-tree o the root are smaller than the key in the root node
✔ All keys in the right sub-tree of the root are greater than the key in the root node
✔ The left and right sub-trees of the root are again binary search trees
Given the following sequence of numbers,
14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5
The following binary search tree can be constructed:
csitnepal

Operations on Binary search tree(BST):
Following operations can be done in BST:
✔ Search(k, T): Search for key k in the tree T. If k is found in some node of tree then
return true otherwise return false.
✔ Insert(k, T): Insert a new node with value k in the info field in the tree T such that the
property of BST is maintained.
✔ Delete(k, T):Delete a node with value k in the info field from the tree T such that the
property of BST is maintained.
✔ FindMin(T), FindMax(T): Find minimum and maximum element from the given
nonempty BST.
Searching through the BST:
•Problem: Search for a given target value in a BST.
•Idea: Compare the target value with the element in the root node.
✔ If the target value is equal, the search is successful.
✔ If target value is less, search the left subtree.
✔ If target value is greater, search the right subtree.
✔ If the subtree is empty, the search is unsuccessful.
BST search algorithm:
To find which if any node of a BST contains an element equal to target:
1. Set curr to the BST’s root.
2. Repeat:
2.1. If curr is null:
2.1.1. Terminate with answer none.
2.2. Otherwise, if target is equal to curr’s element:
2.2.1. Terminate with answer curr.
2.3. Otherwise, if target is less than curr’s element:
2.3.1. Set curr to curr’s left child.
2.4. Otherwise, if target is greater than curr’s element:
2.4.1. Set curr to curr’s right child.
2. end
csitnepal

C function for BST searching:
void BinSearch(struct bnode *root , int key)
{
if(root == NULL)
{
printf(“The number does not exist”);
exit(1);
}
else if (key == root->info)
{
printf(“The searched item is found”):
}
else if(key < root->info)
return BinSearch(root->left, key);
else
return BinSearch(root->right, key);
}
Insertion of a node in BST:
To insert a new item in a tree, we must first verify that its key is different from those of
existing elements. To do this a search is carried out. If the search is unsuccessful, then item is
inserted.
•Idea: To insert a new element into a BST, proceed as if searching for that element. If the
element is not already present, the search will lead to a null link. Replace that null link by a
link to a leaf node containing the new element.
insert(18)
BST insertion algorithm:
To insert the element elem into a BST:
1. Set parent to null, and set curr to the BST’s root.
2. Repeat:
2.1. If curr is null:
2.1.1. Replace the null link from which curr was taken
(either the BST’s root or parent’s left child or parent’s right child) by a
link to a newly-created leaf node with element elem.
2.1.2. Terminate.
2.2. Otherwise, if elem is equal to curr’s element:
2.2.1. Terminate.
2.3. Otherwise, if elem is less than curr’s element:
2.3.1. Set parent to curr, and set curr to curr’s left child.
2.4. Otherwise, if elem is greater than curr’s element:
2.4.1. Set parent to curr, and set curr to curr’s right child.
3.End
csitnepal

C function for BST insertion:
void insert(struct bnode *root, int item)
{
if(root=NULL)
{
root=(struct bnode*)malloc (sizeof(struct bnode));
root->left=root->right=NULL;
root->info=item;
}
else
{
if(item<root->info)
root->left=insert(root->left, item);
else
root->right=insert(root->right, item);
}
}
Deleting a node from the BST:
While deleting a node from BST, there may be three cases:
1. The node to be deleted may be a leaf node:
In this case simply delete a node and set null pointer to its parents those side
at which this deleted node exist.
Suppose node to be deleted is -4
2. The node to be deleted has one child:
In this case the child of the node to be deleted is appended to its parent node.
Suppose node to be deleted is 18
csitnepal

3. the node to be deleted has two children:
In this case node to be deleted is replaced by its in-order successor node.
OR
If the node to be deleted is either replaced by its right sub-trees leftmost node or its
left sub-trees rightmost node.
Suppose node to deleted is 12
Find minimum element in the right sub-tree of the node to be removed. In current
example it is 19.
General algorithm to delete a node from a BST:
1. start
2. if a node to be deleted is a leaf nod at left side then simply delete and set null pointer to
it's parent's left pointer.
3. If a node to be deleted is a leaf node at right side then simply delete and set null pointer
to it's parent's right pointer
4. if a node to be deleted has on child then connect it's child pointer with it's parent pointer
and delete it from the tree
5. if a node to be deleted has two children then replace the node being deleted either by
a. right most node of it's left sub-tree or
b. left most node of it's right sub-tree.
6. End
The deleteBST function:
struct bnode *delete(struct bnode *root, int item)
{
struct bnode *temp;
if(root==NULL)
{
printf(“Empty tree”);
return;
}
else if(item<root->info)
root->left=delete(root->left, item);
else if(item>root->info)
root->right=delete(root->right, item);
else if(root->left!=NULL &&root->right!=NULL) //node has two child
{
temp=find_min(root->right);
root->info=temp->info;
root->right=delete(root->right, root->info);
csitnepal

}
else
{
temp=root;
if(root->left==NULL)
root=root->right;
else if(root->right==NULL)
root=root->left;
free(temp);
}
return(temp);
}
/**********find minimum element function**********/
struct bnode *find_min(struct bnode *root)
{
if(root==NULL)
return0;
else if(root->left==NULL)
return root;
else
return(find_min(root->left));
}
Huffman algorithm:
Our example: text files
-1951, David Huffman found the “most efficient method of representing numbers, letters,
and other symbols using binary code”. Now standard method used for data compression.
In Huffman Algorithm, a set of nodes assigned with values if fed to the algorithm.
Initially 2 nodes are considered and their sum forms their parent node. When a new
element is considered, it can be added to the tree. Its value and the previously calculated
sum of the tree are used to form the new node which in turn becomes their parent.
Let us take any four characters and their frequencies, and sort this list by increasing
frequency.
Since to represent 4 characters the 2 bit is sufficient thus take initially two bits for each
character this is called fixed length character.
csitnepal

character frequencies
E: 10
T: 07
O: 05
A: 03
Now sort these characters according to their frequencies in non-decreasing order.
character frequencies code
A: 03 00
O: 05 01
T: 07 10
E: 10 11
Here before using Huffman algorithm the total number of bits required is
nb=3*2+5*2+7*2+10*2=06+10+14+20=50bits
csitnepal

Left branch is 0
Right branch is 1
Now from variable length code we get following code sequence.
character frequencies code
A: 03 110
O: 05 111
T: 07 10
E: 10 0
Thus after using Huffman algorithm the total number of bits required is
nb=3*3+5*3+7*2+10*1=09+15+14+10=48bits
(50-48)/50*100%=4%
Since in this small example we save about 4% space by using Huffman algorithm. If we
take large example with a lot of characters and their frequencies we can save a lot of space.
csitnepal

Unit 7
Sorting:
a) Introduction
b) Bubble sort
c) Insertion sort
d) Selection sort
e) Quick sort
f) Merge sort
g) Comparison and efficiency of sorting
Introduction:
Sorting
Sorting is among the most basic problems in algorithm design. We are given a sequence
of items, each associated with a given key value. The problem is to permute the items so that
they are in increasing (or decreasing) order by key. Sorting is important because it is often the
first step in more complex algorithms. Sorting algorithms are usually divided into two classes,
internal sorting algorithms, which assume that data is stored in an array in main memory, and
external sorting algorithm, which assume that data is stored on disk or some other device that
is best accessed sequentially. We will only consider internal sorting. Sorting algorithms often
have additional properties that are of interest, depending on the application. Here are two
important properties.
In brief the sorting is a process of arranging the items in a list in some order that is either
ascending or descending order.
Let a[n] be an array of n elements a0,a1,a2,a3........,an-1 in memory. The sorting of the array
a[n] means arranging the content of a[n] in either increasing or decreasing order.
i.e. a0<=a1<=a2<=a3<.=.......<=an-1
consider a list of values: 2 ,4 ,6 ,8 ,9 ,1 ,22 ,4 ,77 ,8 ,9
After sorting the values: 1, 2, 4, 4, 6, 8, 8,9 , 9 , 22, 77
In-place: The algorithm uses no additional array storage, and hence (other than perhaps the
system’s recursion stack) it is possible to sort very large lists without the need to allocate
additional working storage.
Stable: A sorting algorithm is stable if two elements that are equal remain in the same relative
position after sorting is completed. This is of interest, since in some sorting applications you
sort first on one key and then on another. It is nice to know that two items that are equal on the
second key, remain sorted on the first key.
csitnepal

Bubble Sort:
The basic idea of this sort is to pass through the array sequentially several times. Each
pass consists of comparing each element in the array with its successor (for example a[i] with
a[i + 1]) and interchanging the two elements if they are not in the proper order. For example,
consider the following array:
Algorithm
BubbleSort(A, n)
{
for(i = 0; i <n-1; i++)
{
for(j = 0; j < n-i-1; j++)
{
if(A[j] > A[j+1])
{
temp = A[j];
A[j] = A[j+1];
A[j+1] = temp;
}
csitnepal

}
}
}
Time Complexity:
= O(n2)
There is no best-case linear time complexity for this algorithm.
Space Complexity:
Since no extra space besides 3 variables is needed for sorting
Space complexity = O(n)
Selection Sort:
Idea: Find the least (or greatest) value in the array, swap it into the leftmost(or rightmost)
component (where it belongs), and then forget the leftmost component. Do this repeatedly.
Let a[n] be a linear array of n elements. The selection sort works as follows:
pass 1: Find the location loc of the smallest element int the list of n elements a[0], a[1], a[2],
a[3], …......,a[n-1] and then interchange a[loc] and a[0].
Pass 2: Find the location loc of the smallest element int the sub-list of n-1 elements a[1], a[2],
a[3], …......,a[n-1] and then interchange a[loc] and a[1] such that a[0], a[1] are sorted.
…..................... and so on.
Then we will get the sorted list a[0]<=a[1]<= a[2]<=a[3]<= …......<=a[n-1].
Algorithm:
SelectionSort(A)
{
for( i = 0;i < n ;i++)
{
least=A[i];
p=i;
for ( j = i + 1;j < n ;j++)
{
if (A[j] < A[i])
least= A[j]; p=j;
}
}
swap(A[i],A[p]);
}
Time Complexity:
= O(n2)
There is no best-case linear time complexity for this algorithm, but number of swap
operations is reduced greatly.
Space Complexity:
Since no extra space besides 5 variables is needed for sorting
Space complexity = O(n)
csitnepal

Insertion Sort:
Idea: like sorting a hand of playing cards start with an empty left hand and the cards facing
down on the table. Remove one card at a time from the table, and insert it into the correct
position in the left hand. Compare it with each of the cards already in the hand, from right to
left. The cards held in the left hand are sorted
Suppose an array a[n] with n elements. The insertion sort works as follows:
pass 1: a[0] by itself is trivially sorted.
Pass 2: a[1] is inserted either before or after a[0] so that a[0], a[1] is sorted.
Pass 3: a[2] is inserted into its proper place in a[0],a[1] that is before a[0], between a[0] and
a[1], or after a[1] so that a[0],a[1],a[2] is sorted.
….....................................................
pass N: a[n-1] is inserted into its proper place in a[0],a[1],a[2],........,a[n-2] so that
a[0],a[1],a[2],............,a[n-1] is sorted with n elements.
Example:
csitnepal

Quick Sort:
Quick sort developed by C.A.R Hoare is an unstable sorting. In practice this is the
fastest sorting method. It possesses very good average case complexity among all the sorting
algorithms. This algorithm is based on the divide and conquer paradigm. The main idea
behind this sorting is partitioning of the elements.
Steps for Quick Sort:
Divide: partition the array into two nonempty sub arrays.
Conquer: two sub arrays are sorted recursively.
Combine: two sub arrays are already sorted in place so no need to combine.
csitnepal

Example: a[]={5, 3, 2, 6, 4, 1, 3, 7}
(1 3 2 3 4) 5 (5 7)
and continue this process for each sub-arrays and finally we get a sorted array.
Algorithm:
QuickSort(A,l,r)
{
f(l<r)
{
p = Partition(A,l,r);
QuickSort(A,l,p-1);
QuickSort(A,p+1,r);
}
}
Partition(A,l,r)
{
x =l;
y =r ;
p = A[l];
while(x<y)
{
while(A[x] <= p)
x++;
while(A[y] >=p)
y--;
if(x<y)
swap(A[x],A[y]);
csitnepal

}
A[l] = A[y];
A[y] = p;
return y; //return position of pivot
}
Time Complexity:
Best Case:
Divides the array into two partitions of equal size, therefore
T(n) = 2T(n/2) + O(n) , Solving this recurrence we get,
T(n)=O(nlogn)
Worst case:
when one partition contains the n-1 elements and another partition contains only one element.
Therefore its recurrence relation is:
T(n) = T(n-1) + O(n), Solving this recurrence we get
T(n)=O(n2)
Average case:
Good and bad splits are randomly distributed across throughout the tree
T1(n)= 2T'(n/2) + O(n) Balanced
T'(n)= T(n –1) + O(n) Unbalanced
Solving:
B(n)= 2(B(n/2 –1) + Θ(n/2)) + Θ(n)
= 2B(n/2 –1) + Θ(n)
= O(nlogn)
=>T(n)=O(nlogn)
Merge Sort
To sort an array A[l . . r]:
• Divide
– Divide the n-element sequence to be sorted into two sub-sequences of n/2 elements
• Conquer
– Sort the sub-sequences recursively using merge sort. When the size of the sequences
is 1 there is nothing more to do
•Combine
Merge the two sorted sub-sequences
Example: a[]={4, 7, 2, 6, 1, 4, 7, 3, 5, 2, 6}
csitnepal

csitnepal

Time Complexity:
Recurrence Relation for Merge sort:
T(n) = 1 if n=1
T(n) = 2 T(n/2) + O(n) if n>1
Solving this recurrence we get
T(n) = O(nlogn)
Space Complexity:
It uses one extra array and some extra variables during sorting, therefore
Space Complexity= 2n + c = O(n)
csitnepal

Sorting Comparison:
Unit 8: Searching:
a) Introduction
b) Sequential search
c) Binary search
d) Comparison and efficiency of searching
e) Hashing
 probing (Linear and Quadratic)
Introduction:
Searching is a process of finding an element within the list of elements stored in any order or
randomly. Searching is divided into two categories Linear and Binary search.
Sequential Search:
In linear search, access each element of an array one by one sequentially and see
whether it is desired element or not. A search will be unsuccessful if all the elements are
accessed and the desired element is not found.
In brief, Simply search for the given element left to right and return the index of the element, if
found. Otherwise return “Not Found”.
Algorithm:
LinearSearch(A, n,key)
{
for(i=0;i<n;i++)
{
if(A[i] == key)
return i;
}
return -1;//-1 indicates unsuccessful search
}
Analysis:
Time complexity = O(n)
csitnepal

Binary Search:
Binary search is an extremely efficient algorithm. This search technique searches the given
item in minimum possible comparisons. To do this binary search, first we need to sort the array
elements. The logic behind this technique is given below:
✔ First find the middle element of the array
✔ compare the middle element with an item.
✔ There are three cases:
✗ If it is a desired element then search is successful
✗ If it is less than desired item then search only the first half of the array.
✗ If it is greater than the desired element, search in the second half of the array.
Repeat the same process until element is found or exhausts in the search area.
In this algorithm every time we are reducing the search area.
Running example:
Take input array a[] = {2 , 5 , 7, 9 ,18, 45 ,53, 59, 67, 72, 88, 95, 101, 104}
csitnepal

Algorithm:
BinarySearch(A,l,r, key)
{
if(l= = r) //only one element
{
if(key = = A[l])
return l+1; //index starts from 0
else
return 0;
}
else
{
m = (l + r) /2 ; //integer division
if(key = = A[m]
return m+1;
else if (key < A[m])
return BinarySearch(l, m-1, key) ;
else
return BinarySearch(m+1, r, key) ;
}
}
Efficiency:
From the above algorithm we can say that the running time of the algorithm is
T(n) = T(n/2) + Ο(1)
= Ο(logn) (verify).
In the best case output is obtained at one run i.e. Ο(1) time if the key is at middle.
In the worst case the output is at the end of the array so running time is Ο(logn) time.ith
In the average case also running time is Ο(logn).
For unsuccessful search best, worst and average time complexity is Ο(logn).
Hashing:
It is an efficient searching technique in which key is placed in direct accessible address for
rapid search.
Hashing provides the direct access of records from the file no matter where the record is in
the file. Due to which it reduces the unnecessary comparisons. This technique uses a
hashing function say h which maps the key with the corresponding key address or
location.
A function that transforms a key into a table index is called a hash function.
A common hash function is
h(x)=x mod SIZE
if key=27 and SIZE=10 then
hash address=27%10=7
csitnepal

Hash-table principles:
Hash collision:
If two or more than two records trying to insert in a single index of a hash table then such
a situation is called hash collision.
Some popular methods for minimizing collision are:
✔ Linear probing
✔ Quadratic probing
✔ Rehashing
✔ Chaining
✔ Hashing using buckets etc
But here we need only first two methods for minimizing collision
Linear probing:
A hash-table in which a collision is resolved by putting the item in the next empty place
within the occupied array space.
It starts with a location where the collision occurred and does a sequential search through a
hash table for the desired empty location.
Hence this method searches in straight line, and it is therefore called linear probing.
Disadvantage:
Clustering problem
Example:
Insert keys {89, 18, 49, 58, 69} with the hash function
h(x)=x mod 10 using linear probing.
solution:
csitnepal

when x=89:
h(89)=89%10=9
insert key 89 in hash-table in location 9
when x=18:
h(18)=18%10=8
when x=49:
h(49)=49%10=9 (Collision occur)
so insert key 49 in hash-table in next possible vacant location of 9 is 0
when x=58:
insert key 58 in hash-table in next possible vacant location of 8 is 1
(since 9, 0 already contains values).
when x=69:
insert key 69 in hash-table in next possible vacant location of 9 is 2
(since 0, 1 already contains values).
Fig Hash-table for above keys using linear probing
Quadratic Probing:
Quadratic probing is a collision resolution method that eliminates the primary
clustering problem take place in a linear probing.
When collision occur then the quadratic probing works as follows:
(Hash value + 12)% table size
if there is again collision occur then there exist rehashing.
(hash value + 22)%table size
if there is again collision occur then there exist rehashing.
(hash value = 32)% table size
in general in ith collision
hi(x)=(hash value +i2)%size
csitnepal

Example:
Insert keys {89, 18, 49, 58, 69} with the hash-table size 10 using quadratic probing.
solution:
when x=89:
h(89)=89%10=9
when x=18:
h(18)=18%10=8
when x=49:
so use following hash function,
h1(49)=(49 + 1)%10=0
hence insert key 49 in hash-table in location 0
when x=58:
h1(58)=(58 + 1)%10=9
again collision occur use again the following hash function ,
h2(58)=(58+ 22)%10=2
when x=69:
h1(69)=(69 + 1)%10=0
again collision occur use again the following hash function ,
h2(69)=(69+ 22)%10=3
fig:Hash table for above keys using quadratic probing
csitnepal

Unit:9
Graph:
a) Introduction
b) Representation of Graph
✔ Array
✔ Linked list
c) Traversals
✔ Depth first Search
✔ Breadth first search
d) Minimum spanning tree
✔ Kruskal's algorithm
Graph:
A Graph is a pair G = (V,E) where V denotes a set of vertices and E denotes the set of
edges connecting two vertices. Many natural problems can be explained using graph for
example modeling road network, electronic circuits, etc. The example below shows the road
network.
Let us take a graph:
V(G)={v1, v2, v3, v4, v5}
E(G)={(v1,v2),(v2,v3),(v1,v3),(v3,v4),(v4,v5)}
Types of Graph:
Simple Graph:
We define a simple graph as 2 – tuple consists of a non empty set of vertices V and a
set of unordered pairs of distinct elements of vertices called edges. We can represent graph as
G = (V, E). This kind of graph has no loops and can be used for modeling networks that do not
have connection to themselves but have both ways connection when two vertices are connected
csitnepal

but no two vertices have more than one connection. The figure below is an example of simple
graph.
Multigraph:
A multigraph G =(V, E) consists of a set of vertices V, a set of edges E, and a function f
from E to {{u, v}|u, v Î V, u ¹ v}. The edges e1 and e2 are called multiple or parallel edges if
f(e1) = f(e2). In this representation of graph also loops are not allowed. Since simple graph has
single edges every simple graph is a multigraph. The figure below is an example of a
multigraph.
Pseudograph:
A pseudograph G =(V, E) consists of a set of vertices V, a set of edges E, and a
function f from E to {{u, v}|u, v ∈ V}. An edge is a loop if f(e) = {u, u} = {u} for some u ∈ V.
The figure below is an example of a multigraph
Directed Graph:
A directed graph (V, E) consists of a set V of vertices, a set E of edges that are ordered
pairs of elements of V. The below figure is a directed graph. In this graph loop is allowed but
no two vertices van have multiple edges in same direction.
Directed Multigraph:
csitnepal

A directed multigraph G =(V, E) consists of a set of vertices V, a set of edges E, and a
function f from E to {(u, v)|u, v ∈ V}. The edges e1 and e2 are called multiple edges if f(e1) =
f(e2). The figure below is an example of a directed multigraph.
Terminologies:
Two vertices u, v are adjacent vertices of a graph if {u, v} is an edge.
The edge e is called incident with the vertices u and v if e = {u, v}. This edge is also said to
connect u and v. where u and v are end points of the edge.
Degree of a vertex in an undirected graph is the number of edges incident with it, except a
loop at a vertex. Loop in a vertex counts twice to the degree. Degree of a vertex v is denoted by
deg (v).A vertex of degree zero is called isolated vertex and a vertex with degree one is called
pendant vertex.
Example: Find the degrees of the vertices in the following graph.
Solution:
deg(a) = deg(f) = deg(e) = 2 ; deg(b) = deg(c) = 3; deg(d) = 4
Representation of Graph
Generally graph can be represented in two ways namely adjacency lists(Linked list
representation) and adjacency matrix(matrix).
Adjacency List:
This type of representation is suitable for the undirected graphs without multiple edges,
and directed graphs. This representation looks as in the tables below.
csitnepal

If we try to apply the algorithms of graph using the representation of graphs by lists of edges,
or adjacency lists it can be tedious and time taking if there are high number of edges. For the
sake of the computation, the graphs with many edges can be represented in other ways. In this
class we discuss two ways of representing graphs in form of matrix.
Adjacency Matrix:
Given a simple graph G =(V, E) with |V| = n. assume that the vertices of the graph
are listed in some arbitrary order like v1, v2, …, vn. The adjacency matrix A of G, with respect
to the order of the vertices is n-by-n zero-one matrix (A = [aij]) with the condition,
Since there are n vertices and we may order vertices in any order there are n! possible order of
the vertices. The adjacency matrix depends on the order of the vertices, hence there are n!
possible adjacency matrices for a graph with n vertices.
In case of the directed graph we can extend the same concept as in undirected graph as
dictated by the relation
If the number of edges is few then the adjacency matrix becomes sparse. Sometimes it will be
beneficial to represented graph with adjacency list in such a condition.
csitnepal

Solution:Let the order of the vertices be a, b, c, d, e, f
Let us take a directed graph
Solution:
Let the order of the vertices be a, b, c, d, e, f, g
Graph Traversals
Breadth-first search:
This is one of the simplest methods of graph searching. Choose some vertex arbitrarily
as a root. Add all the vertices and edges that are incident in the root. The new vertices added
will become the vertices at the level 1 of the BFS tree. Form the set of the added vertices of
level 1, find other vertices, such that they are connected by edges at level 1 vertices. Follow the
above step until all the vertices are added.
csitnepal

Example:
Use breadth first search to find a BFS tree of the following graph
Algorithm:
BFS(G,s) //s is start vertex
{
T = {s};
L =Φ; //an empty queue
Enqueue(L,s);
while (L != Φ )
{
v = dequeue(L);
for each neighbor w to v
if ( w∉ L and w ∉ T )
{
enqueue( L,w);
T = T U {w}; //put edge {v,w} also
}
}
}
csitnepal

Analysis
From the algorithm above all the vertices are put once in the queue and they are
accessed. For each accessed vertex from the queue their adjacent vertices are looked for and
this can be done in O(n) time(for the worst case the graph is complete). This computation
for all the possible vertices that may be in the queue i.e. n, produce complexity of an
algorithm as O(n2).
Depth First Search:
This is another technique that can be used to search the graph. Choose a vertex as a root
and form a path by starting at a root vertex by successively adding vertices and edges. This
process is continued until no possible path can be formed. If the path contains all the vertices
then the tree consisting this path is DFS tree. Otherwise, we must add other edges and vertices.
For this move back from the last vertex that is met in the previous path and find whether it is
possible to find new path starting from the vertex just met. If there is such a path continue the
process above. If this cannot be done, move back to another vertex and repeat the process. The
whole process is continued until all the vertices are met. This method of search is also called
backtracking.
Example:
Use depth first search to find a spanning tree of the following graph.
csitnepal

csitnepal

Analysis:
The complexity of the algorithm is greatly affected by Traverse function we can write
its running time in terms of the relation T(n) = T(n-1) + O(n), here O(n) is for each vertex at
most all the vertices are checked (for loop). At each recursive call a vertex is decreased.
Solving this we can find that the complexity of an algorithm is O(n2).
Minimum Spanning Trees
A minimum spanning tree in a connected weighted graph is a spanning tree that has the
smallest possible sum of weights of its edges. In this part we study one algorithm that is used to
construct the minimum spanning tree from the given connected weighted graph.
Kruskal’s Algorithm:
The problem of finding MST can be solved by using Kruskal’s algorithm. The idea
behind this algorithm is that you put the set of edges form the given graph G = (V,E) in
nondecreasing order of their weights. The selection of each edge in sequence then guarantees
that the total cost that would from will be the minimum. Note that we have G as a graph, V as a
set of n vertices and E as set of edges of graph G.
Example:
Find the MST and its weight of the graph.
csitnepal

Solution:
The total weight of MST is 64.
Algorithm:
KruskalMST(G)
{
T = {V} // forest of n nodes
S = set of edges sorted in nondecreasing order of weight
while(|T| < n-1 and E !=Æ)
{
Select (u,v) from S in order
Remove (u,v) from E
if((u,v) doesnot create a cycle in T))
T = T È {(u,v)}
}
}
Analysis:
In the above algorithm the n tree forest at the beginning takes (V) time, the creation of
set S takes O(ElogE) time and while loop execute O(n) times and the steps inside the loop take
almost linear time (see disjoint set operations; find and union). So the total time taken is
O(ElogE)
csitnepal

A Complete Note in Data Structure and Algorithms
By Bhupendra Saud
Email: Saud.bhupendra427@gmail.com
coming soon:notes in NM, OS, DAA, C, C++,DBMS.
This Note is prepared for Bsc
Csit 2nd
semester students
according to their course of
study by Bhupendra saud. If
any defects, errors take place
please send your views to
Bhupendra sir or
csitnepal.com
csitnepal

Data structure and algorithm notes

Recommended

More Related Content

What's hot (20)

Similar to Data structure and algorithm notes (20)

Recently uploaded (20)

Data structure and algorithm notes