This document discusses binary search trees and provides source code to implement binary search tree traversal operations. It explains that a binary search tree is a binary tree where the left child of a node is less than the parent and the right child is greater. It provides code for preorder, inorder, and postorder traversals of the binary search tree and outputs the traversal order for a sample tree.

1. DATA STRUCTURES ( C++ )
DATASTRUCTURES PROGRAMMING CONCEPTS
Developed by,
M.V.B.REDDY,CSE
Associate professor,
Dept. of CSE, GST,
GITAM UNIVERSITY.
Email ID : bramhareddy999@gmail.com

2.  Let us take an array a[5] and take a variable top points to -1.
 PUSH:
 To INSERT the element in to stack using top.
 Here we check for the OVERFLOW condition.
STACK USING ARRAYS
 POP:
 To RETRIEVE elements from the stack using top.
 Here we check for the UNDERFLOW condition.
 This concept is nothing but LIFO.
SOURCE CODE:
/* Program To Implement Stack using Array */
#include < iostream.h >
#include < conio.h >
#include < stdlib.h >
#define MAX 10

3. class stack
{
private : int sp, a [ MAX ];
public :
void init ( );
void push ( int );
void pop ( );
void display ( );
void count ( );
};
void stack :: init ( )
{
sp = - 1;
}
void stack :: push ( int data)
{
if (sp = = ( MAX – 1 ) )
{
cout<<"n STACK OVERFLOW.......n";
return;
}

4. sp + + ;
a [ sp ] = data;
}
void stack :: pop ( )
{i
f ( sp < 0 )
{
cout<<"n STACK UNDERFLOW.....n";
return;
}
cout<<"n POPED DATA IS ::: "<<a[sp];
sp - - ;
}
void stack :: display ( )
{
cout << "n DATA PRESENT IN A STACK IS ::: n";
for ( int i = sp ; i > = 0 ; i - -)
cout << a [ i ] <<"t";
}
void stack :: count ( )
{
cout<<"n NUMBER OF ELEMENTS IN A STACK ARE ::: "<<(sp+1);
};

5. void main ( )
{
stack ob;
int data,ch;
clrscr ( );
ob.init ( );
cout<<"n**********STACK OPERATIONS**********n";
cout<<"n1.Push Operation";
cout<<"n2.Pop Operation";
cout<<"n3.Display Operation";
cout<<"n4.Count Operation";
cout<<"n5.Exit Operation";
cout<<"n*************************************n";
do
{
cout<<"n ENTER YOUR CHOICE :: ";
cin>>ch;
switch ( ch )
{
case 1:cout<<"n ENTER ELEMENT TO BE INSERTED :::";
cin>>data;
ob.push ( data ); break;
case 2: ob.pop ( ); break;
case 3: ob.display ( ); break;
case 4:ob.count ( ); break;
case 5: exit ( 0 );
defualt: cout<<"nINVALID CHOICE ";
}}
while ( ch ! = 5 );
getch ( );
}

6. OUTPUT:

7.  We will create a linked list and insert an element ‘10’ and address as ‘0’.using top for
the first node.
 For second node insert data element ‘20’ and insert first node address at second
STACK node address USING field.
LINKED LIST
 For third node insert data element ‘30’ and insert second node address at third node
address field . after thirty we will stop the process .
 If we want to print the elements 30,20,10 will be displayed, Thiss follows LIFO
conceot.

8. Source code:
#include<conio.h>
#include<iostream.h>
class st
{
public:
struct node
{
int data;
struct node *next;
}*start,*temp,*top;
st()
{
start=temp=top=NULL;
}
void create()
{
int d;
cout<<"Enter data";
cin>>d;
if(start==NULL)
{
start=new node;
start->data=d;

9. start->next=NULL;
top=start;
}
else
{
temp=new node;
temp->data=d;
temp->next=top;
top=temp;
}
}
void disp()
{
while(top!=NULL)
{
cout<<top->data<<"t";
top=top->next;
}
} };
void main()
{
st ob;
int ch;
clrscr();

10. while(ch)
{
cout<<"Enter ur choice";
cout<<"0 STOPn1 CREATEn 2 READ";
cin>>ch;
if(ch==1)
ob.create();
else if(ch==2)
ob.disp();
}
}
OUTPUT:

11.  Here we will take an array a[5],and two variables front, rear points to -1.
 WRITE:
QUEUE USING ARRAYS
 Here will insert the element into the queue using rear variable.
 Here check for the Overflow condition.
 READ:
 Here we will retrieve the elements from the queue using front variable.
 Here check for the Underflow condition.
 This follows the FIFO concept.
rear
-1 0 1 2 3 4
front

12. SOURCE CODE:
/* Program To Implement Queue using Array */
#include< iostream.h >
#include< conio.h >
#include< process.h >
#define MAX 10
class queue
{
private : int front, rear, a [ MAX ];
public :
void init ( );
void write ( int );
void read ( );
void count ( );
void display ( );
};
void queue :: init ( )
{f
ront = rear = - 1;
}
void queue :: write ( int data)
{ if ( rear = = ( MAX - 1 ) )
cout<<"n QUEUE IS OVERFLOW......";
else
a [ + + rear ] = data;
}
void queue :: read ( )
{ if( front = = rear )
cout<<"n QUEUE IS UNDERFLOW.....";
else
cout<<"n DELETED ELEMENT IN QUEUE IS :: "<<a[++front];
}

13. void queue :: count ( )
{
cout<<"n NUMBER OF ELEMENTS IN A QUEUE ARE ::
"<<(rear-front);
}
void queue :: display ( )
{
cout<<"n ELEMENTS IN A QUEUE ARE:: ";
for( int i = (front + 1); i < = rear; i + + )
cout<< a [ i ]<<"t";
}
void main ( )
{
queue ob;
int ch,data;
clrscr ( );
ob.init ( );
cout<<"n*****QUEUE OPERATIONS****n";
cout<<"n1.Write ";
cout<<"n2.Read ";
cout<<"n3.Count";
cout<<"n4.Display";
cout<<"n5.Exit";
cout<<"**************************n";

14. do
{
cout<<"n ENTER YOUR CHOICE :: ";
cin>>ch;
switch ( ch )
{
case 1:cout<<"n ENTER ELEMENT TO BE INSERTED IN QUEUE :: ";
cin>>data;
ob.write ( data );
break;
case 2:ob.read ( );
break;
case 3:ob.count ( );
break;
case 4:ob.display ( );
break;
case 5:exit ( 0 );
break;
default :cout<<"n INVALID CHOICE...";
}}
while( ch ! = 5 );
getch ( );
}

15. OUTPUT:

16.  Here we will create linked list with ‘n’ nodes one after another 10,20,30 etc.
 If we try to print the elements it will display as 10,20,30. which follows FIFO
Queue concept.
using linked list

17. SOURCE CODE:
/* Program To Implement Queue using Linked List */
#include < iostream.h >
#include< conio.h >
#include < alloc.h >
#define NULL 0
class node
{i
nt data;
node *next;
public:
void create ( node *);
void print ( node *);
};
void node :: create (node *list)
{
cout<<"n ENTER THE INPUT NO :: ";
cout<<"n TYPE 999 AT THE END :: ";
cin>>list->data;
if(list -> data = = 999)
list->next = NULL;
else
{l
ist -> next = new node;
create( list -> next);
} return;
}

18. void node :: print (node *list)
{i
f( list -> next ! = 0)
{
cout<< list->data;
cout<<"->";
}
else
return;
print( list -> next);
}
void main ( )
{
node *head, ob;
clrscr ( );
head = new node;
ob.create ( head );
cout<<"n QUEUE ELEMENTS ARE:: ";
ob.print( head );
cout<<"999";
getch ( );
}

19. OUTPUT:

20.  A binary tree is a tree data structure in which each node has at most two children. Typically
the child nodes are called left and right. Binary trees are commonly used to implement binary
search trees and binary heaps.
 Starting at the root of a binary tree, there are three main steps that can be performed and the
order in which they are performed define the traversal type.
BINARY TREE USING RECURSION
 There are 3 types of traversals:
 1. Pre-Order
 2. In-Order
 3. Post-Order
 To traverse a non-empty binary tree in preorder, perform the following operations
recursively at each node, starting with the root node:
1. Visit the root.
2. Traverse the left sub tree.
3. Traverse the right sub tree.
 To traverse a non-empty binary tree in in order, perform the following operations recursively
at each node, starting with the root node:
1. Traverse the left sub tree.
2. Visit the root.
3. Traverse the right sub tree.
 To traverse a non-empty binary tree in post order, perform the following operations
recursively at each node, starting with the root node:
1. Traverse the left sub tree.
2. Traverse the right sub tree.
3. Visit the root.

21. 15
7 22
BINARY TREE:
Preorder:- 15,7,22 will be displayed .
Post order:- 7,22,15 will be displayed .
In order:- 7,15,22 will be displayed .

22. SOURCE CODE:
/* Program To Implement Binary Tree Traversing */
#include < iostream.h >
#include < conio.h >
class bstree
{
public:
struct node
{
int data;
node *left;
node *right;
}*head;
void create (node *);
void inorder (node *);
void preorder (node *);
void postorder (node *);
};
void* bstree:: create(node *list)
{
node *temp1,*temp2;
int val;
if(list = = NULL)
{
list = new node;
cout<<"nEnter Data Element:: ";
cin>>list->data;
list -> left = list -> right = NULL;
}
else

23. {
cout<<"n enter the data element";
cin>>val;
temp1 = list;
while( temp1 ! = NULL )
{
temp2 = temp1;
if(temp1 -> data > val)
temp1 = temp1 -> left;
else
temp1 = temp1 -> right;
}
if(temp2 -> data > val)
{
temp2 -> left = new node;
temp2 = temp2 -> left;
temp2 -> data = val;
temp2 -> left = temp2 -> right = NULL;
}
else
{
temp2 -> right = new node;
temp2 = temp2 -> right;
temp2 -> data = val;
temp2 -> left = temp2 -> right = NULL;
}
}
return (list);
}

24. void bstree:: inorder(node *root)
{
if( ! root )
return;
inorder( root -> left );
cout<<root->data<<"t";
inorder( root -> right );
}
void bstree::preorder(node*root)
{
if( ! root )
return;
cout<<root->data<<”t”;
preorder( root -> left );
preorder( root -> right);
}
void bstree::postorder(node*root)
{
if( ! root)
return;
postorder( root -> left );
postorder( root -> right );
cout<<root->data<<”t”;
}

25. void main ( )
{
node n,*head;
head = NULL;
clrscr ( );
cout<<"nCreate A Binary Treen";
head=n.create ( head );
cout<<"n the inorder traversal gives the
following nodes";
n.inorder ( head );
getch ( );
}
OUTPUT:

27. BINARY SEARCH TREE
15
7 22
A tree having left child less than parent and right child grater than the parent.
Traversals are same as binary tree.

28. SOURCE CODE:
/* Program to implement Binary search tree */
#include < iostream.h >
#include < conio.h >
class btree
{
private :
struct btreenode
{
btreenode *leftchild ;
int data ;
btreenode *rightchild ;
} *root;
public:
btree ( ) ;
void buildtree ( int num ) ;
static void insert ( btreenode **sr, int num ) ;
void traverse ( ) ;
static void inorder ( btreenode *sr ) ;
static void preorder ( btreenode *sr ) ;
static void postorder ( btreenode *sr ) ;
static void del ( btreenode *sr ) ;
~btree ( ) ;
} ;

29. btree :: btree ( )
{
root = NULL ;
}
void btree :: buildtree ( int num )
{
insert ( &root, num ) ;
}
void btree :: insert ( btreenode **sr, int num )
{
if ( *sr == NULL )
{
*sr = new btreenode ;
( *sr ) -> leftchild = NULL ;
( *sr ) -> data = num ;
( *sr ) -> rightchild = NULL ;
return ;
}
else // search the node to which new node will be attached
{
// if new data is less, traverse to left
if ( num < ( *sr ) -> data )
insert ( & ( ( *sr ) -> leftchild ), num ) ;
else
// else traverse to right
insert ( & ( ( *sr ) -> rightchild ), num ) ;
}r
eturn ;
}

30. void btree :: traverse( )
{
cout << "nIN - ORDER TRAVERSAL :: " ;
inorder ( root ) ;
cout << "nPRE - ORDER TRAVERSAL :: " ;
preorder ( root ) ;
cout << "nPOST - ORDER TRAVERSAL :: " ;
postorder ( root ) ;
}
void btree :: inorder ( btreenode *sr )
{
if ( sr != NULL )
{
inorder ( sr -> leftchild ) ;
cout << "t" << sr -> data ;
inorder ( sr -> rightchild ) ;
}
else
return ;
}
void btree :: preorder ( btreenode *sr )
{
if ( sr != NULL )
{
// print the data of a node
cout << "t" << sr -> data ;
// traverse till leftchild is not NULL
preorder ( sr -> leftchild ) ;
// traverse till rightchild is not NULL
preorder ( sr -> rightchild ) ;
}

31. else
return ;
}
void btree :: postorder ( btreenode *sr )
{
if ( sr != NULL )
{
postorder ( sr -> leftchild ) ;
postorder ( sr -> rightchild ) ;
cout << "t" << sr -> data ;
}
else
return ;
}
btree :: ~btree( )
{
del ( root ) ;
}
void btree :: del ( btreenode *sr )
{
if ( sr != NULL )
{
del ( sr -> leftchild ) ;
del ( sr -> rightchild ) ;
}
delete sr ;
}

32. void main( )
{
btree bt ;
int req, i = 1, num ;
clrscr();
cout << "n SPECIFY THE NUMBER OF ITEMS TO BE INSERTED :: " ;
cin >> req ;
while ( i + + <= req )
{
cout << "n ENTER THE DATA :: " ;
cin >> num ;
bt.buildtree ( num ) ;
}
bt.traverse( ) ;
getch();
}
OUTPUT:

33. SPARSE MATRIX
AIM: Write a program in C++ to implement ADDITION and MULTIPLICTION of two SPARSE
matrixes.
THEORY:
If a lot of elements from a matrix have a value 0 then the matrix is known as
SPARSE MATRIX. If the matrix is sparse we must consider an alternate way of representing it
rather the normal row major or column major arrangement. This is because if majority of elements
of the matrix are 0 then an alternative through which we can store only the non-zero elements and
keep intact the functionality of the matrix can save a lot of memory space.
Example:
Sparse matrix of dimension 7 x 7.
COLUMNS
0 1 2 3 4 5 6
0 0 0 0 -5 0 0 0
1 0 4 0 0 0 0 7
2 0 0 0 0 9 0 0
ROWS 3 0 3 0 2 0 0 0
4 1 0 2 0 0 0 0
5 0 0 0 0 0 0 0
6 0 0 8 0 0 0 0

35. SOURCE CODE:
/*Program to demonstrate addition and multiplication of Two Sparse Matrix */
#include < iostream.h >
#include < conio.h >
#define x 25
class sparce
{
private:
int a [ x ] [ x ], b [ x ] [ x ], c [ x ] [ x ], m, n, p, q;
public:
void init ( );
void input ( );
void add ( );
void mul ( );
void display ( int [25][25], int, int );
void convert( int [25][25], int, int );
};
void sparce :: init ( )
{
int i, j;
for(i = 0; i < x;i + + )
for( j = 0; j < x; j + +)
c [ i ] [ j ] = 0;
}

36. void sparce :: input()
{
int i,j;
cout<<"nEnter order Of First matrix::";
cin>>m>>n;
cout<<"nEnter order Of Second matrix::";
cin>>p>>q;
cout<<"nEnter"<<m*n<<"Elements Into First Matrixn";
for(i=0;i<m;i++)
for( j = 0; j < n; j + + )
cin>> a[ i ] [ j ];
cout<<"nEnter"<<p*q<<"Elements Into Second Matrixn";
for(i = 0; i < p ; i + + )
for ( j = 0; j < q ; j + + )
cin>>b [ i ] [ j ];
}
void sparce :: add ( )
{
int i, j;
if( m = = p && n = = q )
{
for( i = 0 ; i < m ; i + + )
for( j = 0; j < n; j + + )
c[ i ] [ j ] = a [ i ][ j ] + b [ i ] [ j ];
convert( c, m, n);
}
else
cout<<"nAddition Is Not Possible";
}

37. void sparce :: mul ( )
{
int i, j, k;
if(n = = p)
{
for( i = 0; i < m; i + +)
for( j = 0; j < q; j + + )
for( k = 0; k < n; k + + )
c[ I ] [ j ] + = a [ I ] [ k ] * b [ k ] [ j ];
convert(c, m, n);
}
else
cout<<"n Multiplecation Is Not Possible";
}
void sparce :: display(int c[25][25], int m, int n)
{
int i,j;
for( i = 0 ;i < m; i + + )
{
for( j = 0 ; j < n ; j + + )
cout<<c [ i ] [ j ]<<"t";
cout<<"n";
}
}

38. void sparce :: convert(int c[25][25], int m, int n)
{
int i, j, k = 1,t = 0;
int sp[25][25];
for( i = 0 ; i < m ; i + +)
for( j = 0 ; j < n ; j + + )
if(c [ i ] [ j ] ! = 0 )
{
sp [ k ] [ 0 ] = i;
sp [ k ] [ 1 ] = j;
sp [ k ] [ 2 ] = c [ i ] [ j ];
k + + ;
t + + ;
}
sp[ 0 ] [ 0 ] = m;
sp[ 0 ] [ 1 ] = n;
sp[ 0 ] [ 2 ] = t;
display( sp, k, 3);
}
void main ( )
{
sparce ob;
clrscr ( );
ob.init ( );
ob.input ( );
cout<<"nAddition of Two Sparce Matrixn";
ob.add ( );
ob.init ( );
cout<<"nMultiplecation Of Two Sparce Matrixn";
ob.mul ( );
getch ( );
}

39. OUTPUT:

40. INFIX TO POSTFIX CONVERTIONic
 Suppose Q is an arithmetic expression written in infix notation. This algorithm finds the
equivalent postfix expression P.
 Step 1. Push “(“ onto stack and add “)” to the end of Q.
 2. Scan Q from left to right and repeat step 3 to 6 for each element of Q until the
stack is empty.
 3. If an operand is encountered , add it to p.
 4. If a left parenthesis is encountered ,push it onto stack.
 5 If an operator * is encountered , then:
a. repeatedly pop from stack and top each operator (on the top of stack ) which
has the same precedence or higher precedence than * .
b. Add * to stack.
 6. If a right parenthesis is encountered , then:
a. repeatedly from stack and add to P each operator (on the top of stack) until a left
parenthesis is encountered.
b. remove the left parenthesis [ Do not add the left parenthesis top] [End of if
structure] [End of step 2 loop]
 7. Exit

41. Symbol scanned stack Expression P
1 A ( A
2 + ( + A
3 ( ( + ( A
4 B ( + ( AB
5 * ( + ( * AB
6 C ( + ( * ABC
7 - ( + ( - ABC*
8 ( ( + ( - ( ABC*
9 D ( + ( - ( ABC*D
10 / ( + ( - ( / ABC*D
11 E ( + ( - ( / ABC*DE
12 ^ ( + ( - ( / ^ ABC*DE
13 F ( + ( - ( / ^ ABC*DEF
14 ) ( + ( - ABC*DEF^/
15 * ( + ( - * ABC*DEF^/
16 G ( + ( - * ABC*DEF^/G
17 ) ( + ABC*DEF^/G*-
18 * ( + * ABC*DEF^/G*-
19 H ( + * ABC*DEF^/G*-H
20 ) ABC*DEF^/G*-H*+
(A+(B*C-(D/E^F)*G)*H)

42. SOURCE CODE:
/* Program To implement infix to postfix Expression */
#include < iostream.h >
#include< process.h >
#include < conio.h >
char stack[30], postfix[30], infix[30];
int top = - 1;
int pri( char x )
{i
nt value;
switch ( x )
{
case ')': value=0; break;
case '+': case '-': value=1; break;
case '*': case '/': case '%': value=2; break;
case '^': value=3; break;
case '(': value=4; break;
default: cout<<"INVALID EXPRESSION !!!!!!";
exit(1);
} return value;
}

43. void push ( char x )
{
top = top + 1;
stack [top] = x;
}
char stacktop ( )
{
return stack [ top ];
}
int isalnum (char x)
{
return ( (x>='0' && x<='9') ||( x>='a' &&
x<='z') || ( x>='A' && x<='Z'));
}
char pop( )
{
return stack[top - - ];
}

44. void intopost(char infix[ ], char postfix[ ])
{ int i, j=0;
char c, pc;
for ( i = 0; ( c = infix[ i ] ) != '0' ; i + +)
{ if ( isalnum (c) )
postfix [ j + + ] = c;
else
{
while ( top ! = - 1 && (pri (stacktop () ) >= pri (c) ) )
{
If ( stacktop( ) = = '(' && c! = ')' )
break;
if ( stacktop( ) = = '(' && c = =')' )
{
pop () ;
break;
}
pc = pop( );
if ( pc! = '(' )
postfix [ j + + ] = pc;
else break;
}
if( c! = ')' )
push ( c );
}} while( top ! = -1 )
postfix[ j + + ] = pop( );
postfix [ j ] = '0';
}

45. void main ( )
{
clrscr ( );
cout<<"ENTER INFIX EXPRESSION ::nnttt";
cin>>infix;
intopost( infix, postfix );
cout<<"POSTFIX EXPRESSION ::nnttt ";
cout<<postfix;
getch ( );
}
OUTPUT:

46. POSTFIX EVALUATION
THEORY:
Reverse Polish notation is a mathematical notation wherein every operator
follows all of its operands. It is also known as Postfix notation and is parenthesis free.
In Reverse Polish notation the operators follow their operands; for instance, to add three
and four, one would write “3 4 +” rather than “3 + 4”. If there are multiple operations, the operator
is given immediately after its second operand; so the expression written “3 − 4 + 5” in
conventional infix notation would be written “3 4 − 5 +” in RPN: first subtract 4 from 3, then add 5
to that.
Infix Expression: Any expression in the standard form like "2*3-4/5" is an Infix(In order)
expression.
Postfix Expression: The Postfix(Post order) form of the above expression is "23*45/-".
Postfix Evaluation: In normal algebra we use the infix notation like a+b*c. The
corresponding postfix notation is abc*+. The algorithm for the conversion is as follows:
•Scan the Postfix string from left to right.
•Initialize an empty stack.
•If the scanned character is an operand, add it to the stack. If the scanned character is an
operator, there will be at least two operands in the stack
If the scanned character is an Operator, then we store the top most element of the
stack(topStack) in a variable temp. Pop the stack. Now evaluate
topStack(Operator)temp. Let the result of this operation be retVal. Pop the stack and
Push retVal into the stack.
Repeat this step till all the characters are scanned.
•After all characters are scanned, we will have only one element in the stack. Return
topStack.

47. Example:
Postfix String: 1 2 3 * + 4 - .
Initially the Stack is empty. Now, the first three characters scanned are 1,2 and 3, which are
operands. Thus they will be pushed into the stack in that order.
Stack Expression
Next character scanned is "*", which is an operator. Thus, we pop the top two elements from the
stack and perform the "*" operation with the two operands. The second operand will be the first
element that is popped.
Stack Expression
The value of the expression(2*3) that has been evaluated(6) is pushed into the stack.
Stack Expression

48. Next character scanned is "+", which is an operator. Thus, we pop the top two elements from the
stack and perform the "+" operation with the two operands. The second operand will be the first
element that is popped.
Stack Expression
The value of the expression(1+6) that has been evaluated(7) is pushed into the stack.
Stack Expression
Next character scanned is "4", which is added to the stack.
Stack Expression
Next character scanned is "-", which is an operator. Thus, we pop the top two elements from the
stack and perform the "-" operation with the two operands. The second operand will be the first
element that is popped.

49. The value of the expression(7-4) that has been evaluated(3) is pushed into the stack.
Stack Expression
The value of the expression(7-4) that has been evaluated(3) is pushed into the stack.
Stack Expression
Now, since all the characters are scanned, the remaining element in the stack (there will be only one End result:
Postfix String : 1 2 3 * + 4 -
Result : 3

50. SOURCE CODE:
/*Program To Evaluate Postfix Expression */
#include < iostream.h >
#include < conio.h >
#include < math.h >
#include < string.h >
class postfix
{ private:
int stack[50], len, top;
char post[50];
public:
postfix ( );
void push ( int );
int pop ( );
int pfix ( );
};
void postfix :: postfix ( )
{
top = - 1;
}i
nt postfix :: pfix ( )
{
int a, b, i, temp;
cout<<"nEnter Postfix Expression::";
cin>>post;
len = strlen ( post );
post [ len] = '#';

51. for( i = 0 ; post [ i ] ! = '#' ; i + +)
{
if( post [ i ] <= '9' && post [ i ] >= '0')
push( post [ i ] - 48);
else
{
a = pop ( );
b = pop ( );
switch ( post [ i ])
{
case '+': temp = b + a; break;
case '-': temp = b
- a; break;
case '*': temp = b * a; break;
case '/': temp =
b/a; break;
case '%': temp =
b%a; break;
case '^': temp =
pow( b, a );
}
push ( temp );
}
}
return( pop ( ) );
}

52. void postfix :: push( int x )
{
stack[ + + top ] = x;
}i
nt postfix :: pop ( )
{
int x = stack [ top ];
top- -;
return x;
}
void main ( )
{
int x;
postfix ob;
clrscr ( );
x=ob.pfix ( );
cout<<"nResult Of Postfix Expression Ist"<<x;
getch ( );
}
OUTPUT:

53. Quick Sort
11 7 21 3 46 89 2 34
right
1)When pivot is at left end,
1)Compare a[pivot] with a[right] element
if (a[pivot] < a[right]) then right--
else
swap a[pivot] and a[right]
2)When pivot is at right end,
Compare a[pivot] with a[left] element
if (a[left] < a[pivot]) then left++
else
swap a[left] and a[pivot]
pivot
left

54. STEP1: 11 7 21 3 46 89 2 34
left, pivot right
STEP2: 2 7 21 3 46 89 11 34
left right, pivot
STEP3: 2 7 21 3 46 89 11 34
left right, pivot
STEP4: 2 7 21 3 46 89 11 34
left right, pivot
STEP5: 2 7 11 3 46 89 21 34
left, pivot right
STEP6: 2 7 11 3 46 89 21 34
left, pivot right

55. STEP7: 2 7 11 3 46 89 21 34
left, pivot right
STEP8: 2 7 11 3 46 89 21 34
left, pivot right
STEP9: 2 7 3 11 46 89 21 34
left right, pivot
STEP10: 2 7 3 11 46 89 21 34
left, right, pivot
 Here we will stop the main process as the left and right pointers are equal.
 Now see the elements left to ‘11’ are less than ‘11’ and elements right to ‘11’ are grater than ‘11’.
 Now divide the main list into 2 sub lists such as(2,7,3) and (46,89,21,34) and do the same above
process.

56.  #include<stdio.h>
#include<conio.h>
#define MAXSIZE 500
Source code
void quickSort(int elements[], int maxsize);
void sort(int elements[], int left, int right);
int elements[MAXSIZE];
int main()
{ int i, maxsize;
printf(“nHow many elements you want to sort: “);
scanf(“%d”,&maxsize);
printf(“nEnter the values one by one: “);
for (i = 0; i < maxsize; i++)
{
printf (“nEnter element %i :”,i);
scanf(“%d”,&elements[i]);
}

57. printf(“nArray before sorting:n”);
for (i = 0; i < maxsize; i++)
printf(“[%i], “,elements[i]);
printf (“n”);
quickSort(elements, maxsize);
printf(“nArray after sorting:n”);
for (i = 0; i < maxsize; i++)
printf(“[%i], “, elements[i]);
}
void quickSort(int elements[], int maxsize)
{
sort(elements, 0, maxsize - 1);
}
void sort(int elements[], int left, int right)
{
int pivot, l, r;
l = left;
r = right;
pivot = elements[left];
while (left < right)
{
while ((elements[right] >= pivot) && (left < right))
right—;

58. if (left != right)
{
elements[left] = elements[right];
left++;
}
while ((elements[left] <= pivot) && (left < right))
left++;
if (left != right)
{
elements[right] = elements[left];
right—;
}}
elements[left] = pivot;
pivot = left;
left = l;
right = r;
if (left < pivot)
sort(elements, left, pivot - 1);
if (right > pivot)
sort(elements, pivot + 1, right);
}

59.  Consider the elements as shown,
77 33 44 11 88 22 66 55
77 0
Selection sort
min i
 Here min is compared with a[1]
 as min is > a[1]
 min=a[1]
33 0
min i
 This min is compared with a[2] ,as this is < a[2]
 min is same that is 33
 This min is compared with a[3] ,as this is > a[3]
 min =a[3].
11 0
min I
 Now this is compared with a[4],a[5],a[6],a[7] as min is less than all of these min
remains 33
 At last swap min and a[i] like this continue the process with i=1,2,3……

60. SOURCE CODE:
#include < iostream.h >
#include < conio.h >
class selsort
{public : void sort(int *, int);
};
void selsort::sort(int *a, int n)
{
int i, j, x, min, temp;
for( i = 0 ; i < ( n – 1 ) ; i + + )
{
x = i; min = a [ i ];
for( j = i + 1; j < n; j + + )
{i
f( min > a [ j ] )
{
min = a [ j ];
x = j;
}} temp = a [ i ] ; a [ i ] = a [ x ]; a [ x ] = temp;
}}
void main( )
{int a[50], n, i;
clrscr( );

61. cout<<"n ENTER THE SIZE OF THE ARRAY: nt ";
cin>>n;
cout<<"n ENTER THE ELEMENTS:nt";
for( i = 0 ;i < n ;i + + )
cin>>a [ i ];
cout<<"n ELEMENTS BEFORE SORTING:nt";
for( i = 0 ; i < n ; i + + )
cout<<a[i]<<"t";
selsort obj; obj.sort(a,n);
cout<<"n ELEMENTS AFTER SORTING ARE:nt";
for( i = 0 ; i < n ; i + + )
cout<<a[i]<<"t";
getch();
}
OUTPUT:

62. LINEAR SEARCH
10 20 30 40 50 60 70
0 1 2 3 4 5 6
•Here we want to search for ‘50’.
• So compare ’50’ with a[i] where i=0,1,2,3,….
If (a[i]==50)
Then element is found at location i that is 4
Else
i++
•Here the time complexity is O(n).

63. SOURCE CODE:
#include < iostream.h >
#include < conio.h >
class lsearch
{
private:
int a[50], n, count, key;
public:
void init ( );
void linear ( );
};
void lsearch::init ( )
{
count = 0;
}
void lsearch::linear ( )
{
int i;
clrscr ( );
cout<<"nENTER SIZE OF AN ARRAY :: ";
cin>>n;
cout<<"nnENTER "<<n<<" ELEMENTS INTO AN ARRAY ::";
for( i = 0; i < n; i + +)
cin>> a [ i ];
cout<<"nnENTER SEARCH ELEMENT :: ";
cin>>key;
cout<<"nnELEMENTS IN ARRAY ARE :n";

64. for( i = 0; i < n; i + +)
cout<< a [ i ]<<"t";
for( i = 0; i < n; i + + )
if(a [ i ] = = key)
{
count + +;
break;
}i
f( count = = 1 )
cout<<"nn ELEMENT IS FOUND IN
"<< ( i + 1)<<" LOCATION";
else
cout<<"nELEMENT IS NOT
FOUND....";
}
void main ( )
{
lsearch ob;
clrscr ( );
ob.init ();
ob.linear ( );
getch ( );
}

65. OUTPUT:

66. BINARY SEARCH
•Here elements must be in Ascending/Descending order.
•Consider the elements in ascending order
711 15 23 46 64 71 83
low high
here low=0 and high=7
Then calculate mid=(low+high)/2
•Let us search for k=71
•If (a[mid]==k)
then element is found at ‘mid’ location
•If(k<a[mid])
then high=mid-1
else
low=mid+1
•Repeat the previous steps tell low and high are equal.

67. SOURCE CODE:
/*Program To Implement Binary Search */
#include < iostream.h >
#include < conio.h >
class bsearch
{
private :
int a[50], n , x;
public :
void binary ( );
};
void bsearch::binary ( )
{
int i, j, temp, mid, beg, end;
beg = 0;
cout<<"nnENTER THE SIZE OF THE ARRAY :: ";
cin>>n;
end = n - 1;
cout<<"nnENTER THE ELEMENTS OF THE ARRAY :: ";
for( i = 0; i < n; i + +)
cin>>a[i];
cout<<"nnELEMENTS BEFORE BEFORE SORTING ARE :: ";
for( i = 0; i < n; i + + )
cout<< a [ i ]<<" ";

68. for( i = 0; i < n; i + + )
{
for( j = i + 1; j < n; j + +)
{
if( a[ i ] > a[ j ] )
{
temp = a [ i ];
a[ i ] = a[ j ];
a[ j ] = temp;
}
}
}
cout<<"nnELEMENTS AFTER SORTING ARE :: ";
for( i = 0; i < n; i + + )
cout<<a[ i ]<<" ";
cout<<"nnENTER THE ELEMENT TO BE SEARCHED :: ";
cin>>x;
while ( beg < = end )
{
mid = ( beg + end ) / 2;
if ( a [ mid ] = = x )
{

69. cout<<"nSEARCHING IS SUCCESSFUL AND THE ELEMENTS IS
PRESENT AT "<< ( mid + 1 )<<" LOCATION";
return;
}
else if(x<a[mid])
end = mid - 1;
else beg = mid + 1;
}
cout<<"n SEARCH IS UNSUCCESSFUL";
}
void main ( )
{
bsearch obj;
clrscr ( );
obj . binary ( );
getch ( );
}
OUTPUT:

70. POLINOMIAL ADDITION AND MULTIPLICATION
•1 expression: 3x2+2x+1
Store all the coefficients 1,2,3 into an array1.
•1 expression: 2x2+1x+2
Store all the coefficients 2,1,2 into an array2.
ADDITION:
3x2+2x+1
2x2+1x+2
5x2+3x+3
Store the result expression coefficients in array3

71. SOURCE CODE:
/*Program To Demonstrate Addition And Multiplication Of Two Polynomial Expression */
#include < iostream.h >
#include < conio.h >
#define n 100
class poly
{
private:
int a[n], b[n], add[n], mul[n], p, q, at;
public:
void init ( );
void input ( );
void process ( );
void display ( );
};
void poly :: init ( )
{
int i;
for( i = 0; i < n; i + + )
a[ i ] = b [ i ] = add[ i ] = mul[ i ] = 0;
}

72. void poly :: input ( )
{
int i;
cout<<"nEnter Degree Of First Polynomial::";
cin>>p;
cout<<"nEnter Degree Of Second
Polynomial::";
cin>>q;
cout<<"nEnter Values First
Polynomialn";
for( i = 0; i <= p; i + + )
{
cout<<"nEnter X^"<<i<<"
Th Coefficient";
cin>>a[ i ];
}
cout<<"nEnter Values First
Polynomialn";
for( i = 0; i <= q; i + + )
{
cout<<"nEnter X^"<<i<<"
Th Coefficient";
cin>>b[ i ];
}
}

73. void poly :: process ( )
{
int i, j;
if( p > q )
at = p;
else
at = q;
for ( i = 0; i <= at; i + +)
add[ i ] = a[ i ] + b[ i ];
for( i = 0; i <= p; i + + )
for( j = 0; j <= q; j + + )
mul [ i + j ] + = a [ i ] * b [ j ];
}
void poly :: display ( )
{
int i;
cout<<"Addition Of Two Polynomial Expressions Arenn";
for( i = at; i >=0 ; i - -)
cout<<add[i]<<"X^"<<i<<"+";
cout<<"nnMultiplecation Of Two Polynomial Expressions Arenn";
for( i = p + q; i > = 0; i - -)
cout<<mul[i]<<"X^"<< i <<"+";
}

74. void main()
{
poly ob;
clrscr ( );
ob.init ( );
ob.input ( );
ob.process ( );
ob.display ( );
getch ( );
}
OUTPUT:

75. SINGLE LINKED LIST
THEORY:
Figure shows a Linked List. Each item in the list is called a node and contain two fields, a data
field and a next address field. The data field holds the actual element on the list. The next address
field contains the address of the next node in the list. Such an address which is used to access a
particular node, is known as a pointer. The entire linked list is accesses from an external pointer list,
that points to the first node in the list. The next field of last node in the list contains a special value,
known as NULL. The null pointer is used to signal the end of the list.
The singly-linked list is the most basic of all the linked data structures. A singly-linked list
is simply a sequence of dynamically allocated objects, each of which refers to its successor in the
list. Despite this obvious simplicity, there are myriad implementation variations.
The following code inserts a node after an existing node in a singly linked list. The diagram
shows how it works. Inserting a node before an existing one cannot be done; instead, you have to
locate it while keeping track of the previous node.

76. Similarly, we have functions for removing the node after a given node, and for removing a
node from the beginning of the list. The diagram demonstrates the former. To find and
remove a particular node, one must again keep track of the previous element.

77. SOURCE CODE:
/*Program To Implement Single Linked list */
#include< stdio.h >
#include < iostream.h >
#include < conio.h >
#include < process.h >
#include< alloc.h >
class slist
{
private:
struct list
{i
nt data;
struct list *next;
}*start,*temp,*curr,*add,*tem,*addr;
public:
void init ( );
void create ( );
void disp ( );
list *search ( int );
void insert ( );
void del ( );
};

78. void slist :: init ( )
{
start = temp = curr = NULL;
}
void slist::create ( )
{
char ch;
temp = new list;
cout<<"n ENTER THE DATA TO BE STORED n";
cin>> temp->data;
temp->next = NULL;
start = curr = temp;
cout<<"n DO YOU WANT TO INSERT ANOTHER NODE (Y/N)";
cin>>ch;
while( ch = = 'y' )
{
temp = new list;
cout<<"n ENTER DATA TO BE STORED:n";
cin>>temp->data;
temp->next = NULL;
curr->next = temp;
curr = temp;
cout<<"n DO YOU WANT TO INSERT ANOTHER NODE (Y/N):";
cin>>ch;
}
}

79. void slist :: disp ( )
{
if( start = = NULL)
cout<<"n LIST IS EMPTY";
else
{
cout<<"n DATA PRESENT IN A LIST IS n";
temp = start;
while( temp -> next ! = NULL)
{
cout<<"|"<<temp->data<<"|"<<temp->next<<"|-->";
temp = temp -> next;
}
cout<<"|"<<temp->data<<"|"<<temp->next<<"|";
}
}
slist::list *slist :: search( int key)
{
temp = start;
while( temp -> next ! = NULL)
{
if( temp->data = = key )
return temp;
else
temp = temp->next;
}i
f( temp->next = = NULL )

80. if( temp->data = = key )
return temp;
else
return NULL;
}
void slist:: insert ( )
{
int key;
cout<<"n ENTER DATA AFTER WHICH WE CAN INSERT NEW NODE:";
cin>>key;
add=search(key);
if( add = = NULL )
cout<<"n NODE IS NOT FOUND";
else
{
temp = new list;
cout<<"n ENTER INSERTED ELEMENT";
cin>>temp->data;
if( add->next = = NULL)
{
temp->next = NULL;
add->next = temp;
curr = temp;
}

81. else
{
addr = add->next;
add->next = temp;
temp->next = addr;
}
}
}
void slist :: del ( )
{
int key;
cout<<"n ENTER NODE DATA SHOULD BE DELETE:n";
cin>>key;
add = search ( key );
if( add = = NULL )
cout<<"n NODE IS NOT FOUNDn";
else
if( curr = = add )
{
curr = start;
while( curr->next ! = NULL)
{
temp = curr;
curr = curr->next;
}

82. free ( curr );
curr = temp;
curr->next = NULL;
}
else
if( start = = add )
{
temp = start;
start = start->next;
free( temp );
}
else
{ tem = add->next;
temp = start;
while( temp-> next ! = add)
temp = temp->next;
temp->next = tem;
free( add );
}
}
void main ( )
{
slist ob;
int key, ch;
list *temp;
clrscr ( );
cout<<"n * * * SINGLE LINKED LIST OPERATION * * * n";

83. cout<<"n 1. CREATE n 2. DISPLAY n 3. INSERT n 4. DELETE n 5. SEARCH n 6.EXIT n";
cout<<"n *************************n";
do
{
cout<<"n ENTER YOUR CHOICE n";
cin>>ch;
switch ( ch )
{
case 1: ob.create ( );
break;
case 2: ob.disp ( );
break;
case 3: ob.insert ( );
break;
case 4: ob.del ( );
break;
case 5: cout<<"n ENTER THE ELEMENT TO SEARCH";
cin>>key;
temp=ob.search(key);
if( temp = = NULL)
cout<<"n ELEMENT IS NOT FOUND n";
else
cout<<"n ELEMENT IS FOUND n";
break;
case 6: exit(0);
default: cout<<"n INVALID CHOICE n";
}
}while( ch ! = 0 );
getch ( );
}

84. OUTPUT:

85. SINGLE CIRCULAR LINKED LIST
THEORY:
The linked list that we have seen so far is often know as linear lists. The elements of such
a linked list can be accessed, first by setting up a pointer pointing to the first node in the list and
then traversing the entire list using this pointer. Although a linear linked list is a useful data
structure, it has several shortcomings.

86. SOURCE CODE:
/* Program to implement single circular linked list */
#include<iostream.h>
#include<conio.h>
#include<process.h>
#include<alloc.h>
class clist
{
private:
struct list
{
int data;
struct list *next;
}*start,*temp,*curr,*add,*tem,*addr;
public:
void init(); void creat();
void display(); list *search(int);
void insert(); void del();
};
void clist::init()
{
start=temp=curr=NULL;
}

87. void clist::creat()
{
char ch;
temp=new list;
cout<<"n ENTER ENTER DATA TO BE STORED ::";
cin>>temp->data;
cout<<"nADDRESS OF STARTING NODE :: "<<temp;
temp->next=start;
start=curr=temp;
cout<<"nDO YOU WANT TO INSERT ANOTHER NODE (y/n) :: ";
cin>>ch;
while(ch=='y')
{
temp=new list;
cout<<"n ENTER DATA TO BE STORED :: ";
cin>>temp->data;
temp->next=start;
curr->next=temp;
curr=temp;
cout<<"nDO YOU WANT TO INSERT ANOTHER NODE (y/n) :: ";
cin>>ch;
}
}

88. void clist::display()
{
if(start==NULL)
cout<<"nLIST IS EMPTY.....";
else
cout<<"nDATA PRESENT IN A LIST IS :: n";
temp=start;
while(temp->next!=start)
{
cout<<"|"<<temp->data<<"|"<<temp->next<<"|-->";
temp=temp->next;
}
cout<<"|"<<temp->data<<"|"<<temp->next<<"|";
}
clist::list *clist::search(int key)
{
temp=start;
while(temp->next!=start)
{
if(temp->data==key)
return temp;
else
temp=temp->next;
}

89. if(temp->next==NULL)
{
if(temp->data==key)
return temp;
else
return NULL;
} return NULL;
}
void clist::insert()
{
int key;
cout<<"n ENTER DATA AFTER WHICH WE CAN INSERTED NEW NODE ::
";
cin>>key;
add=search(key);
if(add==NULL)
cout<<"n NODE IS NOT FOUND ....";
else
{
temp=new list;
cout<<"n ENTER INSERTED ELEMENT :: ";
cin>>temp->data;

90. if(add->next==start)
{
temp->next=start;
add->next=temp;
curr=temp;
}
else
{
addr=add->next;
add->next=temp;
temp->next=addr;
}
}
}
void clist::del()
{
int key;
cout<<"nEnter node to deleted:";
cin>>key;
add=search(key);
if(add==NULL)
cout<<"nNode is not found";
else if(curr==add)

91. {
curr=start;
while(curr->next!=start)
{
temp=curr;
curr=curr->next;
} free(curr);
curr=temp;
curr->next=start;
}
else if(start==add)
{
temp=start;
start=start->next;
free(temp);
}
else
{
tem=add->next;
temp=start;
while(temp->next!=add)
temp=temp->next;
temp->next=tem;
free(add);
}
}

92. void main()
{
clist ob;
int key,ch;
clist::list *temp;
clrscr();
cout<<"nCIRCULAR LINKED LIST n";
cout<<"n1.Createn2.Displayn3.Insertn4.Deleten5.Searchn6.Exitn";
do
{
cout<<"nEnter your choice";
cin>>ch;
switch(ch)
{
case 1:ob.creat();
break;
case 2:ob.display();
break;
case 3:ob.insert();
break;
case 4:ob.del();
break;

93. case 5:cout<<"nEnter search element";
cin>>key;
temp=ob.search(key);
if(temp==NULL)
cout<<"nElement is not found";
else
cout<<"nElement is found";
break;
case 6:exit(0);
default:cout<<"Invalid choice";
}
}while(ch!=6);
getch();
}

94. OUTPUT:

95. DOUBLE LINKED LIST
AIM: Write a program in C++ to implement DOUBLE LINKED LIST
THEORY:
A two-way list is a linear collection of data elements, called nodes, where each node N is
divided into three parts:
•An item data field.
•A pointer field next which contains the location of the next node in the list.
•A pointer field prev which contains the location of the previous node in the list.
The list requires two list pointer variables: FIRST, which points to the first node in the list, and
LAST, which points to the last node in the list. The figure contains a schematic diagram of such a
list. Observe that the null pointer appears in the next field of the last node in the list and also in the
prev field of the first node in the list. Observe that, using the variable FIRST and the pointer field
next, we can traverse a two-way list in the forward direction as before. On the other hand, using the
variable LAST and the pointer field prev, we can also traverse the list in the backward direction.

96. OPERATION ON TWO-WAY LISTS:
1. Traversing.
2. Searching.
3. Deleting
4Inserting

97. SOURCE CODE:
/* Program to implement Double linked list */
#include<iostream.h>
#include<conio.h>
#include<process.h>
#include<alloc.h>
class dlist
{
private:
struct list
{
int data;
struct list *next,*prev;
}*start,*temp,*curr,*add,*addr,*tem;
public:
void init();
void creat(); void display();
list *search(int);
void insert(); void del();
};
void dlist::init()
{
start=temp=curr=NULL;
}

98. void dlist::creat()
{
char ch;
temp=new list;
cout<<"nENTER DATA TO BE STORED :: ";
cin>>temp->data;
cout<<"n STARTING NODE ADDRESS :: "<<temp<<"n";
temp->next=NULL;
temp->prev=NULL;
start=curr=temp;
cout<<"n DO YOU WANT TO INSERT ANOTHER NODE
(y/n) :: ";
cin>>ch;
while(ch=='y')
{
temp=new list;
cout<<"nENTER DATA TO BE STORED :: ";
cin>>temp->data;
temp->next=NULL;
temp->prev=curr; curr->next=temp;
curr=temp;
cout<<"nDO YOU WANT TO INSERT ANOTHER
NODE (y/n) :: ";
cin>>ch;
}
}

99. void dlist::display()
{
if(start==NULL)
cout<<"n LIST IS EMPTY....";
else
{
cout<<"n DATA PRESENT IN A LISTn:::";
temp=start;
while(temp->next!=NULL)
{
cout<<"|"<<temp->prev<<"|"<<temp->data<<"|"<<temp->next<<"|--
>";
temp=temp->next;
}
cout<<"|"<<temp->prev<<"|"<<temp->data<<"|"<<temp->next<<"|";
}
}
dlist::list *dlist::search(int key)
{
temp=start;
while(temp->next!=NULL)
{
if(temp->data==key)

100. return temp;
else
temp=temp->next;
}i
f(temp->next==NULL)
{
if(temp->data==key)
return temp;
else
return NULL;
} return NULL;
}
void dlist::insert()
{
int key;
cout<<"nENTER DATA AFTER WHICH WE CAN INSERT A
NEW NODE :: ";
cin>>key;
add=search(key);
if(add==NULL)
cout<<"n NODE IS NOT FOUND.....";
else
tem=new list;

101. cout<<"n ENTER ELEMENT TO BE SEARCHED :: ";
cin>>tem->data;
if(add->next==NULL)
{
tem->next=NULL;
tem->prev=add;
add->next=tem;
curr=tem;
}
else
{
addr=add->next;
add->next=tem;
tem->next=addr;
tem->prev=add;
}
}
void dlist::del()
{
int key;
cout<<"n ENTER NODE DATA TO BE DELETED :: ";
cin>>key;
add=search(key);
if(add==NULL)
cout<<"n NODE IS NOT FOUND :: ";
else

102. if(curr==add)
{
curr=start;
while(curr->next!=NULL)
{
temp=curr;
curr=curr->next;
}
free(curr);
curr=temp;
curr->next=NULL;
}
else if(start==NULL)
{
temp=start;
start=start->next;
free(temp);
}
else
{
tem=add->next;
temp=start;
while(temp->next!=add)
temp=temp->next;
temp->next=tem;
free(add);
}
}

103. void main()
{
dlist ob;
int key,ch;
dlist::list *temp;
clrscr();
cout<<"**********DOUBLE LINKED LIST**********";
cout<<"n1.Createn2.Displayn3.Insertn4.Deleten5.Searchn6.Exitn";
do
{
cout<<"nENTER YOUR CHOICE :: ";
cin>>ch;
switch(ch)
{
case 1:ob.creat();
break;
case 2:ob.display();
break;
case 3:ob.insert();
break;
case 4:ob.del();
break;

104. case 5:cout<<"n ENTER SEARCH ELEMENT :: ";
cin>>key;
temp=ob.search(key);
if(temp==NULL)
cout<<"n ELEMENT IS NOT FOUND....";
else
cout<<"n ELEMENT IS FOUND.....";
break;
case 6:exit(0);
default:cout<<"n INVALID CHOICE....";
}
}while(ch!=6);
getch();
}

105. OUTPUT:

106. GRAPH TRAVERSING: DEPTH FIRST SEARCH
THEORY:
•DFS is an uninformed search that progresses by expanding the first child node
of the search tree that appears and thus going deeper and deeper until a goal node
is found, or until it hits a node that has no children. Then the search backtracks,
returning to the most recent node it hadn't finished exploring. In a non-recursive
implementation, all freshly expanded nodes are added to a LIFO stack for
exploration.
•Space complexity of DFS is much lower than BFS (breadth-first search). It also
lends itself much better to heuristic methods of choosing a likely-looking branch.
Time complexity of both algorithms are proportional to the number of vertices plus
the number of edges in the graphs they traverse (O(|V| + |E|)).

107. SOURCE CODE:
/*Program To Implement Depth First Search */
#include < iostream.h >
#include < conio.h >
#define MAX 20
class depth
{
private: int a[MAX][MAX], visited[MAX];
int n, top;
public: void init ( );
void input ( );
void dfs ( int );
};
void depth::init ( )
{ int i, j;
for( i = 0; i < MAX; i + + )
{ visited[ i ] = 0;
for( j =0; j < MAX ; j + + )
a[ i ] [ j ] = 0;
} top = - 1;
}
void depth::input ( )
{ int i, j;
cout<<"nENTER NUMBER OF NODES IN A GRAPH :: ";

108. cin>>n;
cout<<"nENTER ADJACENCY MATRIX FOR A GRAPH :: n";
for( i = 1; i <= n; i + +)
for( j = 1; j <= n; j + + )
cin>>a[ i ][ j ];
}
void depth::dfs ( int v)
{ int i;
visited[v] = 1;
cout<<v<<"->";
for( i = 1; i <= n; iI + + )
if( a [ v ] [ i ] = = 1 && visited [ i ] = = 0)
dfs ( i );
}
void main ( )
{ depth ob;
int start;
clrscr ( );
ob.init ( );
ob.input ( );
cout<<"nSTARTING NODE FOR DFS TRAVERSING :: ";
cin>>start;
cout<<"nDEPTH FIRST SEARCH TRAVERSING IS ::nn";
ob.dfs ( start );
getch ( );
}

109. OUTPUT:

110. GRAPH TRAVERSING: BREADTH FIRST
THEORY:
•BFS is an uninformed search method that aims to expand and examine all nodes of
a graph or combinations of sequence by systematically searching through every solution. In other
words, it exhaustively searches the entire graph or sequence without considering the goal until it
finds it.
•From the standpoint of the algorithm, all child nodes obtained by expanding a node are added
to a FIFO queue. In typical implementations, nodes that have not yet been examined for their
neighbors are placed in some container (such as a queue or linked list) called "open" and then once
examined are placed in the container "closed".
Algorithm for Breadth First Search
1.Enqueue the root node.
2.Dequeue a node and examine it.
•If the element sought is found in this node, quit the search and return a result.
•Otherwise enqueue any successors (the direct child nodes) that have not yet been
discovered.
3.If the queue is empty, every node on the graph has been examined – quit the search and
return "not found".
4.Repeat from Step 2.

111. SOURCE CODE:
/*Program To Implement Breadth First Search */
#include < iostream.h >
#include < conio.h >
#define MAX 20
class breadth
{
private:
int a[MAX][MAX], visited[MAX], queue[50];
int n, front, rear;
public:
void init ( );
void input ( );
void bfs ( );
};
void breadth::init ( )
{
int i, j;
for( i = 0; i < MAX; i + + )
{
visited [ i ] = 0;
for( j = 0; j < MAX; j + + )
a[ i ] [ j ] = 0;
} front = rear = - 1;
}

112. void breadth::input ( )
{
int i, j;
cout<<"nENTER NUMBER OF NODES IN A GRAPH :: ";
cin>>n;
cout<<"nENTER ADJACENCY MATRIX FOR A GRAPH :: n";
for( i = 1;i <= n; i + + )
for( j = 1; j <= n; j + + )
cin>>a[ i ][ j ];
}
void breadth::bfs ( )
{
int i, start;
cout<<"nSTARTING NODE FOR BFS TRAVERSING :: ";
cin>>start;
cout<<"n BREADTH FIRST SEARCH TRAVERSING IS:: n t";
cout<<start;
visited[ start ] = 1;
rear + +;
front + +;
queue[ rear ] = start;
while(front <= rear)
{
start = queue[front];
front + +;

113. for( i =1; i <= n; i + + )
{
if(a[ start ][ i ] = =1 && visited[ i ] = = 0)
{
cout<<"->"<<i;
visited[ i ] =1;
rear + +;
queue [ rear ] = i;
}
}
}
}
void main ( )
{
breadth ob;
int start;
clrscr ( );
ob.init ( );
ob.input ( );
ob.bfs ( );
getch ( );
}

114. OUTPUT:

115. SHORTEST PATH FOR GRAPH
THEORY:
Shortest path is nothing but the path which lies between two nodes with the lowest cost. In a
graph contain so many paths are existed between two nodes (source node to destination node) to
choose the lowest cost path to reach from source node to destination node is nothing but shortest
path algorithm. Shortest path algorithm was first proposed by E. W. DIJKSTRA.
SOURCE CODE:
/*Program To Implement Shortest Path for Graph */
#include < iostream.h >
#include < conio.h >
#define INF 9999
class stpath
{
private:
int i, j, k;
public:
void spath(int [ ][20], int );
void display(int [ ][20], int );
};

116. void stpath::spath(int a[ ][20], int n)
{
for( i = 0 ;i < n; I + + )
for( j = 0; j < n; j + + )
if(a[ i ] [ j ] = = 0)
a[ i ][ j ] = INF;
cout<<"nADJACENCY MATRIX OF COST OF EDGES ARE ::";
display( a, n );
for( k = 0; k < n; k + + )
for( i = 0; i < n; i + + )
for( j = 0; j < n; j + + )
if( a[ i ][ j ] > a[ i ] [ k] + a[ k ][ i ])
a[ i ][ j ] = a[ i ][ k ] + a[ k ][ j ];
cout<<"nADJACENCY MATRIX OF LOWEST COST OF EDGES ARE ::n";
display(a, n);
}
void stpath::display(int a[ ] [20], int n)
{
for( i = 0; i < n; i + + )
{
for( j = 0; j < n; j + + )
cout<<a[ i ][ j ]<<"t";
cout<<"n";
}
}

117. void main()
{
int i, j , n , a[20][20];
stpath ob;
clrscr();
cout<<"nENTER NUMBER OF NODES IN A GRAPH :: ";
cin>>n;
cout<<"nENTER ADJACENCY MATRIX ::n";
for( i = 0; i < n; i + + )
for( j = 0; j < n; j + + )
{
cout<<"Enter "<<i+1<<" To "<<j+1<<" Node Distance";
cin>>a[ i ] [ j ];
}
ob.spath(a, n);
getch ( );
}

118. OUTPUT: