ࡱ> WYLMNOPQRSTUV^g[ Cm bjbj .v ΐΐ(zPPw'w'w'''''-'BtCG(kGkGkGkGHHH$XvEw' HkGkG#333` kGw'kG333^Ů}"1kGpd'["| h}0B,+1w'1L H:3:6HHH2HHHBHHHHHHHHHP Y: @Data Structures using C MCA- 25 TopicHrs1Introduction to data structures10Information and meaning Abstract Data Types, Sequences as value definitions, ADT for varying length character strings, Data types in C, Pointers I n C, Data structures and C.Arrays in C The Array as an ADT, Using one-dimensional arrays, Implementing one-dimensional arrays, Arrays as parameters, Character strings in C, Character string operationsStructures in C Implementing structures, Unions, Implementation of unions, Structure parameters, Representing other data structures, Rational numbers, Allocation of storage and scope of variables.Dynamic memory allocation and cancellation in C2The Stack5Definition and examples Primitive operation, Example, Stack as an ADTRepresenting stacks in C Implementing the POP operation, Testing for exceptional conditions, Implementing the PUSH operationExample: Infix , Postfix and Prefix Basic definitions and examples, Evaluation a postfix expression, Program to evaluate a postfix expression, Converting an expression from infix to postfix, Program to converting an expression from infix to postfix 3Recursion4Recursive definition and processes Factorial function, Multiplication of natural numbers, Fibonacci Sequence, Binary Search, Properties of recursive definition or algorithmRecursion in C Factorial in C, Fibonacci numbers in C, Binary Search in C and Towers of Hanoi problem.4Queues and Lists10Queue and it sequential representation Queue as an ADT, C implementation of queues, Insert operation, Priority Queue, Array implementation of a priority.Linked Lists Inserting and removing nodes from a list, Linked implementation of stacks, Getnode and Free node operations, Linked implementation of queue, Linked list as a data structure, Example of list operations, Header nodes.Lists in C Array implementation of lists, Limitations of Array implementation, Allocating and freeing dynamic variables, Linked lists using dynamic variable, Queues as lists in C, Examples of list operations in C, Non integer and non-homogeneous lists.Other list structures Circular lists, Stack as a circular list, Queue as a circular list, Primitive operations on circular lists, Doubly linked lists5Trees10Binary trees Operations on binary trees, Applications of binary treesBinary tree representation Node representation of binary tree, Internal end external nodes, Implicit array representation of binary trees, Choosing a binary tree representation, Binary tree traversals in C, Threaded Binary treesTrees and their application C representation of trees, Tree traversals, General expressions as trees, Evaluating an expression tree, Constructing a tree.6Sorting8Exchange sort Bubble sort  Quick sortSelection and tree sorting Binary tree sort HeapsortInsertion sorts Simple insertion Shell sort Address Calculation sortMerge and Radix sort Radix Sort7Searching5Basic Search Techniques Algorithmic notation, Sequential searching, Searching an ordered table, Indexed sequential search, Binary search, Interpolation searchTree Searching Inserting into a binary search tree, Deleting from a binary search treeHashing Resolving hash clashes by open addressing, Choosing a hash function. Chapter 2. The Stack Syllabus The StackDefinition and examples Primitive operation, Example, Stack as an ADTRepresenting stacks in C Implementing the POP operation, Testing for exceptional conditions, Implementing the PUSH operationExample: Infix , Postfix and Prefix Basic definitions and examples, Evaluation a postfix expression, Program to evaluate a postfix expression, Converting an expression from infix to postfix, Program to converting an expression from infix to postfix STACK It is an ordered collection of data items in which insertions and deletions are made at one end called the top. It is known as a Last-In-First-Out (LIFO) list. The size of the stack changes dynamically. As elements are entered into the stack its size grows and as elements are removed the size decreases.    An abstract data type in stack includes the following operations MAKENULL(S) Stack S is made empty. TOP(S) The element at the top of the stack S is returned. POP(S) The top element of the stack S is deleted. PUSH(x, S) The element x is inserted into the stack S. EMPTY(S) It returns true if stack S is empty otherwise it returns false. Stack Representation An array can be used to represent a stack. One dimensional array can be used. Ex: Stack[max_stack_size] where max_stack_size is the maximum number of entries. The first element of the stack is stored in stack[0], the second in stack[1] and the ith element in stack[i-1]. The variable top points to the top element of the stack. Initially top=-1 to indicate an empty stack. The operations on a stack are Push inserting an element into the stack. Pop deleting an element from the stack.  max_stack _size top (i) max_stack _size top (ii) max_stack _size top (iii)(i) normal stack (ii) inserting an element (iii) deleting an element Each time an element is entered in to the stack top is incremented. The elements can be entered into the stack until top >= max_stack_size - 1. When a new insertion is attempted after this condition it becomes the STACK OVERFLOW condition. Each time an element is deleted from the stack top is decremented. The elements can be deleted from the stack until top = - 1. When a new deletion is attempted after this condition it becomes the STACK UNDERFLOW condition. Representing Stack in C It can be declared as a structure using two objects: Array to hold the elements of the stack Integer to indicate the position of the current stack top within the array #define STACKSIZE 100 struct stack { int top; int items[STACKSIZE]; };Actual stack S may be declared as Struct stack s; Algorithm for PUSH and POP PUSH(stack, top, max_stack_size, item)This algorithm inserts the element item into the stack stack implemented using an array. Top points to the top element of the stack. The stack can contain at most max_stack_size elements i.e. item.If(top=max_stack_size-1) Print  OVERFLOW  ReturnEnd ifTop ! top+1Stack[top] !itemReturn POP(stack, top, data)This algorithm removes the top element from the stack stack implemented using an array. The top element of the stack is assigned to stack.If(top = -1) Print  UNDERFLOW  ReturnEnd ifData ! Stack[top]Top ! top-1Return PUSH Operation: Step 1: [Overflow Check] If TOP = MAXSTACK 1 THEN "STACK OVERFLOW. Step 2: [Increment top Value] TOP = TOP +1 Step 3: [Insert the item] S[TOP] = item POP Operation: Step 1: [Under Flow Check] If TOP = -1 THEN STACK UNDERFLOW Step 2: [Delete the item] Item = S[TOP] Step 3: [Decrement the TOP] TOP = TOP -1 DISPLAY Operation: Step 1: [Under Flow Check] If TOP = -1 THEN STACK UNDERFLOW Step 2: [Display the item] From i = TOP to i = 0 Print S[i] PROGRAM: #includeHeader files#include#define N 5Declaration of the max number of elements in the stack.struct stackDeclaration of the stack as a structure containing two objects - Array to hold the elements of the stack- data - Integer to indicate the position of the current stack top within the array-top{ int top; int data[N];};struct stack s;Declaration of actual stackvoid push(struct stack*,int);Declaration of the functions used to push, pop and display elements of the stack. Push and display have the return type as void. Pop has the return type as int since the function returns the items stored in the stack. The items stored are of the data type int.int pop(struct stack*);void display(struct stack*);void main()Main function{ int ch, item, a; clrscr(); s.top = -1;Initializing top = -1i.e stack is empty  printf("1.Push\n");Displaying the menu printf("2.Pop\n"); printf("3.Display\n"); printf("4.Exit\n"); while(1) { printf("\n Enter your choice:"); scanf("%d",&ch); switch(ch) { case 1 : printf("Enter the item:");If the choice is the push operation scanf("%d",&item); Push(&s,item);Call the function push. The base address of stack and the item to be pushed are passed to the function. Break; case 2 : a=pop(&s);The choice is pop. The function pop is called and the return value is assigned to a. The base address of the stack is passed to the function. if (a) printf("The item popped : %d\n", a); break; case 3 :display(&s);The choice is display. The function display is called. The base address of the stack is passed to the function. break; case 4 : exit();The choice is exit. The system defined function exit is called. Break; default: printf("\n Enter correct choice."); } }}// The function pushvoid push(struct stack *ps, int x)The base address of the stack is passed as the value of ps. The item is passed as x.{ If(ps->top == N-1)If the top of the stack is equal to max size of stack -1 i.e. it has reached the maximum limit, no more elements can be stored in the stack. The over flow condition is printed. printf("Stack Overflow\n"); ElseIf not The top is incremented by 1 value. The array data will now point to the top and will store the value of x.  { (ps->top)++; ps->data[ps->top]=x; }}//The function popint pop(struct stack *ps)The base address of the stack is passed as the value of ps.{ int temp; If(ps->top == -1)If the top of the stack is equal to -1 i.e. the stack is completely empty, no element can be deleted from the stack. The under flow condition is printed. { printf("Underflow\n"); return(0); } Else {If not The temporary variable temp is assigned the value of the item present in the stack at the top of the stack. Top is decremented by 1. The value of temp is returned to the main function. temp = ps->data[ps->top]; (ps->top)--; return(temp); }}// The function displayvoid display(struct stack *ps)The base address of the stack is passed as the value of ps.{ int i; If(ps->top == -1)If the top of the stack is equal to -1 i.e. the stack is completely empty, no element can be displayed from the stack.  printf("Stack is Empty\n"); Else { printf("The stack contains:\n");If not The top most element is first displayed and so on until the index I of the stack is greater than or equal to zero. for(i=ps->top; i>=0;i--) { printf("| %d |\n",ps-> data[i]); } }} Evaluation of Expressions Infix Expression Ex: A+B*C-D To evaluate this expression We need to have the knowledge of the precedence of all the operators used in an expression. The parenthesis has to be written. Such an expression is known as the infix expression. The order of precedence of the binary operators is Exponentiation Multiplication / Division Addition / Subtraction When we have the operators of the same precedence scanned without the parenthesis the order is assumed left to right. In the case of exponentiation the order is assumed to be right to left. Ex: A + B + C => (A + B) + C A ^ B ^ C => A ^ (B ^ C) By using parenthesis we override the default precedence. Prefix Expression or Polish Notation Ex: *+ABC This equivalent to (A+B)*C The operator symbols are placed before the operands. InfixPrefixA + B+ ABA + B C+AB - C - + AB(A+B)*(C-D)* (A + B) (C - D) * + AB (C - D) * + AB - CD((A+B)*C-(D+E))/(F+G)(+ AB * C - (+ DE)) / ( + FG) (* + ABC - ( + DE)) / ( + FG) ( - * + ABC + DE) / ( + FG) / - * + ABC + DE + FGA B / (C * D ^ E)A B / ( C * ^ DE) A B / ( * C ^ DE) A - / B * C ^ DE - A / B * C ^ DEA^B*C-D+E/F/(G+H)A ^ B * C - D + E / F / ( + GH) ^ AB * C D + E / F / ( +GH) * ^ ABC D + E / F / ( + GH) - * ^ ABCD + E / F / ( + GH) - * ^ ABCD + / EF / ( + GH) - * ^ ABCD + / / EF + GH + - * ^ ABCD // EF + GH((A+B) * C ( D E)) ^ ( F+G)( ( + AB ) * C ( - DE)) ^ (+ FG) ( * + ABC ( - DE)) ^ ( + FG) ( - * + ABC DE) ^ ( + FG) ^ - * + ABC DE + FG Postfix Expression or Reverse Polish Notation Ex: AB + C * Is equivalent to (A+B)*C The operator symbols are placed after the operands. InfixPostfixA + BAB +A + B CAB + -C AB + C -(A+B)*(C-D)(AB +) * (CD -) AB + CD - * ((A+B)*C-(D+E))/(F+G)((AB +) * C ( DE +)) / (FG +) ( AB + C* - (DE +)) / (FG + ) (AB + C * DE + - ) / (FG +) AB + C * DE + - FG + /A^B*C-D+E/F/(G+H)A^ B * C D + E / F / (GH +) AB ^ * C D + E / F / (GH +) AB ^ * C D + EF / / (GH+) AB ^ * C D + EF / GH + /((A+B) * C ( D E)) ^ ( F+G)( (AB +) * C ( DE - ) ) ^ ( FG + ) ( AB + C * - (DE - )) ^ ( FG + ) ( AB + C * DE - - ) ^ ( FG + ) AB + C * DE - - FG + ^A B / (C * D ^ E)A B / ( C * DE ^ ) A - B / ( C DE ^ * ) A B C DE ^ * / A B C DE ^ * / - Program to evaluate a given postfix expression. ALGORITHM: Step 1: Add # at the end of P. Step 2: Repeat scanning P from left to right and repeat step 3 and 4 for each element of P until # is encountered . Step 3: If an operand is encountered, push it onto STACK. Step 4: If an operator @ is encountered: Remove the two top elements of the STACK. If A is the top element, B is the next top element. Evaluate B @ A. [NOT A @ B] Push the result of (2) of step 4 back on to STACK. Step 5: Evaluated value is equal to the top of the STACK. Step 6: Exit. Program #includeHeader files#include#include#include#include struct stack { int top; int data[20]; }s1; Declaration of the stack as a structure containing two objects - Array to hold the elements of the stack- data - Integer to indicate the position of the current stack top within the array - top void push(struct stack *s, int ch); int pop(struct stack *s);Declaration of the functions used to push, pop and display elements of the stack. Push and display have the return type as void. Pop has the return type as int since the function returns the items stored in the stack. The items stored are of the data type int.void main()Main function{ char ch, postfix[20];Postfix is a character array which stores the postfix expression. Int i,op1,op2,result; Struct stack s;Declaration of actual stack clrscr(); printf("Enter the valid postfix expression\n"); scanf("%s", postfix);Entering the valid postfix expression for(i = 0;i <= strlen(postfix);i++)Strlen(postfix) gives the total length of the expression.  { ch = postfix[i]; if(isdigit(ch))Checking whether it is a digit push(&s1,ch -'0'); Else { op1 = pop(&s1); op2 = pop(&s1); switch(ch) { case '+' : result = op1 + op2; break; case '-' : result = op1 - op2; break; case '*' : result = op1 * op2; break; case '/' : result = op1 / op2; break; case '^' : result = pow(op2 ,op1); } push(&s1,result); } } printf("\n OPERATOR PRECEDENCE:"); printf("\n[3] ^ [RIGHT TO LEFT] HIGHEST."); printf("\n[2] * , / [LEFT TO RIGHT] MID."); printf("\n[2] + , - [LEFT TO RIGHT] LOWER."); printf("\n\n POSTFIX EXPRESSION : %s ",postfix); printf("\n\n EVALUATED RESULT = %d",pop(&s1)); getch();}void push(struct stack *s, int x){ ++(s->top); s->data[s->top] = x;}int pop(struct stack *s){ int temp; temp = s->data[s->top]; --(s->top); return(temp);} To convert an infix expression to postfix expression. ALGORITHM Step 1: PUSH the left parenthesis ( on to STACK and add right parenthesis ) at the end of Q. Step 2: Scan Q from left to right and repeat step 3 to 6 for each element of Q, until the attack is empty. Step 3: If an operand is encountered, add it to P. Step 4: If ( is encountered , push it onto stack and this ( can be removed or popped only when ) is encountered . Step 5: If an operator @ is encountered Repeatedly pop from the STACK and add to P each operator ( on the top of stack, which has the same precedence or higher precedence than operator @. PUSH to stack. Step 6: If ) is encountered Repeatedly pop from the STACK and add to P each operator (on the top of stack ) until left parenthesis ( is encountered . Remove the left parenthesis (. [Do not add left parenthesis to P]. Program #include #include #include #define N 10struct stack { int top; char data[N]; };void push(struct stack*,char);char pop(struct stack*);int precedence(char);void main(){ struct stack s; char infx[10],post[10]; char ch; int i,j=0; s.top=-1; Clrscr(); s.top = -1; Printf("Enter the Infix Expression:\n"); scanf("%s",infx); for(i=0;i= precedence(ch))&&(s.top != -1)) post[j++] = pop(&s); push(&s,ch); } } While(s.top != -1) post[j++] = pop(&s); post[j]='\0'; Printf("\nPostfix Expression is %s",post); Getch();}void push(struct stack *s,char x){ s->top++; s->data[s->top] = x;}char pop(struct stack *s){ int temp; Temp = s->data[s->top]; s->top--; return(temp);}int precedence(char ch){ switch(ch) { case '+' : case '-' : return(1); break; case '*' : case '/' : return(2); break; case '^' : return(3); break; case '(' : return(0); } return;} Chapter 3. Recursion Syllabus RecursionRecursive definition and processes Factorial function, Multiplication of natural numbers, Fibonacci Sequence, Binary Search, Properties of recursive definition or algorithmRecursion in C Factorial in C, Fibonacci numbers in C, Binary Search in C and Towers of Hanoi problem. RECURSION It is defined as the process in which a procedure invokes itself or invokes other procedures that eventually invoke the first procedure. When a procedure contains a call to itself it is called direct recursion and when a procedure calls another procedure, which in turn calls the first procedure it is called as indirect recursion. FACTORIAL FUNCTION Given a positive integer n, n factorial is defined as the product of all integers between n and 1. Ex: 5 factorial is 5 * 4 * 3 * 2 * 1 = 120 ! is often used to denote the factorial function. n ! = 1 if n == 0 n ! = n * ( n 1) * ( n 2 ) * ( n 3 ) * ------- * 1 if n > 0 The pseudocode can be written as product = 1; for( i = n ; i > 0 ; i - - ) product = product * i; return(product); This is called as an iterative procedure. It calls for the explicit repletion of a process until a certain condition is satisfied. It can be written as a function that returns the product n ! when n is the input. To write this using the recursive definition the algorithm is Factorial (Fact, N) Step 1: if (N = 0) Fact = 1 Return Step 2: Factorial (Fact, N -1) Step 3: Fact = Fact * N #include #include #includeint fact(n);void main(){ Int n, f = 1;  clrscr(); printf(\n Enter the number\n); scanf(%d, &n); f= fact (n) printf(\n The factorial of %d is %d, n, f); getch();}//The function factint fact( n){ int f; if (n < 0) { printf( \n Factorial of negative numbers not possible); exit; } else  If ( n == 0) return(1); Else { f = n * fact(n 1); return(f); }} Multiplication of natural numbers Iterative Definition The product a * b can be defined as a added to itself b times where a and b are positive integers. Recursive Definition a * b = a if b == 1 a * b = a * ( b 1 ) + a if b > 1 #include #includeint prod(int,int);void main(){ int a,b; clrscr(); printf("\n Enter the two numbers\n"); scanf("%d%d",&a,&b); printf("\nThe product of two numbers %d and %d is %d",a,b,a*b); printf("\nThe product of two numbers %d and %d is %d",a,b,prod(a,b)); getch();}int prod(int a,int b){ int p; if(a==0||b==0) return(0); else if(b==1) return(a); Else { p=a+prod(a,b-1); return(p); }} Fibonacci Series The fibonacci sequence is 0,1,1,2,3,5,8,13,21,34,. Each element in this sequence is the sum of the two preceding elements. It is written as 0 + 1 = 1 1 + 1 = 2 1 + 2 = 3 2 + 3 = 5 3 + 5 = 8.. Recursive definition Fib(n) = n if n == 0 or n == 1 Fib(n)= fib( n - 2 ) + fib( n 1 ) if n > = 2 Program to get the fibonacci sum #include #includeint fib(int);void main() { int f,n; clrscr(); printf("\n ENTER THE VALUE OF N - - - > "); scanf("%d",&n); f=fib(n); printf("\n The Fibonacci Sum is - - - > "); printf("%d",f); getch();}fib(int n){ int f; if(n==0||n==1) return(n); Else { f=fib(n-2)+fib(n-1); return(f); }} Program to get the fibonacci series #include #includevoid main(){ int fib[20]; int i,n; clrscr(); printf("\n ENTER THE VALUE OF N - - - > "); scanf("%d",&n); fib[0]=0; fib[1]=1; printf("\n The Fibonacci Series is - - - > \n"); for(i=1;i<=n;i++) { fib[i+1]=fib[i]+fib[i-1]; printf("%d\t",fib[i]); } getch();} Binary Search It is an efficient method of finding out a required item from a given list, provided the list is in order. The process is: First the middle item of the sorted list is found. Compare the item with this element. If they are equal search is complete. If the middle element is greater than the item being searched, this process is repeated in the upper half of the list. If the middle element is lesser than the item being searched, this process is repeated in the lower half of the list. Step 1: [If the item is not found] If First > Last Return ( -1 ). Else go to Step 2 Step 2: [Calculate the middle] Mid = (First + Last) / 2 Step 3: If item < List [mid] Binary_search (List [ ], item, First, Mid -1) Step 4: Else if Item > List [Mid] Binary_search (List [ ], item, Mid + 1, Last) Step 5: Else Return (Mid + 1). Step 6: Stop. Ex: Array a contains the elements 1,3,4,5,7,9,31,33 The item to be searched is 7. This is a sorted list. First has the value 0. Last has the value 7. The middle position in this list is (First + Last) / 2 = (0+7) / 2 = 3. The middle element is a[3] = 5. If Item < a [Mid] Binary_Search (a [ ], Item, First, Mid -1) Else if Item > a [Mid] Binary_Search (a [ ], Item, Mid + 1, Last) Is 7 < 5 then search the list between the first element i.e. 0 and the element with the value Mid 1 i.e. 2 Is 7 > 5 then search the list between the element with the value Mid + 1 i.e. a[4] and the last element i.e. a[7]. The search process is repeated in the lower half of the list. The list is 7,9,31,33 The middle position in this list is now ( 4 + 7 ) / 2 = 5 and the middle element in this new list is 9. Is 7 < 9 or 7 > 9? Is 7 < 9 then search the list between the element at position 4 and the element with the value Mid 1 i.e. 4 . The list is 7,9. The middle position in this list is (4 + 4 ) / 2 = 4 and the middle element is now 7. The search returns the element 7. Program #include #include#define N 20int a[N];int binary(); void sort();void main(){ Int ch, n,no,i,j,c;  clrscr();  printf("Enter the size of Array:"); scanf("%d",&n); printf("Enter the Array Elements :\n"); for(i=0;i a[j]) { temp=a[i]; a[i]=a[j]; a[j]=temp; } } }} Tower of Hanoi Problem Three pegs A, B and C exist. Disks are placed on A such that the larger disk is always below a smaller disk. The task is to move the five disks to peg C using peg B as a temporary storage. The rules are: Only the top disk of any peg can be moved to the other peg. The larger disk cannot sit on the smaller disk. The solution: General case : n disks. If n = 1 the solution is to move the disk from A and place it on peg C. If n = 2 the solution is Move disk 1(smaller disk) from peg A and place it on peg B. Move disk 2(larger disk ) from peg A and place it on peg C. Move disk 1 from peg B and place it on the larger disk on peg C. If n = 3 the solution is Move disk 1 from peg A and place it on peg C. Move disk 2 from peg A and place it on peg B. Move disk 1 from peg C and place it on peg B. Move disk 3 from peg A and place it on peg C. Move disk 1 from peg B and place it on peg A. Move disk 2 from peg B and place it on peg C. Move disk 1 from peg A and place it on peg C. If n = 4 the solution is Move disk 1 from peg A and place it on peg B. Move disk 2 from peg A and place it on peg C. Move disk 1 from peg B and place it on peg C. Move disk 3 from peg A and place it on peg B. Move disk 1 from peg C and place it on peg A. Move disk 2 from peg C and place it on peg B. Move disk 1 from peg A and place it on peg B. Move disk 4 from peg A and place it on peg C. Move disk 1 from peg B and place it on peg A. Move disk 2 from peg B and place it on peg A. Move disk 1 from peg C and place it on peg A. Move disk 3 from peg B and place it on peg C. Move disk 1 from peg A and place it on peg B. Move disk 2 from peg A and place it on peg C. Move disk 1 from peg B and place it on peg C. The solution for n disks can be written as If n = 1move the single disk from A to C. Move the top n 1 disks from A to B using C as temporary storage. Move the remaining disk from A to C. Move the n 1 disks from B to C using A as temporary storage. Repeat the process until all disks are placed on C. Program #include #includeHeader filesvoid toh(int, char, char, char);Function declarationvoid main(){ int n;Number of disks clrscr(); printf("\n Enter the number of disks\n"); scanf("%d", &n); toh(n,'A','B','C');Function call with number of disks and the pegs A, B, C getch();}// Function tohvoid toh(int n, char source, char temp, char dest)Parameters: number of disks, source, dest, temp{ if(n==1)If number of disks is 1 then the disk is directly moved from peg A to peg C. { printf("\n Move disk %d from peg %c to peg %c", n, source,dest); return; } toh(n-1, source , dest, temp);Call the function with n 1 disks printf("\n Move disk %d from peg %c to peg %c", n, source,dest); toh(n-1, temp, source, dest);Call the function with n 1 disks with temp as the source and source as the temporary storage.}Result Enter the number of disks 4 Move disk 1 from peg A to peg B Move disk 2 from peg A to peg C Move disk 1 from peg B to peg C Move disk 3 from peg A to peg B Move disk 1 from peg C to peg A Move disk 2 from peg C to peg B Move disk 1 from peg A to peg B Move disk 4 from peg A to peg C Move disk 1 from peg B to peg C Move disk 2 from peg B to peg A Move disk 1 from peg C to peg A Move disk 3 from peg B to peg C Move disk 1 from peg A to peg B Move disk 2 from peg A to peg C Move disk 1 from peg B to peg C Properties of Recursive definitions It should not generate an infinite sequence of calls of itself. There should be a non recursive exit. Chapter 6. Sorting Syllabus SortingExchange sort Bubble sort  Quick sortSelection and tree sorting Binary tree sort HeapsortInsertion sorts Simple insertion Shell sort Address Calculation sortMerge and Radix sort Radix SortSorting It is the operation of arranging items in an array in a sequential order according to some criterion. Exchange Sort Bubble sort Quick sort Bubble sort It can be used for small arrays. It is easy to understand. It is easy to program. Ex: Array A contains: 4, 10, 5, 6, 3, 9, 14, 12, 8, 15 Array410563914128154 is compared with 10. 4 < 10, no exchange 10 is compared with 5. 10 > 5, exchange 5 with 10 Temp Array4510639141281510 is compared with 6. 10 > 6, exchange 6 with 10 Temp Array4561039141281510 is compared with 3. 10 > 3, exchange 3 with 10 Temp Array4563109141281510 is compared with 9. 10 > 9, exchange 9 with 10 Temp Array4563910141281510 is compared with 14. 10 < 14, no exchange 14 is compared with 12. 14 > 12, exchange 12 with 14 Temp Array4563910121481514 is compared with 8. 14 > 8, exchange 8 with 14 Temp Array 4563910128141514 is compared with 15. 14 < 15, no exchange. Pass 1 45639101281415Similarly, 6 is compared with 3. 6 > 3, exchange 6 with 3. Temp Array 4536910812141512 is compared with 8. 12 > 8, exchange 8 with 12. Pass 2 45369108121415Now 5 is compared with 3. 5 > 3, exchange 3 with 5 Temp Array4356910812141510 is compared with 8. 10 > 8,exchange 8 with 10. Pass 3 43569810121415Now 4 is compared with 3. 4 > 3, exchange 4 with 3. Temp Array 345698101214159 is compared with 8. 9 > 8, exchange 9 with 8. Temp Array 34568910121415Final array Pass 4 34568910121415Algorithm If a[i] > a[i+1] Exchange a[i] with a[i+1] Program #include #includeHeader filesvoid bubble(int a[10], int n);Function declarationvoid main(){ int n,I, a[10]; Clrscr(); printf("\n Enter the number of elements in the array:\t");n number of elements in the array Scanf("%d",&n); printf("\n Enter the elements\n");a[ ]- the array of elements to be sorted for(i=0;ia[i+1])If a[i] > a[i+1] exchange the elements using a temporary variable temp. exc = 0 implies exchange has taken place. { exc=0; temp=a[i]; a[i]=a[i+1]; a[i+1]=temp; } } printf("Pass No:%d\t",pass);Count the number of passes } printf("\nThe sorted array is\n"); for(i=0;i= a[i]) Step 4.1: i = i + 1. Step 5: Repeat step 5.1 while (key < a [ j ] ) Step 5.1: j = j 1. Step 6: If (i < j) temp = a[ i ] a[ i ] = a[ j ] a[ j ] = temp Step 7: else temp = a[low ] a[ low ] = a[ i ] a[ j ] =temp return (j) Step 8: Stop. Program #include #includeHeader filesvoid quick(int a[],int, int); int part(int a[],int, int);Function declaration of quick and partint count;Global variable to count the partitionsvoid main(){ int a[10], i, j, n, low=0, high; clrscr(); printf("How many numbers you want to enter:");n- number of elements to be entered into the array scanf("%d",&n); high=n-1;Upper range is n 1 printf("Enter the number\n"); for(i=0;i= a[i]) )If I < upper range and key is greater than a[i]  i++; while(key < a[j])If key is less than a[j] j--; if(i j Swap the values of a[low] and a[j] { temp = a[low]; a[low]=a[j]; a[j]=temp; } Return(j);Return the value of j }}Result How many numbers you want to enter: 10 Enter the number - 10 15 12 8 6 4 17 19 16 14 Partition 1 Between the elements 0 and 9 Partition 2 Between the elements 0 and 4 Partition 3 Between the elements 0 and 3 Partition 4 Between the elements 0 and 2 Partition 5 Between the elements 0 and 1 Partition 6 Between the elements 6 and 9 Partition 7 Between the elements 6 and 8 Partition 8 Between the elements 6 and 7 Sorted List - 4 6 8 10 12 15 14 16 17 19 Selection and tree sorting The successive elements are selected in order and placed in their sorted positions. The elements of the input may have to be preprocessed to make the ordered selection possible. Binary tree sort This uses a binary search tree. Heapsort Insertion sorts It sorts the records by inserting records into an existing sorted file. Simple insertion It is the basic of all insertion sorts. It takes and places one element after another from array and inserts each element into its proper position. Ex: Let the array have the elements 10, 5, 8, 15, 3, 6. The first element is considered- 10. 10 is compared with the next element- 5. Since 10 is greater than 5,10 is placed after 5. The list will now look as 5, 10, 8, 15, 3, 6 The second element in the array- 10 is now compared with the third element- 8. 10 is greater than 8. 8 is placed above 10. The list will now look as 5, 8, 10, 15, 3, 6. The third element in the array 10 is compared with the fourth element-15. 10 is less than 15. No exchange takes place The process starts with the first element again. 5 is compared with all the elements in the array till an element lesser than 5 is found i.e. 3. 3 is now moved to the first place in the array. The list now looks like 3, 5, 8, 10, 15, 6 3 is compared with each element in the array. No element is found to be lesser than 3. Similarly 5 is compared with all the elements. No element is found to be lesser than 5. 8 is compared with the other elements. 6 is found to be lesser than 8. It is moved to the position before 8. The list will now look as 3, 5, 6, 8, 10, 15 Program #include #includeHeader filesvoid insertion(int x[ ],int n);Fn declaration of insertionint count=0;Global variable count to count the number of passesvoid main(){ int n,i, a[10];Array a clrscr(); printf("Enter the number of elements \n");n- number of elements in the array  scanf("%d",&n); printf("\nEnter the elements\n");Array a contains the elements to be sorted for(i=0;i=0&&y=0 and the temporary stored element is less than the current element, a[i+1] will now have a[i] a[i+1]=a[i]; a[i+1]=y;a[i+1] will store the value of y } printf("The number of passes are %d \n",count);Printing the number of passes printf("The sorted list is\n");The sorted list is now stored in a[ ] and printed for(i=0;i> Queue(eltype); abstract empty(q) Queue(eltype)q; postcondition empty==(len(q)==0); abstract eltype remove(q) Queue(eltype)q; precondition empty(q)==FALSE; postcondition remove==first(q); q==sub(q, 1, len(q) 1); abstract insert(q, elt) Queue(eltype)q; eltype elt; postcondition q == q + ; C implementation of Queues Solution 1 Implement it as an array to hold the elements of the queue and use two variables front and rear. These variables are used to hold the positions of the front and last elements of the queue. #define MAXSIZE 100struct queue{ int elements[MAXSIZE]; Int front, rear;} que;It can be declared as The use of an array introduces the concept of overflow - the queue grows larger than the max size of the array. The operation insert (queue, element) can be implemented as que.elements[++que.rear] = element The operation delitem = remove (queue) can be implemented as delitem = que.elements[que.front++] Initially que.rear = -1 and que.front = 0. The queue is always empty whenever que.rear < que.front. The number of elements in the queue will always be equal to the value of que.rear que.front + 1   An array of six elements is used to represent a queue. The queue is empty. Therefore que.rear = -1 and que.front = 0.  Items A,B,C have been inserted. Since elements are inserted at the rear end que.rear=2 and que.front=0 since no deletion has taken place.   Items A, B are deleted from the queue. Since deletion takes place at the front end front gets incremented twice. Therefore que.front = que.rear = 2.  Three new items D, E, F are inserted into the queue. Since elements are inserted at the rear end, the rear value is incremented thrice. Therefore que.rear = 5 and que.front = 2. To insert new elements into the queue que.rear must be increased to 6. This is not possible since this is an array of only five elements. The insertion cannot be made. This situation is also possible when the queue is empty.  Since the elements are deleted at the front end, value of front gets incremented thrice. Therefore que.rear = que.front = 5. Modify the remove operation. Whenever the item is deleted the entire queue is shifted to the beginning of the array. delitem=remove(queue) can be rewritten as Delitem=que.elements[0];delitem stores the contents of the queue at the position 0for(i=0;i #include #define N 5Header files The size of the queue is 5int q[N],frnt=0,rear=-1,item;Global variables the array, front, rear, itemvoid main(){ int ch; clrscr(); while(1) { printf("\n1.Insert\n"); printf("2.Delete\n"); printf("3.Display\n"); printf("4.Exit\n");Menu items printf("Enter your choice : "); scanf("%d", &ch);Your choice switch(ch) { case 1 : qinsert(); break;Call to insert function case 2 : qdelete(); break;Call to delete function case 3 : qdisplay(); break;Call to display function case 4 : exit(0); break;Exit function default: printf("Enter proper choice\n");If wrong choice enter correct choice } } }//Function qinsertqinsert()Insert function{ int i, temp; if(rear == N-1)If queue is full print overflow printf("\n Queue Overflow\n"); ElseIf not { printf("Enter the element: "); scanf("%d", &item);Enter the element rear++;Increment rear  q[rear]=item;The item is stored at the rear position in the queue } return;Return to main function}//Function qdeleteqdelete()Delete function{ if(rear == frnt - 1)If the queue is empty print underflow condition printf("\n Queue Underflow\n"); else if(rear == frnt)If rear is equal to front, there is only one element in the queue. Print the last element. Make front = 0 and rear = -1. { printf("\n The last item in queue"); printf("\n The deleted item is %d", q[ frnt ]); frnt=0; rear=-1; } ElseIf not delete the item at the front position and increment q.front. { printf("\n Deleted item is %d",q[frnt]); frnt++; } return;}//Function qdisplayqdisplay()Display function{ int i; if(rear==frnt-1)If rear =front 1 the queue is empty  printf("\n No items in queue\n"); ElseIf not Print the items from front to rear position. { printf("\n\n"); for(i=frnt; i<=rear; i++) printf(" %d |",q[i]); printf("\n\n Front of queue is %d", q[frnt]);Print the value of front and rear. printf("\n Rear of queue is %d", q[rear]); } return;}Result 1. Insert 2. Delete 3. Display 4. Exit Enter your choice : 1 Enter the element: 11. Insert 2. Delete 3. Display 4. Exit Enter your choice : 1 Enter the element: 21. Insert 2. Delete 3. Display 4. Exit Enter your choice : 1 Enter the element: 21. Insert 2. Delete 3. Display 4. Exit Enter your choice : 1 Enter the element: 41. Insert 2. Delete 3. Display 4. Exit Enter your choice : 1 Enter the element: 51. Insert 2. Delete 3. Display 4. Exit Enter your choice : 1 Queue overflow1. Insert 2. Delete 3. Display 4. Exit Enter your choice : 3 1 | 2 | 3 | 4 | 5 | Front of queue is 1 Rear of queue is 51. Insert 2. Delete 3. Display 4. Exit Enter your choice : 2 Deleted item is 1 1. Insert 2. Delete 3. Display 4. Exit Enter your choice : 2 Deleted item is 21. Insert 2. Delete 3. Display 4. Exit Enter your choice : 2 Deleted item is 31. Insert 2. Delete 3. Display 4. Exit Enter your choice : 2 Deleted item is 31. Insert 2. Delete 3. Display 4. Exit Enter your choice : 2 Deleted item is 41. Insert 2. Delete 3. Display 4. Exit Enter your choice : 2 The last item in queue The deleted item is 51. Insert 2. Delete 3. Display 4. Exit Enter your choice : 2 Queue underflow 1. Insert 2. Delete 3. Display 4. Exit Enter your choice : 3 No item in queue 1. Insert 2. Delete 3. Display 4. Exit Enter your choice : 4  Solution 2 The array that holds the queue has to be viewed as a circular implementation rather than a straight line. The first element i.e. position 0 follows the last element i.e. position 1 immediately. This implies that even if the last position is occupied, a new element can be accommodated at position 0.  The queue contains 4 elements F, E, D, C in a 6 element array. The value of que.rear = 5 and que.front = 2.   The array is not full but the last position is occupied. To insert the new element G, it can be placed in the position 0. The elements start at position que[2], que[3], que[4], que[5], que[0].  Three elements are deleted from the queue. que.front=5. Since no new elements are inserted into the queue, que.rear = 0.  A new item H is inserted into the queue. Since elements are inserted at the rear end, the rear value is incremented once. Therefore que.rear = 1 and que.front = 5.    Since the elements are deleted at the front end, value of front gets incremented once. Since it is a continuous cycle, que.front = 0 and que.rear = 1.  The problem with this representation is the difficulty in determining when the queue is empty. The condition que.rear < que.front will not be valid. The situations 2, 3, 4 have the que.rear < que.front, but the queue is not empty. The convention used to solve this problem is to assume that the value of que.front is the array index immediately precceding the first element of the queue rather than the index of the first element itself. A queue of integers can be declared and initialized as -. The values of q.rear and q.front are initialized to the last index of the array.#define MAXSIZE 100struct queue { Int elements[MAXSIZE]; Int front, rear;} q;q.front =q.rear = MAXSIZE 1;empty(queue) int empty( struct queue *pq)The argument to this function is a pointer variable pq of the type struct queue{If pq->front is equal to pq->rear the function returns a True value which means the queue is empty. return((pq->front==pq->rear)?TRUE: FALSE);}remove(q) int remove(struct queue *pq) The arguments of this function is a pointer variable pq to struct queue{ if(empty(pq))If the queue is empty i.e. the function empty returns a true value print it is an underflow condition { printf(Queue underflow);  exit(1); }If the pq->font is pointing to the last index i.e. it is the last element of the array it is made to contain the value 0. if(pq->front==MAXSIZE-1) pq->front=0; ElseIf not remove the element by incrementing the value of pq->front (pq->front)++; return(pq->elements[pq->front]);Return the element.}insert(struct queue *pq, int element) The function involves the testing for overflow condition which occurs when the array is full.  The queue contains 4 elements F, E, D, C in a 6 element array. The elements are at the positions que[2], que[3], que[4], que[5]. The value of que.rear = 5 and since the first element of the queue is at 2 the value of que.front = 1.   To insert the new element G, it can be placed in the position 0. The elements start at position que[2], que[3], que[4], que[5], que[0]. The value of que.rear = 0 and que.front = 1.   One more element is inserted into the queue. que.front=5. The queue is completely full at this point. Therefore que.rear = que.front=0. This situation causes an overflow condition. The condition is that que.rear = que.front. This is also the condition for the underflow condition. The solution for this can be to sacrifice one element of the array. This makes the queue contain only MAXSIZE -1 elements in the array.The function can be written as void insert(struct queue*pq, int ele) The arguments are a pointer variable pq and an element ele.{ if(pq->rear==MAXSIZE-1)If pq->rear is pointing to the last element then make pq->rear=0. pq->rear=0; ElseOtherwise increment pq->rear (pq->rear)++; if(pq->rear==pq->front)If queues rear and front value are equal i.e. the queue is full, print the overflow condition. { printf(Queue Overflow); exit(1); } pq->elements[pq->rear]= ele;The ele is stored at the position of pq->rear. return;} Algorithm Insert Operation QINSERT ( Q, FRONT, REAR, N, ITEM ): Step 1 : [ Overflow check ] If FRONT = ( REAR + 1 ) % N Then write ( overflow ) Return. Step 2 : [ Increment rear pointer ] If FRONT = -1 FRONT = REAR = 0. Else REAR = (REAR +1) % N. Step 3 : [ Insert the item ] Q[REAR] = ITEM Step 4 : Return.Delete Operation QDELETE ( Q, FRONT, REAR, ITEM ) Step 1 : [ Underflow check ] If FRONT = -1 Then write ( Underflow ) Return. Step 2 : [ Delete the item ] ITEM = Q[FRONT] Write The ITEM is Deleted . Q[ FRONT ] = NULL. Step 3 : If FRONT = REAR [ When there is only one item ] FRONT = REAR = -1. Else FRONT = ( FRONT + 1 ) % N. Step 4 : Return.Display Operation QDISPLAY ( Q, FRONT, REAR, ITEM ): Step 1: [ Underflow check ] If FRONT = -1 Then write ( QUEUE is empty ) Return. Step 2: [ Display the items ] If ( FRONT <= REAR ) From I = FRONT to I <= REAR : Write Q [I]. Else if ( FRONT > REAR ) For I = FRONT to I<=N Write Q [I]. For I = 0 to I <= REAR Write Q [ I ]. Step 3: Write front of queue is Q[FRONT] Write front of queue is Q[REAR] Step 4: Return. Program #include #include #define N 5Header files Max size of queue is 5.int q[N], item, front= -1,rear=-1;Global variables the array q[N], front, rear, itemvoid main()Main function{ int ch; clrscr(); while(1) { printf("\n1.Insert\n"); printf("2.Delete\n"); printf("3.Display\n"); printf("4.Exit\n"); Menu items printf("Enter ur choice:"); scanf("%d",&c);Read the choice of menu item switch(c) { case 1 : qinsert(); break;Call qinsert() case 2 : qdelete(); break;Call qdelete() case 3 : qdisplay(); break;Call qdisplay() case 4 : exit(0); break;Call exit() default: printf("Enter Proper Choice:"); } }}//Function qinsertqinsert(){ if( front == ((rear + 1) % N)) If the queue is full print the overflow condition.  { printf("Queue Overflow\n"); return; } Else { printf("Enter the item :"); scanf("%d", &item);Read a new item if(front == -1) If the queue is empty Initialize front and rear = 0 front = rear =0; Else rear = (rear + 1) % N ; q[rear] = item;Store item in q[rear] } return;}//Function qdeleteqdelete(){ int num; if(front == -1) If queue is empty print the Underflow condition. { printf("Queue Underflow\n"); return; } Else { item = q[front];Store the item at q[front] printf("Deleted item is %d", q[front]); if(front == rear) front = rear = - 1; Else front = ( (front + 1) % N ); } return;}//Function qdisplayqdisplay(){ int i; if(front == -1) If front =-1 the queue is empty { printf("Queue Empty\n"); return; } Else { Print the elements printf("The status of queue is:\n\n"); for(i = front; i <= rear; i++) printf(" %d|", q[i]); } if(front > rear) { for(i = front; i <= N;i++) printf(" %d|", q[i]); for(i=0;i<=rear; i++) printf(" %d|", q[i]); } printf("\n Front of Queue %d\n", q[front]); printf("Rear of Queue %d\n", q[rear]); return;}Result 1. Insert 2. Delete 3. Display 4. Exit Enter your choice : 1 Enter the element: 11. Insert 2. Delete 3. Display 4. Exit Enter your choice : 1 Enter the element: 21. Insert 2. Delete 3. Display 4. Exit Enter your choice : 1 Enter the element: 21. Insert 2. Delete 3. Display 4. Exit Enter your choice : 1 Enter the element: 41. Insert 2. Delete 3. Display 4. Exit Enter your choice : 1 Enter the element: 51. Insert 2. Delete 3. Display 4. Exit Enter your choice : 1 Queue full1. Insert 2. Delete 3. Display 4. Exit Enter your choice : 3 The status of queue is: 1 | 2 | 3 | 4 | 5 | Front of queue is 1 Rear of queue is 51. Insert 2. Delete 3. Display 4. Exit Enter your choice : 2 Deleted item is 1 1.Insert 2.Delete 3.Display 4.Exit Enter your choice: 3 The status of queue is: 2| 3| 4| 5| Front of Queue 2 Rear of Queue 5 1.Insert 2.Delete 3.Display 4.Exit Enter your choice: 1 Enter the item :6 1.Insert 2.Delete 3.Display 4.Exit Enter your choice:3 The status of queue is: 2| 3| 4| 5| 0| 6| Front of Queue 2 Rear of Queue 61.Insert 2.Delete 3.Display 4.Exit Enter your choice:2 Deleted item is 2 1.Insert 2.Delete 3.Display 4.Exit Enter your choice:2 Deleted item is 3 1.Insert 2.Delete 3.Display 4.Exit Enter your choice:3 The status of queue is: 4| 5| 0| 6| Front of Queue 4 Rear of Queue 6 1.Insert 2.Delete 3.Display 4.Exit Enter your choice:2 Deleted item is 4 1.Insert 2.Delete 3.Display 4.Exit Enter your choice: 2 Deleted item is 5 1.Insert 2.Delete 3.Display 4.Exit Enter your choice:2 Deleted item is 61.Insert 2.Delete 3.Display 4.Exit Enter your choice:3 Queue Empty1.Insert 2.Delete 3.Display 4.Exit Enter your choice:2 Queue Underflow1.Insert 2.Delete 3.Display 4.Exit Enter your choice:4 Priority Queue It is a data structure which has the intrinsic ordering of the elements. These determine the results of the basic operations. It is a collection of elements such that each element is assigned a priority and there is an order in which the elements are deleted and processed. The rules are: An element of higher priority is processed before any element of lower priority. Two elements with the same priority are processed according to the order in which they were added to the queue. There are two kinds of priority queuesascending priority queue and descending priority queue. Ascending Priority Queue The elements can be inserted at any position in the queue but only the smallest element can be removed from the queue. ascpqinsert(ascpq,x) inserts the element x into the queue ascpq. ascpqdelete(ascpq) removes the smallest element from the queue ascpq and returns its value to the user. Deleting Priority Queue The elements can be inserted at any position in the queue but only the largest element can be removed from the queue. descpqinsert(descpq, x) inserts the element x into the queue descpq. This is logically equivalent to the insert operation of ascpqinsert(ascpq, x). descpqdelete(descpq) removes the largest element from the queue descpq and returns its value to the user. Array Representation A separate queue for each level of priority is maintained. Each such queue will appear in its own circular array and will have its own pair of pointers FRONT and REAR. FRONTREAR EMBED Equation.3 2213005144 Program #include #include #define N 3Header Files Max size of queue 3int q[3][N]; int front[3] = { 0, 0, 0 }; int rear[3] = { -1, -1,-1 }; int item, pr; Global variablesvoid main()Main Function{ int ch; clrscr(); while(1) { printf("\n1.Insert\n"); printf("2.Delete\n"); printf("3.Display\n"); printf("4.Exit\n"); Menu items printf("Enter ur choice:"); scanf("%d",&ch);Enter the choice of menu item switch(ch) { Case 1 : printf("Enter the priority number:"); scanf("%d", &pr); if(pr > 0 && pr < 4) pqinsert(pr-1); else printf("Only 3 priority exits\n"); break;Read Priority number Priority number should be between 1 and 3. Call pqinsert() function Case 2 : pqdelete(); break;Call pqdelete() function Case 3 : display(); break;Call display() function Case 4 : exit(0);Call exit function } }}//Function pqinsertpqinsert(int pr)Argument to the function is pr.{ if(rear[pr] == N-1) The queue may be full. printf("\n Queue overflow \n"); Else { printf("Enter the item"); scanf("%d",&item);Read the item to be inserted. rear[pr]++;Increment rear of the pr. Q[pr][rear[pr]] = item;Store the item in queue. } return;}//Function pqdeletepqdelete(){ int I; For(i=0;i<3;i++)For each of the priority numbered queue { if(rear[i] == front[i]-1)If rear = front the queue is empty Printf("\n Queue %d is empty",i+1); Else {Print the deleted item printf("\n Deleted item is %d of queue %d\n", q[i][front[i]], i+1); front[i]++;Increment front. return; } }}//Function displaydisplay(){ int i, j; for(i=0;i<3;i++)For all the priority numbered queues { if(rear[i]==front[i]-1)If rear = front -1 the queue is empty Printf("\n No items in queue %d\n",i+1); Else { Printf("\n Queue %d |",i+1); for(j=front[i];j<=rear[i];j++)Print the elements printf("%d|", q[i][j]); Printf("\n front of queue is %d\n", q[i][front[i]]); Printf("\n rear of queue is %d\n", q[i][rear[i]]); } } return;} Classification of Queues       Operations on Queue Insertion It allows the user to insert an element into the queue. It has to test for overflow condition. If the queue is not full, the rear is incremented by 1 and the item will be inserted at the rear end. Qinsert(Q, Front, Rear, N, Item) if Rear = N -1 Display Overflow Return Rear = Rear + 1 Q[Rear] = item Return Deletion The element is deleted from the front end. This operation cannot take place if the queue is empty. The condition is called underflow. The front is incremented by 1. Qdelete(Q, Front, Rear, Item) if rear = front -1 Display Underflow Return item = Q[Front] if Front = Rear Front = 0, Rear = -1 Else Front = Front + 1 Return QEmpty It returns a 1 when the queue is empty and it returns a 0 when the queue is not empty. It is used to check for the underflow condition. QEmpty() If Rear = Front -1 Return (1) Else Return (0) QFull It returns a 1 when the queue is full and it returns a 0 when the queue is not full. It is used to check for the underflow condition. QFull() If Rear = N -1 Return (1) Else Return (0) Display It allows the user to display the items in the queue. It has to check for the underflow condition. Display() If Rear = Front 1 Display Stack is empty Else Repeat next step from front of stack to rear of stack. Display the elements of the stack Return Structure Implementation of Linear Queue #include #includeHeader Files#define MAX 5Size of queue-5struct queue { int items[MAX]; int front; int rear; }q;Declaration of the queue as a structure containing three objects - Array to hold the elements of the queue- items - Integers to indicate front and rear of queue front, rear void main(){ int c; q.front=0; q.rear=-1;Initializing the front and rear of the queue clrscr(); printf("\n--------------------------------------"); printf("\n Queue Implementation Using Structures"); printf("\n--------------------------------------"); while(1) { printf("\n 1.Insert"); printf("\n 2.Delete"); printf("\n 3.Display"); printf("\n 4.Exit"); Menu Items printf("\n Enter u re choice"); scanf("%d", &c);Read the choice switch(c) { case 1: qinsert(&q); break;Call fn qinsert() case 2: qdelete(&q); break;Call fn qdelete() case 3: qdisplay(&q); break;Call fn qdisplay() case 4: exit(0);exit() } }  getch();}//Function to insert elementsqinsert(struct queue *q)A pointer variable q of the structure is used as the argument.{ int item; if((q->rear)==MAX-1) The Overflow condition printf("\n OVERFLOW "); Else { printf("\n Enter the element"); scanf("%d",&item);Read the element (q->rear)++; q->items[q->rear]=item;Increment the rear and store item }}//Function to deleted an itemqdelete(struct queue *q)A pointer variable q of the structure is used as the argument.{ if((q->rear)==(q->front)-1)The Underflow condition printf("\n UNDERFLOW "); else if(q->rear==q->front)If the last element is going to be deleted { printf("\n The last element to be deleted is %d", q->items[q->front]);Display the last element q->front=0; q->rear=-1;Reinitialize front and rear } Else { printf("\n The deleted item is %d",q->items[q->front]);Display the deleted item q->front++;Increment front } return;}//Function to display the elementsqdisplay(struct queue *q)A pointer variable q of the structure is used as the argument.{ int i; if(q->rear==q->front-1)The Underflow condition printf("\n Queue is empty"); Else { printf("\n The elements are:"); for(i=q->front;i<=q->rear;i++) printf("\n %d \t",q->items[i]);Display the elements printf("\n The front of queue is %d",q->items[q->front]); printf("\n The rear of queue is %d",q->items[q->rear]);Display the front and rear of the queue. } return;} Circular Queue Ex: Size of circular queue (N) is 4. Sl. No.OperationStatus of QueueFront and Rear 1. Insert 100 1 2 3 10  Front= 0, Rear=0 2. Insert 110 1 2 3 10 11  Front= 0, Rear=1 3. Insert 120 1 2 3 10 11 12  Front= 0, Rear=24. Insert 140 1 2 3 10 11 12 14  Front= 0, Rear=35.Insert 150 1 2 3 10 11 12 14 Queue Overflow Front= 0, Rear=3 The condition Front=(Rear+1)%N is true6.Delete0 1 2 3 11 12 14  Front =1, Rear=37.Delete0 1 2 3 12 14  Front =2, Rear=38.Insert 150 1 2 3 15 12 14 Queue Overflow Front= 2, Rear=0 Rear=(Rear+1)%N is true Rear=(3+1)%4=09.Delete0 1 2 3 15 14  Front =3, Rear=010.Delete0 1 2 3 15  Front =0, Rear=0 Priority Queue Insertion Operation First specify the priority and the item to be inserted. If the queue of the given priority is full display the condition Queue Overflow. If the queue is not full insert the elements at the rear end of the given priority queue. If rear[priority]=N-1 Display Queue Overflow Exit rear[priority]=rear[priority]+1 queue[priority][rear[priority]]=item Deletion Operation Scan the first non-empty queue and delete the first element of that queue. Scan all the queues till a non empty queue is reached. if queue[ ][ ] is empty Display Queue Underflow Exit Delete the element which is in front of the first non empty queue. DEQUES It is a double ended queue. It is maintained by a circular queue. Input restricted Deque It allows insertions only at one end of the queue but allows deletion at both the ends. The insertions are possible only at the rear end. Output restricted Deque It allows deletions only at one end of the queue but allows insertion at both the ends. The deletion is possible only at the front end. Linked Lists It is a data structure where the data items are stored by explicit ordering. Each item in the list is called a node and contains two fields. The fields are the information field info-it contains the actual element and an address field next- it contains the address of the next element. The address is used to access a particular node is called a pointer. The entire list is accessed from an external pointer list that contains the address of the first node in the list. The next field of the last node in the list contains a special value known as null. It is used to signal the end of a list. The list with no nodes is called an empty list or null list. Inserting and Removing Nodes from a list Inserting Nodes The list is a dynamic structure. Info next Info next Info next List Fig: 1  Info next P Info next Info next Info next Fig: 2  List  Info next P  Info next Info next Info next List Fig: 3  Info next  info next info next info next P List Fig: 4   P List Fig: 5  List Fig: 6  Fig: 1 consists of a list which contains three nodes. 1. We need to insert the integer 6 into the list at the front of the list. To do this we need a mechanism that obtains empty nodes and then adds them to the existing nodes. The mechanism for obtaining the empty nodes is p=getnode();. It obtains an empty node and then sets the contents of the variable p to the address of that node. The value of p is now a pointer to this newly allocated table. Fig: 2 consists of a new node P. This is to be inserted into the list at the front of the list. 2. To insert the integer 4 into the info portion of the allocated node. info(p)=4; Fig: 3 consists of the new node P with the value 4 in the info field. 3. The next field of the node P should be set. Since this node has to be inserted at the beginning of the list this node should point to the current first node i.e. the node having the value 1. next(p)=list; This operation places the value of list into the next field of node(p). Fig: 4 shows this operation. P points to the list. 4. The external pointer list should now point to the new additional node since it is at the beginning of the list. list = p; This changes the value of list to the value of p. Fig: 5 shows this operation. This shows p as an auxiliary variable which is used during the process of modifying the list. 5. The value of p is not necessary after the process of modifying. The value of p may be changed now. Fig: 6 shows the final list. The pseudo code for this insertion in general is p=getnode(); info(p) =x; next(p)=list; list = p; where x is the value of the node to be inserted. Removing nodes The process is to remove the first node of a nonempty list and store the value of the info field into the variable x. Info next Info next Info next List Fig: 1   P   List Fig: 2 X = 1 Info next  info next info next P  List Fig: 3 X = 1  Info next P Fig: 3 Info next Info next   List Fig: 4  X = 1 Info next Info next Fig: 5  List  We first have the list as shown in Fig: 1. We have to assign the value of the external pointer list to the new pointer p. This is done using the operation p =list; This changes the value of p to the value of list as shown in Fig: 2. Now p contains the address of the list. Since we are removing the first node of the list, the external pointer list should point to the second node. This cane be done by the operation list = next(p); This operation places the value of the next field of node(p) into the list. This is shown in Fig: 3. The value of the info node of the first node is now set as the value of the variable x. x=info(p); This operation sets the value of x as the value of the info field of the node i.e. 1. This is shown in Fig: 4. At this point the first node has been removed from the list and x has been set to the desired value i.e. 1. p still points to the node that was present at the first position in the list. This node is not necessary as it is not present in the list and its information is stored in x. The node(p) should be able to be reused again. freenode(p) frees the node. After this operation node(p) cannot be referenced as the node is not allocated any more. Any reference to p is also illegal. The node can be reallocated again and a pointer to it can be reassigned to p by the operation p=getnode(). Algorithm Insert at Beginning Step 1: Create a new node that is to be inserted NEWNODE = getnode( ); Step 2: Enter the item into the INFO field of the NEWNODE INFO [ NEWNODE ] = ITEM Step 3: Assign the item into INFO field of the NEWNODE LINK [ NEWNODE ] = Start. It makes a link from NEWNODE to the previous first node. Now both start and NEWNODE are pointing to the original first node of the linked list Step 4 : Reassign start with the NEWNODE so that a link is developed start and new node start = NEWNODE Step 5: ExitDelete a node based on an item. Step 1: Check for the position whether the deletion is at first position If ( POS = 1 ) DELETE_AT_BEGINING ( ); Else GO TO Step 2. Step 2: Set CURRPTR to start CURRPTR = start Step 3: Set PV to NULL PV = NULL Step 4: If the item to be deleted is not at first position From I = 0 to I < POS PV = CURRPTR CURRPTR = CURRPTR ( LNK Step 5: LINK of PV is assigned with LINK of CURRPTR i.e. make a link between the PV and the next node of CURRPTR PV ( LINK = CURRPTR(LINK Step 6: Exit. Search a node based on an item. Step 1: Set CP to START CP = START Step 2: Start searching from the first node WHILE CP(VAL != ITEM and CP != NULL repeat step3 Step 3: Assign CP to next node of the list CP = CP(LINK Step 4: Check if item is found If CP = NULL WRITE Item not in the list Else WRITE Item found Step 5: ExitDisplay all the contents of the list. Step 1: Set CP to START CP = START Step 2: Start displaying from the first node WHILE CP not = NULL repeat step3 through step 4 Step 3: Display the contents WRITE CP( VAL Step 4: Assign CP to next node of the list CP = CP(LINK Step 5: Exit  Program #include #includeHeader Filesstruct list { int id; char name[10]; int sem; struct list *link; }; Declaration of list as a structure variable. The list consists of id of type int, name a char array, sem of type int.typedef struct list NODE; NODE *start = NULL;Assigns the symbol name Node to the data type definition struct list. void insrt_beg(); void delinfo(); void display(); void search(); Function declarationsvoid main(){ int ch; clrscr(); while(1) { printf("1.Insert at beginning\n"); printf("2.Delete Id info\n"); printf("3.Search ID\n"); printf("4.Display\n"); printf("5.Exit\n"); Menu items printf("Enter ur choice\n"); scanf("%d",&ch);Read choice of menu  switch(ch) { case 1 : insrt_beg(); break;Call fn insrt_beg() case 2 : delinfo(); break;Call fn delinfo() case 3 : search(); break;Call fn search() case 4 : display(); break;Call fn display() case 5 : exit(0); break;Call exit() default : printf("Not a valid choice\n"); } End of switch statement } End of while loop}End of main fn// Function delinfovoid delinfo(){ NODE *curptr,*pptr; int idno; curptr = start; pptr = NULL;Pointer variables curptr and pptr of the type NODE curptr points to start pptr points to NULL printf("Enter Id to be deleted\n"); scanf("%d", &idno);Read the id to be deleted if(start == NULL) printf("List is Empty\n");If start is null the list is empty else if (start->id == idno)If start is pointing to the id that is equal to the idno  { start =start->link; printf("\n id deleted\n"); free(curptr); return;Start is pointed to link The curptr is freed from memory } Else { while((curptr->id != idno) && (curptr !=NULL))If the curptr is not pointing to the idno and it is not equal to NULL, store the value of curptr in pptr and curptr points to the next node { pptr = curptr; curptr = curptr->link; } if(curptr == NULL) printf("\n id no not found\n");If curptr is pointing to NULL the id is not found. Else {Else the item is deleted. Value of curptr->link is stored in ptr->link printf("\n id deleted \n"); pptr->link = curptr->link; }End of else  }End of else}End of function//Function insrt_begvoid insrt_beg(){ NODE *newnode; newnode = (NODE *)malloc(sizeof(NODE));Initialize newnode and allocate space to it printf("Enter ID:"); scanf("%d",&newnode->id);Read the id to be inserted fflush(stdin); printf("\nEnter Name:"); gets(newnode->name); printf("\nEnter semester:"); scanf("%d",&newnode->sem); Read name and semester newnode->link =start; start=newnode;Since the id is inserted at the beginning newnode is linked to start}//Function Searchvoid search(){ NODE *curptr; int e, c=0; curptr=start; if(start==NULL) printf("List is Empty\n");If start has the value of NULL the list is empty  ElseIf not { printf("Enter Id to be searched:"); scanf("%d", &e);Read the id to be searched while(curptr != NULL)If curptr not equal to NULL { if(curptr->id == e)If the id pointed by curptr is equal to the id { c=1; printf("\n Id No : %d Found \n", curptr->id); break; Print that the id is found } curptr=curptr->link;Point to the next link } if(c==0) printf("\nNO such ID number in the list\n");No id is found }End of else}//Function Displayvoid display(){ NODE *curptr; curptr=start; if(start==NULL) printf("List Empty\n");If start has the value of NULL the list is empty  ElseIf not { printf("\n LIST \n"); printf("---------------------------\n"); printf("STU_ID\tNAME \tSEMESTER"); printf("\n------\t------\t---------"); while(curptr != NULL)If curptr not equal to NULL { printf("\n%d",curptr->id); printf("\t%s\t%d",curptr->name,curptr->sem); curptr=curptr->link; Print the information }End of while }End of else printf("\n \n");} Result 1.Insert at beginning 2.Delete Id info 3.Search ID 4.Display 5.Exit Enter ur choice: 1 Enter ID:12 Enter Name: Lekha Enter semester:41.Insert at beginning 2.Delete Id info 3.Search ID 4.Display 5.Exit Enter ur choice: 1 Enter ID:13 Enter Name: Manish Enter semester:41.Insert at beginning 2.Delete Id info 3.Search ID 4.Display 5.Exit Enter ur choice: 4 LIST -------------------------------------- STU_ID NAME SEMESTER ---------- --------- --------------- 13 Manish 4 12 Lekha 41.Insert at beginning 2.Delete Id info 3.Search ID 4.Display 5.Exit Enter ur choice: 3 Enter Id to be searched: 12 Id No : 12 Found1.Insert at beginning 2.Delete Id info 3.Search ID 4.Display 5.Exit Enter ur choice: 2 Enter Id to be deleted: 12 id deleted1.Insert at beginning 2.Delete Id info 3.Search ID 4.Display 5.Exit Enter ur choice: 3 Enter Id to be searched:12 NO such ID number in the list1.Insert at beginning 2.Delete Id info 3.Search ID 4.Display 5.Exit Enter ur choice: 4 LIST -------------------------------------- STU_ID NAME SEMESTER ---------- --------- --------------- 13 Manish 4 1.Insert at beginning 2.Delete Id info 3.Search ID 4.Display 5.Exit Enter ur choice: 5  Linked Implementation of Stacks The operation of adding the element in the beginning of the list is similar to pushing the element into a stack. The stack can be accessed only thru the top element. The list can be accessed only from the pointer to its first element. The operation of deleting the first element from the linked list is similar to popping the element from the stack. The only immediately accessible element is removed and the next element becomes immediately accessible. The stack can be represented as a linear linked list. The first node of the list is the top of the stack. If the external pointer s points to the linked list, the operation push(s, x) can be implemented as push(s,x)p =getnode();info(p)=x;next(p)=s;s=p; The operation empty(s) will test whether s is equal to null. The operation x = pop(s) removes the first node from a non empty list and signals the underflow condition if the list is empty. if (empty(s)){ printf(Stack underflow); exit(1);}Else{p = s; s = next(p);x = info(p);freenode(p);} The stack implemented as a linked list. Info next Info next Info next Stack  Stack  The advantage of the implementation of stacks as linked lists is that all the stacks used by the program can share the same available list. When any stack needs a node, it can obtain it from a single available list. When the stack no longer needs a node it returns the node to that available list. This is possible only when the total amount of space needed by all the stacks at any one time is less than the amount of space initially available to them. The stacks are able to grow and shrink to any size. There is no space preallocated to any single stack. No stack will be using any space which it does not need. Getnode and Freenode operations Computers do not have an infinite amount of storage and cannot manufacture extra storage for immediate use. The function of freenode is to make a node that is no longer used currently be reused in another context. The nodes cannot be accessed by the programmer without the getnode and freenode operations. The nodes that are available for access is implemented as a stack using linked list. The list is linked together by the next field in each node. The getnode operation removes the first node from this list and makes that node available for use. The freenode operation adds the unused node to the front of the list and makes the node available for reallocation by the getnode. The list of available nodes is called available list. if(avail==null){ printf(Overflow); exit(1); }p= avail;avail = next(avail);When the available list is empty it means that all the nodes are currently in use and a new allocation cannot be done. When a program calls the getnode in this situation there is an overflow condition. p = getnode(); can be implemented as next(p) = avail;avail = p;freenode(p); can be implemented as Program to implement stack as a linked list #include #includeHeader Filesstruct list { int info; struct list *link; };Declaration of the list as a structure containing - Integer to hold the elements of the list- infoint c=0;global variabletypedef struct list NODE; NODE *start = NULL,*curptr;Assigning the node to the data type definition struct list void push(); void pop(); void display();void main(){ int ch; clrscr(); printf("\n Stack Size = 4"); while(1) { printf("\n1.Push"); printf("\n2.Pop"); printf("\n3.Display"); printf("\n4.Exit"); printf("\nEnter choice:"); scanf("%d",&ch); switch(ch) { case 1 : if(c<4) push(); else printf("\n Stack Overflow"); break; case 2 : pop(); break; case 3 : display(); break; case 4 : exit(0); break; default : printf("\n Not a Valid Choice."); } }}void push(){ NODE *newnode; newnode = (NODE *) malloc(sizeof(NODE)); C=c+1; printf("Enter the information:"); scanf("%d",&newnode->info); newnode->link =start; start=newnode;}void pop(){ NODE *curptr, *preptr; c=c-1; if(start!=NULL) printf("The deleted item is %d\n",start->info); if(start==NULL) printf("Stack is empty\n"); Else { curptr=start; start = start->link; free(curptr); }}void display(){ NODE *curptr; clrscr(); curptr=start; if(start==NULL) printf("Stack Empty\n"); Else { printf("\nNumber of elements :%d\n",c); printf("\nStack\n"); printf("------\n\n\n"); printf("Start "); while(curptr!=NULL) { printf(" --> %d",curptr->info); curptr=curptr->link; } printf("\n\n"); }} Linked List Implementation of Queues The items are deleted from the front of the queue and inserted at the rear. The pointer to the first element of a list represents the front of the queue. The pointer to the last element of the list represents the rear of the queue.  rear Front  The queue q consists of a list and two pointers q.front and q.rear. The operations empty(q) and x=remove(q) are similar to empty(s) and x=pop(s) with s replacing q. When the last element is removed from the queue both q.front and q.rear should be set to null. if(empty(q)){ printf(Queue Underflow); exit(1);}p = q.front;x = info(p);q.front = next(p);if(q.front==null) q.rear=null;freenode(p);return(x); x = remove(q); can be implemented as p= getnode();info(p)=x;next(p)=null;if(q.rear==null) q.front=p;Else next(q.rear)=p;q.rear=p; insert(q, x) is implemented as After the insertion the queue now has a new entry. rear  Front  Disadvantages It occupies more storage than an array. This is because a node in the list contains two fields info and next. The array contains only one field to store the information. Additional time is spent in managing the available list. Each addition and deletion of the element from a stack or a queue involves the corresponding addition and deletion to the available list. Advantages All stacks and queues of a program will have the access to the same available free nodes list. Nodes not used by a stack may be used by another as long as the total number of nodes in use at any time is not greater than the total number of nodes available for that program. Linked list as a data structure An item is accessed in a linked list by traversing from the beginning. The advantage of a list over an array occurs when it is necessary to insert or delete an element in the middle of a group of other elements. Linked Lists Inserting and removing nodes from a list, Linked implementation of stacks, Getnode and Free node operations, Linked implementation of queue, , Example of list operations, Header nodes. 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