Prepared By:- Dinesh Sharma Asstt. Professor, CSE & IT Deptt. ITM Gurgaon

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1 Data Structures &Al Algorithms Prepared By:- Dinesh Sharma Asstt. Professor, CSE & IT Deptt. ITM Gurgaon

2 What is Data Structure Data Structure is a logical relationship existing between individual elements of data the logical or mathematical model of a particular organization of data is called "Data Structure". A means of storing collection of data. It is the specification of the elements of the structure, the relationship between them, and the operations that may be performed on them.

3 Data Structure Continued... Data may be organized in many different ways; The choice of a particular data model depends on two considerations: 1. It must be rich enough in structure to reflect the actual relationships of the data in the real world. 2. The structure should be simple enough that one can effectively ec eyprocess the data a when necessary. Algorithm + Data structure Program

4 Classification of Data Structure Data Structure Primitive Data Structure Non primitive data structure integer Float Character Pointer Array Lists Files Linear List Non linear List Stacks Queues Graphs Trees

5 Classification of Data Structure Primitive Data Structure: These are Basic structures and are directly operated upon by machine instuructions, Have different representation on different computers. Examples : Real, Float, integer, character, pointer etc.

6 Classification of Data Structure Non primitive Data Structures: These are more sophisticated data structures. they are derived from the primitive data structure. The non primitive data structure emphasize on structuring of a group of homogeneous or heterogeneous data items. Examples: Arrays, List and Files.

7 Operations on Data Structures The particular data structure t thatt one chooses for a given situation depends largely on the nature of specific operations to be performed. The following are the four major operations associated with any data structure: 1. Traversing 2. Searching 3. Insertion 4. Deletion 5. Sorting 6. Merging

8 Operations on Data Structure Traversing : Accessing each record exactly once so that certain items in the record may be processed. Searching : Finding the location of the record with a given key value, or finding the locations of all records which satisfy one or more conditions. Inserting : Adding a new record to the structure. Deleting : Removing a record from the structure. Sorting : Arranging the record in some logical order. Merging : Combining the records in two sorted files into a single file.

9 Abstract Data Type ADT is a mathematically specified entity that defines a set of its instances, with: A specific interface a collection of signatures of operations, that can be invoked on an instance. A set of axioms (preconditions & Postconditions) that defines the semantics of operation(i.ee what the operations do to instances of ADT, not how)

10 Abstract Data Types Types of Operations: Constructors Manipulation Procedures Access Functions

11 Abstract Data Type Why do we need to talk about ADT in data structure??? They serve as specifications of requirements for the building blocks of solution to algorithmic problems. Provides a language to talk of on a higher level of abstraction. ADT s encapsulate data structures and Algorithms that implement them. Separates the issues of correctness and efficiency.

12 Memory Allocation in C There are two types of memory allocation possible in C language 1. Compile time/static 2. Run-time/Dynamic

13 Memory Allocation in C C provides four dynamic allocation and de-allocation functions: 1. Malloc() 2. Calloc() 3. Free() 4. Realloc()

14 Memory Allocation in C Malloc() Eg int ptr; Calloc() eg Free() Realloc() Ptr=(int *)malloc(10*sizeof(int)) int *ptr; ptr=(int *)calloc(10,2); Eg free(ptr) Eg ptr=realloc(ptr,new_size)

15 Example to show use of Malloc() Void main() { } int *ptr, I,n; printf( enter no of element u want to store in array ); Scanf( %d,&n) ptr=(int *)malloc(sizeof(int)*n); n); printf( \n Enter the elements ); For(i=0;i<n;i++) scanf( %d,ptr+i); printf( u have entered\n ); for(i=0;i<n;i++) printf( %d \t,*(ptr+i)); getch();

16 Stacks Non primitive linear data structure

17 Stacks Outline Stacks Definition Basic Stack Operations Array Implementation of Stacks

18 What is a stack? It is an ordered group of homogeneous items of elements. Elements are added to and removed from the top of the stack (the most recently added items are at the top of the stack). The last element to be added is the first to be removed (LIFO: Last In, First Out).

19 BASIC STACK OPERATIONS 1. Push(): the process of adding a new element to the top of the stack is called PUSH operation. 2. POP(): the process of deleting an element from the top of the stack is called pop operation.

20 Stack Terminology Overflow: This is the situation when stack becomes full and no more elements can be pushed on to the stacks. at this point the stack top is present at the highest location of the stack. Underflow: When stack contains no element. The TOP is present at the bottom of the stack.

21 Stack Terminology TOP: this term refers to the top of the stack. The stack TOP is used to check stack overflow and underflow. Initially the TOP stores -1. We use this term to refer to the maximum size of the stack. MAXSIZE:

22 Array Implementation of the Stacks The stacks can be implemented by the use of arrays and linked lists. One way to implement the stack is to have a data structure where a variable called top keeps the location of the elements in the stack (array) An array is used to store the elements in the stack

23 Stacks

24 Push Operation Push an item onto the top of the stack (insert an item)

25 Pop operation Pop an item off the top of the stack (delete an item)

26 Function To Implement PUSH() void push() { int pushed_item; if(top == (MAX-1)) printf("stack Overflow\n"); else { printf("enter t the item to be pushed in stack : "); scanf("%d",&pushed_item); top=top+1; top+1; stack_arr[top] = pushed_item; } }/*End of push()*/

27 Function to implement POP() void pop() { if(top == -1) printf("stack Underflow\n"); else { printf("popped element is : %d\n",stack_arr[top]); top=top-1; } }/*End of pop()*/

28 Function to display the Stack elements void display() { int i; if(top == -1) printf("stack is empty\n"); else { printf("stack elements :\n"); for(i = top; i >=0; i--) printf("%d\n", stack_arr[i] ); } }/*End of display()*/

29 Main Program in C #include<stdio.h> #include<conio.h> #define MAX 5 int top = -1; int stack_arr[max]; main() { int choice; while(1) { printf("1.push\n"); printf("2.pop\n"); printf("3.display\n"); printf("4.quit\n"); it\ ") printf("enter your choice : "); scanf("%d",&choice); switch(choice) { case 1 : push(); break; case 2: pop(); break; case 3: display(); break; case 4: exit(1); default: printf("wrong choice\n"); }/*End of switch*/ }/*End of while*/ }/*End of main()*/

30 EXAMPLE The following program implements stack of size 10 using array representation. The 10 elements entered by the user are pushed into the stack and then the elements are popped back and displayed.

31 #include<stdio.h> #include<stdlib.h> #define STACKSIZE 10 typedef struct{ int count; int top; int items[stacksize]; /*stack can contain up to 10 integers*/ }STACK; void push(stack *, int); int pop(stack *); int main() { int p, i; STACK s; s.top = -1; /*indicates that the stack is empty at the beginning*/ s.count = 0; /* 0 items are in the stack*/

32 /*reading and pushing items into the stack*/ printf("enter the %d stack items\n",stacksize); for(i=0;i<= i< STACKSIZE-1;i++){ scanf("%d",&p); push(&s,p); } /*popping and printing the items in the stack*/ printf("\n\nstack contains the following items\n"); for(i=0;i<= STACKSIZE-1;i++){ p=pop(&s); pop(&s); printf("%d\t",p); } return 0; }

33 void push(stack *Sptr, int ps) /*pushes ps into stack*/ { if(sptr->top == STACKSIZE-1){ printf("stack is full\n"); printf("there are %d items in the stack\n", Sptr- >count); return; /*return back to main function*/ } else { Sptr->top++; Sptr->items[Sptr->top]= ps; Sptr->count++; } }

34 int pop(stack *Sptr) { int pp; if(sptr->top == -1){ printf("stack is empty\n"); return -1 /*exit from the function*/ } else { pp = Sptr->items[Sptr->top]; Sptr->top--; } } Sptr->count--; return pp;

35 void display() { int i; if(top == -1) printf("stack is empty\n"); else { printf("stack elements :\n"); for(i = top; i >=0; i--) printf("%d\n", stack_arr[i] ); } }/*End of display()*/

36 Applications Of Stack

37 Problems That Uses Stack The runtime stack used by a process (running program) to keep track of methods in progress Search problems Undo, redo, back, forward

38 Problems That Uses Stack Contd... Recursion Reversing R i A string Calculation of a Postfix Expression.

39 Types of Notations Infix Notation eg e.g A+B Prefix Notation(Polish notation) e.g +AB Postfix P Notation ti (Reverse( Polish Notation) e.g AB+

40 Advantages of Postfix Notation No parenthesis required Simple Algo to evaluate a postfix expression No need to remember precedence and associativity rules of various operators.

41 Infix to Postfix Conversion Q 1 A+B/C-D Ans: Step 1: A+(BC/)-D Step 2: A+T-D Step 3: (A+T)-D Step 4: (AT+)- D Step 5: S D Step 6: SD- Step 7: AT+D- Step 8: ABC/+D- Q 2 (A+B) * C/D Ans: Step 1: (AB+) * C/D Step 2: T* C/D Step 3: (T*C)/D Step 4: (TC*)/D Step 5: S/D Step 6: SD/ Step 7: TC*D/ Step 8: AB+C*D/

42 Algo: Postfix(Q, P) [Suppose Q is an arithmatic expression written in infix notation. This algo finds the equivalent postfix expression P] 1. Push ( on to the stack and add ) to the end of expression. 2. Scan Q from left to right,repeat step 3 to 6 untill stack is empty 3. If operend ADD it to P 4. If (, push it onto thestack. 5. If operator then 1. Repeatedly pop from stack and add P each operator that has the same precedence or higher precedence then the operator encountered. 2. Add operator to stack 6. If ) then 7. Exit (a) Pop from stack and add to P each operator untill a left parenthesis is encountered. (b) remove left parenthesis.

43 Convert A+B/C D to postfix Sr. Symbol Stack Postfix No scanned Expr 1 ( 2 A ( A 3 + ( + A 4 B (+ AB 5 / ( + / AB 6 C ( +/ ABC 7 - ( - ABC / + 8 D ( - ABC/ + D 9 ) ABC/ + D -

44 Exercise Covert the following infix expression to Postfix. Show the Contents of the stack at various steps: P= A+(B * C-( D / E^F) * G ) * H

45 Evaluation of Postfix Expression Algo: [To evaluate a Postfix expression] 1. Add a right parenthesis ) at the end of P. 2. Scan P from left to right and repeat step 3 to 6 untill ) is encountered. 3. If an operend is encountered, put it on the stack. 4. If operator is encountered a) Pop top two elements from the stack A,B b) Evaluate A op B c) Push the result of(b) back on the stack (end of if) (end of step 2 loop) 5. Set value equal to the top element of the stack. 6. Exit.

46 Binary Search The binary search algorithm is naturally recursive. That is, the action that we perform on the original list is the very same as the action that we perform on the sublists. As a consequence, it is very easy to write a recursive binary search function.

47 Binary Search Binary search uses a recursive method to search an array to find a specified value The array must be asorted array: a[0] a[1] a[2]... a[finalindex] If the value is found, its index is returned If the value is not found, -1 is returned Note: Each execution of the recursive method reduces the search space by about a half

48 Binary Search An algorithm to solve this task looks at the middle of the array or array segment first If the value looked for is smaller than the value in the middle of the array Then the second half of the array or array segment can be ignored This strategy is then applied to the first half of the array or array segment

49 Binary Search If the value looked for is larger than the value in the middle of the array or array segment Then the first half of the arrayor array segment can be ignored This strategy is then applied to the second half of This strategy is then applied to the second half of the array or array segment If the value looked for is at the middle of the array or array segment, then it has been found If the entire array (or array segment) has been searched in this way without finding the value, then it is not in the array

50 Binary search Algo 1. [initialize i i variables] Set beg=0,end=ub, mid=(beg+end)/2 2. Repeat 3 and 4 while(beg<=end) and a[mid]!=item. 3. If(item<a[mid]) Set end=mid-1; Else Set beg-mid+1 4. Set mid=int(beg+end)/2 5. if(a[mid]==item) Set loc=mid Else loc=null 6. Exit

51 Iterative Binary Search #include<stdio.h> #include<conio.h> void main() { int a[10],item,beg,end; end; Int n,loc,mid,i,f=0; clrscr(); printf("enter the array size"); scanf("%d",&n); printf("\n enter array elements"); for(i=0;i<n;i++) scanf("%d",&a[i]); printf("\n eneter item to be searched"); scanf("%d",&item); loc=0;beg=0;end=n-1; while((beg<=end)&&(a[mid]!=item)) { mid=(beg+end)/2; if (item==a[mid]) { loc=mid; f=1; printf("\n element found at position %d",loc+1); break; } else if(item< a[mid]) end=mid-1; else beg=mid+1; } if(f==0) printf("element not found"); getch(); }

52 Execution of Binary search

53 Execution of Binary search

54 BinarySearch(int a[], int value, int left, int right) { // See if the search has failed if (left > right) return 1; // See if the value is in the first half int middle = (left + right)/2; if (value < a[middle]) return BinarySearch(a, value, left, middle 1); // See if the value is in the second half else if (value > a[middle]) return BinarySearch(a, value, middle + 1, right); // The value has been found elsemiddle; }

55 Recursive Binary Search In C int bsearch(int a[ ], Int lb, I nt ub, int item) { int mid; if(lb>ub) return(-1); mid=(int) (lb+ub)/2; if(a[mid]==item) return(mid); if(a[mid]>item) bsearch(a,lb,mid-1,item); 1item); else bsearch(a,mid+1,ub,item);,, return; }

56 Binary Search in C void main() { int n,i,item,loc; int a[10]; printf("enter the size of array"); scanf("%d",&n); printf("\n enter elements"); for(i=0;i<n;i++) scanf("%d",&a[i]); printf("\nenter item to be searched"); scanf("%d",&item); loc= bsearch(a,0,n- 1,item); if( loc<0) printf("element not found"); else printf("element found at location %d",loc); getch(); }

57 Array Representation in Memory 1. Row Major Representation 2. Column Major Representation

58 Composites: Arrays Implementation: ti 2D array (1) Storage Layout var A : array[l 1..U 1 ] of [L 2..U 2 ] of T (a) ROW-Major Layout A[L 1 ] A[L 1 +1]... A[U 1 ] [L 2 ] [L 2 +1] [L 2 +2]... [U 2-1] [U 2 ] Memory is a contiguous block Base 58

59 Composites: Arrays Implementation: 2D array (1) Storage Layout (b) COLUMN-Major Layout var A : array[l 1..U 1 ] of [L 2..U 2 ] of T A[L 1 ] A[L 1 +1]... A[U 1 ] [L 2 ] [L 2 +1] [L 2 +2]... [U 2-1] [U 2 ] Base Memory is a contiguous block 59

60 Row Major representation Address of A[i][j]= B + W(n(i-l1)+(j-l2)) Where B base address W size of each array element N number of columns(u2-l2+1)

61 Implementation: 2D array (2) Operation: Array subscripting. Eg. ROW-major j Layout var A : array[l 1..U 1 ] of [L 2..U 2 ] of T A[L 1 ] A[L 1 +1]... A[U 1 ] 1 [L 2 ] [L 2 +1] [L 2 +2]... [U 2-1] [U 2 ] Memory is a contiguous block Base address Skip # of rows row size Skip # of L-value of A[i,j] + (i-l 1 1) )*(U( 2 -L 2 +1)*cellsize ) + (j-l 2 )*cellsize Go to the correct row Go to the correct 61 cell

62 Example: Given Array Int a[4..7,-1..3]. Calculate the address of element at location a[6][[2]. Given that base address is 100. Base address B=100 Size of each array element n=2 Lower bound of row L1=4 Lower bound of column L2=-1 Upper bound of row U1=7 Upper bound of row U2=3 i=6, j=2 N=u2-l2+1 Address of A[i][j]= B + W(n(i-L1)+(j-L2)) ANS=?????

63 Queries

The program simply pushes all of the characters of the string into the stack. Then it pops and display until the stack is empty.

The program simply pushes all of the characters of the string into the stack. Then it pops and display until the stack is empty. EENG212 Algorithms & Data Structures Fall 0/07 Lecture Notes # Outline Stacks Application of Stacks Reversing a String Palindrome Example Infix, Postfix and Prefix Notations APPLICATION OF STACKS Stacks