Arrays. Elementary Data Representation Different Data Structure Operation on Data Structure Arrays

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1 Arrays Elementary Data Representation Different Data Structure Operation on Data Structure Arrays

2 Elementary Data Representation Data can be in the form of raw data, data items and data structure. Raw Data Raw data are the facts or they are simply values or set of values Data Item Data items represents single unit of values of certain types. Data Structure A Data Structure is a named group of data of different data types which can be processed as a single unit. A data structure has well-defined operation behavior and properties. Record and Array is the example of the data structure.

3 Primitive and Non Primitive Data Types Data type can be either primitive or non primitive type Primitive This data type are those which are not composed of other data types. C++ named primitive data type as standard or fundamental data types which are int, float, long int, double long double and char. Non Primitive This data type are those which are composed by the primitive data types. Usually class, structure or union are the non primitive data types. Pointer These data types enables the storage the pointers (address) of some data location.

4 Different Data Structures Data Structure is very important in Computer System. As these not only allow the user to combine various data types in a group but also allow processing of the group as a single unit. Data structure be in two types. Simple Data Structure These data structure are normally built from primitive data types like integer, real char, Boolean etc These are ARRAY STRUCUTRE

5 Compound Data Structure Simple data structure can be combined in various ways to form more complex structure called Compound Data Structure. These are also classified into two ways. Linear Data Structure These data structure are single level data structure. A Data Structure is said to be linear if its elements form a sequence. Following are the linear data structures. Stack Queue Linked List Non Linear Data Structure These are the multilevel data structure. TREE is the example of Non linear data structure.

6 Kapil Sehgal PGT Computer. Science Ankleshwar Gujarat Ch 9 Data Structure - Architecture

Each of the data elements can be referenced respectively by a set of

7 Array Array refers to a named list of a finite number n of similar data elements. Each of the data elements can be referenced respectively by a set of consecutive numbers usually 0,1,2,3,.n-1 if total element is n. Array can be one dimensional, two dimensional and multidimensional. One Dimensional Array of 10 elements

8 Structure Structure refers to named collection of variables of different data types. A structure stores the different types of data items. The elements of the structure is refers with (.) dot operators. If the structure name is STRUC contains (Roll no, name, Marks, Grade) then individual field will be referenced as STRUC filed name.

Stack Stack is a Linear Data Structure, in which elements are

In other words Stack is a linear list in which elements are add

The process of add element into stack is called PUSH.

9 Stack Stack is a Linear Data Structure, in which elements are processed in the LAST IN FIRST OUT (LIFO) fashion. In other words Stack is a linear list in which elements are add and removed at single end. The process of add element into stack is called PUSH. The process of delete element from stack is called POP The Recently added and that element which will remove first is called TOP element

10 Queue Queue is a Linear Data Structure, in which elements are processed in the FIRST IN FIRST OUT (FIFO) fashion. In other words Queue is a linear list in which elements are add and removed at different end. The recent added element is called REAR and that next removing element are called FRONT.

11 Linked List Linked lists are special lists of some data elements linked to one another. The logical ordering is represented by having each element pointing to the next element. Each element is called NODE, which have two or more parts, in which one of them is pointer part for next node and rest of all parts are data parts. Linked list may be single linked list or doubly linked list.

Top most node is called the root of the tree and bottom most

12 Tree Tree is a multilevel non linear data structure having a hierarchical relationship among its elements called nodes. Top most node is called the root of the tree and bottom most nodes are called the leaves of the tree. Root Node Leaf node

13 Operation on Data Structure The basic operation that has performed on the data structure are as follows Insertion It means addition of a new data element in a data structure. Deletion It means removal of a data element from the data structure. This element is searched for before its removal. Searching It involves searching for the specified data element in a data structure. Traversal Traversal of the data structure means processing all the data elements of it. Sorting Arranging data elements of a data structure in a specified order is called sorting. Merging Combining elements of two similar data structure to form a new data structure of same type

14 What is Array A Linear data structure is that whose elements form a sequence. When elements of linear data structure are represent in memory by means of sequential memory location, these linear data structure is called Array. Array is a simplest data structure and are easy to traverse, sort and search etc. Array is a list of finite number of homogeneous data elements. The number n is called length or size or range of an array. In C++ the lower bound is always 0 and the upper bound is size-1.

15 Need For Array Let us consider a situation, where we have to write a program which can accept marks of 50 students of a class, calculate average marks and difference of each student s marks with their average marks. If solve this problems by marking the variables, we need 50 variables for marks of students and 1 variable for average and 50 variables for their marks difference, overall 101 variables is needed for handles this situation, which is not an easy tasks. Due to this situation, we need a Data Structure which can hold multiple elements within a single variable called array.

16 Types of Array Array are of different types. First is One-Dimensional array in which finite homogeneous elements. Second is - Multi-Dimensional array in which each element are itself is an array. Here in multidimensional array means only two dimensional array.

17 Implementation of One Dim Array In memory, one dimensional arrays are implemented by allocating a sequence of addressed location. The address of starting element or address of 0 th element of array is called base address, which can never changes during program execution. Address of the i th element of array = base address + ES(i L), where ES is the size of element and l is the lower bound of array. int a[25], what is the address of a[17] where the base address is 1000? Ans = (17-0) = = 1034

18 Searching The are two searching algorithm Linear Search or sequential search In this type of search, We read a number for search and Compare with each element of array, wherever is found, then search successful and return otherwise search unsuccessful. Complexity of Linear search is O(n). Binary Search It is Mid point subdivision method, entire array is divided into two parts and compare the searching element with mid element, if found search successful otherwise the check the mid number is less or greater than searching. In that base the half list is ignored and again repeat this process until solution is found or search become unsuccessful. It complexity is O(log n)

19 Algorithm of Linear Search of n elements Let the Array of n elements is ARR and n is the size of array. 1. Set i = 0 2. Read the size of Array 3 Read (n); 4 Set i = 0 ; 5. Repeat step 6 & 7 n times 6 Set i = i Read (ARR(i)); 8 Print "Read a number of search "; 9 Read(s); 10 Set i = 0 10.Repeat step 12 & 13 n times 11 Set i = i If (ARR(i) = s) Print "Search Successful " & Return 13 Print "Search Unsuffessful "; 14 End.

20 Program for Linear Search for n elements #include<iostream.h> #include<conio.h> int main() { int n,arr[100],i,s; clrscr(); cout<<endl<<"enter the size of Array "; cin>>n; cout<<endl<<"enter"<<n<<" Elements of Array\n"; for(i-0;i<n;i++) cin>>arr[i]; cout<<endl<<"enter no to search "; cin>>s; for(i=0;i<n;i++) if (ARR[i]==s) {c cout<<endl<<"search is successful at "<<i+1<<"position"; return 0; } cout<<endl<<"search is unsuccessful "; return 0; }

21 Algorithm of Binary Search of n elements Let the Array of n elements is ARR and n is the size of array. 1. Set i = 0 2. Read the size of Array 3 Read (n); 4. Repeat step 5 & 6 n times 5 Set i = i Read (ARR(i)); 7. Sort the ARR elements by any sorting algorithm 8 Set beg = 1 and end = n; 9 Repeat the steps 10 to 13 until (beg >end) 10. Set mid = (beg + end) / 2 11 if (ARR[mid] = S) then Print "Search Successful at " Mid "Position" & return 12 if (ARR[mid] > s) then end = mid if (Arr[mid] < s) then beg = mid Print "Search Unsuffessful "; 15 End.

22 Program for Binary Search for n elements #include<iostream.h> #include<conio.h> int main() { int arr[100],n,beg,end,mid,s,i,j,t; clrscr(); //Read the size of Array cout<<endl<<"enter Size of Array "; cin>>n; //Read An Array cout<<endl<<"enter Array elements \n"; for(i=0;i<n;i++) cin>>arr[i]; //Sorting of Array for(i=0;i<n-1;i++) for(j=i+1;j<n;j++) if(arr[i]>arr[j]) { t=arr[i]; arr[i]=arr[j]; arr[j]=t; } cout<<endl<<"enter No. to search "; cin>>s; // Searching beg=1; end=n; while(beg<=end) { mid=(beg+end)/2; if (arr[mid]==s) { cout<<"\n Search successful at postion "; getch(); return 0; } if (arr[mid]>s) end = mid - 1; if (arr[mid]<s) beg = mid + 1; } cout<<"\n Search Unsuccessful "; getch(); return 0; }

Insertion into Array Insertion of new element in

If the array is sorted then new element is added at

24 Insertion into Array Insertion of new element in array can be done in two ways. If the array is unordered, the new element is inserted at the end of the array. If the array is sorted then new element is added at their appropriate position without altering the order and to achieve this. Rest of elements are shifted.

25 Write a program to insert an element at end of array for unsorted array #include<iostream.h> #include<conio.h> void main() { int ar[100],i,n,s; clrscr(); // Read the size of array cout<<endl<<"enter Size of Array "; cin>>n; //Read the Array cout<<endl<<"enter Array Elements \n"; for(i=0;i<n;i++) cin>>ar[i]; //Read Inserting Elements cout<<endl<<"enter Element to Insert "; cin>>s; //Insert Element at end of array " ar[n]=s; //Print New Array cout<<endl<<"array Elements are \n"; for(i=0;i<=n;i++) cout<<ar[i]<<" "; getch(); }

26 Write an algo. to insert an element at end of array for unsorted array Let AR is the array of unsorted n elements and s is the element to insert at end of array AR. 1. Read(n) // size of array 2. Set I = 0 3. Repeat Step 4 and 5 n times 4. Set I = I Read AR[i]; 6. Read (s) // New Element 7. Set AR[n+1] = s; 8. Set I = 0 9. Repeat Step 10 and 11 n+1 times 10.Set I = I Print AR[i]; 12.Exit

27 Write a program to insert an element into sorted array. #include<iostream.h> #include<conio.h> void main() { int ar[100],i,n,s; clrscr(); // Read the size of array cout<<endl<<"enter Size of Array "; cin>>n; //Read the Sorted Array cout<<endl<<"enter Array Elements \n"; for(i=0;i<n;i++) cin>>ar[i]; //Read Inserting Elements cout<<endl<<"enter Element to Insert "; cin>>s; //Insert Element at their position " for(i=n-1;i>=0;i--) if (ar[i]>s) ar[i+1]=ar[i]; else break; ar[i+1]=s; //Print New Array cout<<endl<<"array Elements are \n"; for(i=0;i<=n;i++) cout<<ar[i]<<" "; getch(); }

28 Write an algorithm to insert an element into sorted array. Let AR is the array of unsorted n elements and s is the element to insert at end of array AR. 1. Read(n) // size of array 2. Set I = 0 3. Repeat Step 4 and 5 n times 4. Set I = I Read AR[i]; 6. Read (s) // New Element 7. Set I = n 8. Repeat Step 9 and 10 n times 9. If (AR[i]>s) then Set AR[i+1]=AR[i] else break; 10.Set I = I 1 11.Set I = 0 12.Repeat step 13 & 14 n+1 times 13.Set I = I Print AR[i] 15.Exit

Deleting an Element from Array The element

29 Deleting an Element from Array The element to be deleted is first searched for in the array using one the search techniques either linear or binary. If search successful then delete the element rest are shifted. If the elements are shifted downward or towards right side, then unused elements are available in the beginning of the array.

30 Write a program to delete an element from unsorted array. #include<iostream.h> #include<conio.h> int main() { int arr[100],n,i,d,p=0; clrscr(); // Clear Screen //Read the size of array cout<<endl<<"enter size of Array "; cin>>n; //Read Array elements cout<<endl<<"enter Array elements "; for(i=0;i<n;i++) cin>>arr[i]; //Read the element to delete cout<<endl<<"enter element to delete "; cin>>d; //Find the element Location for(i=0;i<n;i++) if(arr[i]==d) p=i+1; // Check Un availability if (p==0) { cout<<endl<<"element Not Found Deletion Not Possible"; getch(); return 0; } else { // Deleting for(i=p-1;i<n;i++) arr[i]=arr[i+1]; // Print New Array cout<<endl<<"new Array "; for(i=0;i<n-1;i++) cout<<arr[i]<<" "; } getch(); return 0; }

31 Write a program to delete an element from sorted array. #include<iostream.h> #include<conio.h> int main() { int arr[100],n,i,d,j; clrscr(); // Clear Screen //Read the size of array cout<<endl<<"enter size of Array "; cin>>n; //Read Array elements cout<<endl<<"enter Array elements "; for(i=0;i<n;i++) cin>>arr[i]; i=0; //Read the element to delete cout<<endl<<"enter element to delete "; cin>>d; //Find the element Location while(arr[i]<d) i++; // Check Un availability if (arr[i]!=d) { cout<<endl<<"element Not Found Deletion Not Possible"; getch(); return 0; } Else { // Deleting for(j=i;j<n;j++) arr[j]=arr[j+1]; // Print New Array cout<<endl<<"new Array "; for(i=0;i<n-1;i++) cout<<arr[i]<<" "; } getch(); return 0; }

32 Sorting Array Elements Sorting of an array means arranging the array elements in a specified order. Either Ascending or Descending There are various SORTING mechanisms are available like Selection, Bubble, Insertion, Quick, Merge, Heap. The target of all mechanisms are same to sort the array, but the complexity and techniques are different. Ex - Oiginal Array Sorted Array

33 Kapil Sehgal PGT Computer. Science Ankleshwar Gujarat Ch 9 Selection Sort - Methodology

34 Write a program to sort array element by Selection Sort #include<iostream.h> #include<conio.h> void main() { int oa[100],sa[100],i,j,n,se,p; clrscr(); //Read the size of Array cout<<"\n Enter Array Size "; cin>>n; //Read Array Elements for(i=0;i<n;i++) cin>>oa[i]; //Print Original Array cout<<endl<<"original Array \n "; for(i=0;i<n;i++) cout<<oa[i]<<" "; //Sorting Logic j=-1; while(j<n) { se=32000; for(i=0;i<n;i++) if (oa[i]<se) {se=oa[i];p=i;} oa[p]=32500; j++; sa[j]=se; } //Print Sorted Array cout<<endl<<"sorted Array \n "; for(i=0;i<n;i++) cout<<sa[i]<<" "; getch(); }

35 Write a algo. to sort array element by Selection Sort 1 small = AR[L]; 2 For I = l to u { 3 small = AR[I]; 4 For J = I to U { 5 If AR[J] < small then { 6 small = AR[J] ; 7 pos = J; } // swap the smallest elements 8 temp = AR[I]; 9 AR[i] = small 10 AR[pos] = temp; 11 Exit

36 Bubble Sort It is to Compare two adjoining values and exchange them

37 Algorithm for Bubble Sort LOGIC 1. For I = L to U 2. { For J = L to (U - I - 1) 3. { if AR[J] > AR[J+1] then { 4. temp = AR[J]; 5. AR[J] = AR[J+1]; 6. AR[J+1] = temp; } } } 7. Exit

38 Write a program to sort array element by Bubble Sort #include<iostream.h> #include<conio.h> void main() { int ar[100],n,i,j,t; clrscr(); //Read the size of Array cout<<endl<<"emter Size of Array "; cin>>n; //Read the Array Elements; cout<<endl<<"enter Array elements :"; for(i=0;i<n;i++) cin>>ar[i]; //Print Original Array clrscr(); cout<<endl<<"original Array \n "; for(i=0;i<n;i++) cout<<ar[i]<<" "; //Bubble Sort Logic for(i=0;i< n-1;i++) for(j=0;j<(n-i-1);j++) if(ar[j]>ar[j+1]) { t=ar[j]; ar[j]=ar[j+1]; ar[j+1]=t; } //Print Original Array cout<<endl<<"sorted Array \n "; for(i=0;i<n;i++) cout<<ar[i]<<" "; getch(); }

39 Insertion Sort In this we insert a minimum possible value at 0 th Index This sorting algorithm is frequently used when n is small. For Ex This algorithm is very popular with bridge players when they are first sorting their cards.

40 Algorithm for Insertion Sort LOGIC 1. A[0]= Minimum Integer Value 2. Repeat step 3 through 8 for k = 1,2,.. N-1 { 3. temp = A 4. ptr = k-1 5. Repeat steps 6 to 7 while temp < A[ptr] { 6. A[ptr+1] = a[ptr]; 7. ptr = ptr 1; } 8. A[ptr + 1 ] = temp; 9. Exit

41 Write a program to sort array element by Insertion Sort #include<iostream.h> #include<conio.h> void main() { int ar[100],n,i,j,t; clrscr(); //Read the size of Array cout<<endl<<"emter Size of Array "; cin>>n; //Insert Minimum possible value at index 0 ar[0]=-32000; //Read the Array Elements; cout<<endl<<"enter Array elements :"; for(i=1;i<=n;i++) cin>>ar[i]; //Print Original Array clrscr(); cout<<endl<<"original Array \n "; for(i=0;i<n;i++) cout<<ar[i]<<" "; //Insertion Sort Logic for(i=1;i<=n;i++) for(j=0;j<=i;j++) if (ar[j]>ar[i]) { t=ar[j]; ar[j]=ar[i]; ar[i]=t; } //Print Sorted Array cout<<endl<<"sorted Array \n "; for(i=1;i<=n;i++) cout<<ar[i]<<" "; getch(); }

42 Merging of Arrays Merging means combining elements of two arrays to form a new array. Simplest way of merging two arrays is that first copy all the elements of first array into new arrays and then copy all the elements of second arrays at the end of new array and then sort new arrays if needed. Another popular technique to have a sorted resultant array is sorting while merging using merge sort. This technique of merge sort is more efficient than others techniques needs sorting of resultant array.

43 Algorithm - Merging of two unsorted arrays Resultant array is also unsorted Let A is the first array of size n and B is the second array of size m and C is the new array of size n+m elements. 1. Read(n) // size of array A 2. Read(m) // size of Array B 3. Set I = 0; 4. Repeat Step 5 and 6 n times 5. Set I = I Read A[i]; 7. Set I = 0; 8. Repeat Step 9 and 10 n times 9. Set I = I Read B[i]; 11.Set I = 0; 12.Repeat Step 13 and 14 n times 13.Set I = I C[I]=A[I]; 15.Set I = 0 16.Repeat step 17 & 18 m times 17.Set I = I C[n+I]= B[I]; 19.Set I = 0; 20.Repeat Step 21 and 22 n+m times 21.Set I = I Print C[I]; 23.Exit

44 Program - Merging of two unsorted arrays Resultant array is also unsorted #include<iostream.h> #include<conio.h> void main() { int a[100],b[100],c[100],m,n,i; clrscr(); //Read the size of Array A cout<<endl<<"emter Size of Array "; cin>>n; //Read the Array A Elements ; cout<<endl<<"enter Array elements :"; for(i=0;i<n;i++) cin>>a[i]; //Read the size of Array B cout<<endl<<"emter Size of Array "; cin>>m; //Read the Array B Elements ; cout<<endl<<"enter Array elements :"; for(i=0;i<m;i++) cin>>b[i]; //Merge Logic for(i=0;i<n;i++) c[i]=a[i]; for(i=0;i<m;i++) c[i+n]=b[i]; //Print Original Array A cout<<endl<<"array A elements \n "; for(i=0;i<n;i++) cout<<a[i]<<" "; //Print Original Array B cout<<endl<<"array B elements \n "; for(i=0;i<m;i++) cout<<b[i]<<" "; //Print Merged Array C cout<<endl<<"array Merged elements \n "; for(i=0;i<n+m;i++) cout<<c[i]<<" "; getch(); }

45 Algorithm - Merging of two Sorted arrays Resultant array is also Sorted 1.ctrA = L1 ; ctrb = l2 ; ctrc = L3 2.While ctra <= U1 and ctrb <= U2 perform steps 3 through 10 3.{ if A[ctrA] <= B[ctrB] then 4. { C[ctrC] = A[ctrA] 5. ctrc = ctrc ctra = ctra + 1 } 7. else 8. { C[ctrC] = B[ctrB] 9. ctrc = ctrc ctrb =ctrb + 1 } }/* End of while loop */ 11. if ctra > U1 then 12.{ while ctrb <= U2 perform steps 13 through { C[ctrC] = B[ctrB] 14. ctrc = ctrc ctrb = ctrb + 1 } } 16.if ctrb >U2 then 17.{ while ctra <= U1 perform steps 18 trough { C[ctrC] = A[ctrA] 19. ctrc = ctrc ctra = ctra + 1 } }

46 Program - Merging of two Sorted arrays Resultant array is also Sorted #include<iostream.h> #include<conio.h> void main() { int a[100],b[100],c[100],m,n,i,j,k; clrscr(); //Read the size of Array A cout<<endl<<"emter Size of Array "; cin>>n; //Read the Array A Elements ; cout<<endl<<"enter Array elements :"; for(i=0;i<n;i++) cin>>a[i]; //Read the size of Array B cout<<endl<<"emter Size of Array "; cin>>m; //Read the Array B Elements ; cout<<endl<<"enter Array elements :"; for(i=0;i<m;i++) cin>>b[i]; //Merge Logic i=j=k=0; while(i<=n && j<=m) if (a[i]<=b[j]) c[k++]=a[i++]; else c[k++]=b[j++]; if (i>n) while(j<=m) c[k++]=b[j++]; else while(i<=n) c[k++]=a[i++]; //Print Original Array A cout<<endl<<"array A elements \n "; for(i=0;i<n;i++) cout<<a[i]<<" "; //Print Original Array B cout<<endl<<"array B elements \n "; for(i=0;i<m;i++) cout<<b[i]<<" "; //Print Merged Array C cout<<endl<<"array Merged elements \n "; for(i=0;i<n+m;i++) cout<<c[i]<<" "; getch(); }

Implementation of 2D array in Memory While

determined by Calculating number of rows

47 Implementation of 2D array in Memory While storing the element of 2D array in memory, these are needed contiguous locations. They are two alternative to achieve either Row Wise or Column Wise. The number of elements in 2-D array can be determined by Calculating number of rows and number of columns and then multiplying number of rows and columns.

48 Row Major Implementation In this technique stores firstly the first row of the array, then the second row of the array and so on. The computer does not keep trace of the address of all the elements of the array, but keep track the Base Address (Address of first element) and calculate the addresses of the element when required. The formula as follows. Address of in Row Major Order [I,J] = B + W [n(i-1) + (J - 1)] Where m are the number of rows and n are number of columns and W is the size of data type. And I & J are the Element subscript number

49 1. A 2-D array defined as A[4.. 7, ] requires 2 words of storage space for each element if the array is stored in row-major form calculate the address of A[6,2] given the base address as 100(one hundred). sol: Base address B = 100 Element size W = 2 bytes Lr = 4; Lc=-1; i =6; j=2 n,number of rows =Ur - Lr + 1 = = 4 c,number of columns =Uc - Lc + 1 = 3-(-1)+1 = 5 Address[I,J] =B+w(c(i-Lr) + (j -Lc)) =100+2(5(6-4)+2-(-1)) =100+2(5*2+3)=100+2*(13) =100+26= X is 2-D array [10*5].Each element of the array is stored in 2 memory location.if X[1,1] begins at address 150,find the location of X[3,4].use the formula for calculation (The arrangement is in row major). sol: Base address B=150 Element size W=2 number of columns n=5 Lr = Lc =1 Location X[3,4] = (5(3-1)+(4-1)) = (10+3) = = 176.

50 Column Major Implementation In this technique stores firstly the first column of the array, then the second column of the array and so on. The computer does not keep trace of the address of all the elements of the array, but keep track the Base Address (Address of first element) and calculate the addresses of the element when required. The formula as follows. Address of in Column Major Order [I,J] = B + W [m(j-1) + (I - 1)] Where m are the number of rows and n are number of columns and W is the size of data type. And I & J are the Element subscript number

51 1. Each element of an array A[ , ] requires one byte of storage. if the array is stored in column major order beginning location 500,determine the location of A[0,30]. sol: Base address B =500 Element size W = 1byte r (number of rows) are 20 - (-20)+1=41 1r = -20; 1c =10 Address of A[0,30] = [(0-(-20) +41(30-10))] = (20+41*20) = [20+820]= = Each element of an array A[ , ] requires one byte of storage.if the array is stored in column major order begining location 1000, determine the location of A[0,40]. sol: Base address B = 1000 Element size W = 1 byte Number of rows = 20-(-15)+1=36 1r=-15, 1c=20 Address of A[0,40] = [0-(-15))+36(40-20)] = [ * 20] = [15+720] = = 1735.

Chapter4: Data Structures. Data: It is a collection of raw facts that has implicit meaning.

Chapter4: Data Structures. Data: It is a collection of raw facts that has implicit meaning. Chapter4: s Data: It is a collection of raw facts that has implicit meaning. Data may be single valued like ID, or multi valued like address. Information: It is the processed data having explicit meaning.