Array

DATA STRUCTURE

D S

GraphTree

Non - Linearchar

Non - PrimitivePrimitive \ Fundamental

Linearint float double

Array Stack Queue Link List

Linear Array• An array is a list of a finite number n of

homogenous data elements.– Elements of array are referenced by an index set

that consist of n consecutive numbers.

– Elements of array are stored in successive memory locations.

• N is the length or size of the array.

Length = UB – LB + 1

where UB – Upper Bound

LB – Lower Bound

Linear ArrayRepresentation of elements of an array A can be

as follows:

1.Subscript Notations: A1 , A2 , A3 , ……… , An

2.Parenthesis Notation: A(1), A(2), A(3), …… , A(n) [ Used in FORTAN, PL\1 & BASIC)

3.Bracket Notation: A[1], A[2], A[3],….. , A[n]Generally subscript & bracket notations are

used. k in A[k] is subscript or an index & A[k] is

subscripted variable.

• The elements of array are stored in successive memory cells.

• The computer need not to keep track of the address of every element but keeps the track of the address of the first element of the array. Denoted : Base (A)

• By using this address Base(A), computer calculates the address of any element of A .

Linear Array…

Linear Array….

• Formula for calculating address is : Loc (A[k]) = Base (A) + w (k – lower bound)

or

= Base (A) + w * k

where: w – no. of words per memory cell for the array.

Loc(A[k]) – Address of the element A[k] of the array.

Base(A) - Base address of A.

Declaration of Linear Array• Syntax : <Data type> <Array name> [size];

• Where: Array name – denotes the name of the array & can be any valid C identifier.

Data type – data type of the element of array.

size - specifies the no. of elements that can be stored in the array. It can be a positive integer constant or constant integer expression.

• Examples : int age[10];

float sale[10];

Char grade[10];

When the array is declared, the compiler allocates space in memory sufficient to hold all the elements of the array, so the size of the array should be known at the compile time.

Declaration of Linear Array…

• The symbolic constant can be used to specify the size of the array.

• Example: # define SIZE 10

main()

int size 15;

float sale[SIZE];

int marks[size];

The use of symbolic constant to specify the size of array makes it convenient to modify the prig. If the size has to be changed only at one place, i.e. #define directive

Declaration of Linear Array…

Accessing Linear Array Element• The elements of an array can be accessed by

specifying the array name followed by subscript in brackets.

• In C the array subscript starts from 0.

• The last valid subscript is one less than the size of the array i.e. n-1

• The subscript can be any expression that yields an integer value like integer constant, integer variable, integer expression or return value (int) from a function call.

Accessing Linear Array Element…

Example : If a & b are two arrays of size 5, 10int a[5];

float b[10];

int I;

scanf (“%d”, & a[1]);

printf (“%f”, b[3]);

a[4] = 20;

a[4]++;

b[5]+ = 400;

sum = a[0] + a[1] + a[2] + a[3] + a[4];

i = 2;

scanf (“%f”, & a[i]);

printf (“%f”, b[i]);

printf (“%f”, b[i++]);

Processing Linear Array• For processing arrays generally we use a for loop 7

the loop variable is used at the place of subscript.

• The initial value of loop variable is taken as ‘0’ as array

• The loop variable is increased by 1 each time so that we can access & process the next element in the array.

• The total no. of passes in the loop will be equal to the no. of elements in the array & in each pass we will process 1 element.

• Reading values in a: for (i = 0; i<10; i++)

scanf(“%d”, &a[i]);

• Displaying values of a: for (i=0; i<10; i++)

printf(“%d”, a[i]);

• Adding all the elements of a: sum = 0;

for (i=0; i<10; i++)

sum + a[i];

Processing Linear Array…

WAP to input values into an array & display them.main( )

{

int a[5], i;

for (i=0; i<5; i++)

{

printf (“Enter the value for a[%d] : “, i);

scanf (“%d”, & a[i]);

}

printf (“The array elements are : \n”);

for (i=0; i<5; i++)

printf (“%d\t”, a[i]);

printf(“\n”) ;

}

Processing Linear Array…

Initialization of Linear Array• After declaration, the elements of a local array have

garbage value while the elements of global & static array are automatically initially to 0.

• Syntax : <data_type> <array_name> [size] = { value1, value2,….., valueN);

• Example: int marks [5] = { 50,85,70,65,95};

• We can’t copy all the elements of an array to another array by simply assigning it to the other array.

• Example: int a[5] = {1,2,3,4,};

int b[5];

b = a; /*not valid/

• But we have to copy all the elements of array 1 by 1 by using for loop : for(i=0; i<5; i++)

b[i] = a[i];

Program to find the largest & smallest no. in an array.main()

{

int i , a[10] = {2,5,4,1,8,9,11,6,3,7};

int small, int large;

for (i=1; i<10; i++)

{

if(a[i] < small)

small = a[i];

if(a[i] > large)

large = a[i];

}

printf(“Smallest = %d, Largest = %d\n”, small,large);

}

Initialization of Linear Array…

Initialization of Linear Array…Program to reverse the elements of an array.main()

{

int i, j, temp,a[10] = {1,2,3,4,5,6,7,8,9,10};

for (i =0 ; j = 9; i< j ; i++, j-- )

{

temp = a[i];

a[i] = a[j];

a[j] = temp;

}

printf(“After reversing, the array is : “);

for ( i = 0; i< 10; i++)

printf(“%d “, a[i]);

printf(“\n”);

}

Operations on Linear Arrays• Traversal Operation

• Search Operation

• Insert Operation

• Delete Operation

• Sort Operation

• Merge Operation

• Traversing is the process of visiting each element of the array exactly once.

• Elements of Linear Array can be accessed directly but we have to vary index from lower bound to upper bound in steps of one to access individual elements in order.

• For an array of size n, for loop executes n times & thus traversal operation on array will be O(n) operation

Operations on Linear Arrays…

Operations on Linear Arrays…Program for traversing the elements of given array. main( )

{

int i , int a[10], ;

printf (“ Enter the elements of the array which you want to traverse”);

for ( i = 0; i < 10 ; i++)

{

scanf (“%d”, & a[i]);

}

printf (“the traverse array is”);

for (i = 0; i<10; i++)

{

printf (“%d\n”, a[i]);

}

}

}

Operations on Linear Arrays…• Searching is the process of finding the location of

given element in the array.

• There are 2 approaches to search operation:

– Linear search: If the elements are in random order, then Linear search technique is used. In worst case O(n) operations

– Binary search : If the array elements are in sorted order, then it’s preferable to use Binary search. The complexity of binary search in O(log2n).

Program to find the element in the given array.#include<stdio.h> main(){ int array[100], search, c, number; printf("Enter the number of elements in array\n"); scanf("%d",&number); printf("Enter %d numbers\n", number); for ( c = 0 ; c < number ; c++ ) scanf("%d",&array[c]); printf("Enter the number to search\n"); scanf("%d",&search);

for ( c = 0 ; c < number ; c++ )

{

if ( array[c] == search ) /* if required element found */

{

printf("%d is present at location %d.\n", search, c+1);

break;

}

}

if ( c == number )

printf("%d is not present in array.\n", search);

 

return 0;

}

}

Program to find the element in the sorted array.#include<stdio.h>

 

main()

{

int c, first, last, middle, n, search, array[100];

 

printf("Enter number of elements\n");

scanf("%d",&n);

 

printf("Enter %d integers\n", n);

 

for ( c = 0 ; c < n ; c++ )

scanf("%d",&array[c]);

 

printf("Enter value to find\n");

scanf("%d",&search);

 

first = 0;

last = n - 1;

middle = (first+last)/2;

 

while( first <= last )

{

if ( array[middle] < search )

first = middle + 1;

else if ( array[middle] == search )

{

printf("%d found at location %d.\n", search, middle+1);

break;

}

else

last = middle - 1;

 

middle = (first + last)/2;

}

if ( first > last )

printf("%d is not present in the list.\n", search);

 

return 0;

}

Operations on Linear Arrays…• Insertion refers to the operation of adding an element

to existing list of elements.

• After insertion the size of the array is increased by factor of 1.

• Insertion is possible only if the memory space allocated is large enough to accommodate the additional element.

• The complexity of insertion operation is O(n).

Operations on Linear Arrays…Program to insert an element at position k.main( )

{

int *n , int item, int k, int k, int a[10] , int j;

j = *n – 1;

while ( j >= k){

a[j+1] = a[j];

j--;

}

a[k] = item;

(*n)++;

}

Program to insert an element at position k in the sorted array.

#include<stdio.h>

main()

{

int array[100], position, c, n, value;

 

printf("Enter number of elements in array\n");

scanf("%d", &n);

 

printf("Enter %d elements\n", n);

 

for ( c = 0 ; c < n ; c++ )

scanf("%d", &array[c]);

 

printf("Enter the location where you wish to insert an element\n");

scanf("%d", &position);

 

printf("Enter the value to insert\n");

scanf("%d", &value);

 

for ( c = n - 1 ; c >= position - 1 ; c-- )

array[c+1] = array[c];

 

array[position-1] = value;

 

printf("Resultant array is\n");

 

for( c = 0 ; c <= n ; c++ )

printf("%d\n", array[c]);

 

return 0;

}

Operations on Linear Arrays…• Sorting is the process of arranging the elements of

the array in some logical order.

• This logical order can be ascending or descending incase of numeric values or dictionary order in case of alphanumeric value.

• One of the popular technique is Bubble sort

• The complexity of bubble sort is O(n2).

Program to sort the array ascending order.#include<stdio.h> main(){ int array[100], n, c, d, swap; printf("Enter number of elements\n"); scanf("%d", &n); printf("Enter %d integers\n", n); for ( c = 0 ; c < n ; c++ ) scanf("%d", &array[c]); for ( c = 0 ; c < ( n - 1 ) ; c++ ) { for ( d = 0 ; d < n - c - 1 ; d++ ) {

if ( array[d] > array[d+1] )

{

swap = array[d];

array[d] = array[d+1];

array[d+1] = swap;

}

}

}

 

printf("Sorted list in ascending order:\n");

 

for ( c = 0 ; c < n ; c++ )

printf("%d\n", array[c]);

 

return 0;

}

Operations on Linear Arrays…• Merging is the process of combining the elements of

2 similar arrays into a single array.

• One approach is to join them end to end & then sort the combined array, but this approach is not efficient & economical.

• Best approach is to compare the elements of the given array & based on this comparison , decide which element should come first in the third array.

• The merging algorithm runs in linear time.

Program two merge 2 arrays into one array.void merging ( int*a, int m, int *b, int n, int *c)

{

int na, nb, nc;

na = nb = nc = 0;

while ( ( na < m) && (nb < n))

{

if (a[na] < b[nb])

c[nc] = a[na++];

else

c[nc] = a[nb++];

nc++;

}

if (na = = m) /* list ‘a’ is exhausted*/

{

while (nb < n)

c[nc++] = b[nb++];

}

else if ( nb == n) /* list b is exhausted*/

{

while (na < m )

c[nc++] = a[na++];

}

}

Array Applications• Given a list of test scores, determine the maximum

and minimum scores.

• Read in a list of student names and rearrange them in alphabetical order (sorting).

• Given the height measurements of students in a class, output the names of those students who are taller than average.

Limitations of Linear Arrays• The prior knowledge of number of elements in the

linear array is necessary.

• These are static structures. Static in the sense that whether memory is allocated at compilation time or run time, the memory used by by them can’t be reduced or extended.

• As the elements are stored in consecutive locations, the insertions & deletion in the arrays are time consuming.

Multidimensional Arrays

• Most programming languages allow two-dimensional and three-dimensional arrays, i.e., arrays where elements are referenced, respectively, by two and three subscripts. In fact, some programming languages allow the number of dimensions for an array to be as high as 7.

• A two dimensional m×n array A is a collection of m.n data elements such that each element is specified by a pair of integers (such as J, K), called subscripts, with the property that

• 1 ≤ J ≤ m and 1 ≤ K ≤ n

Multidimensional Arrays (Contd.)

• The element of A with first subscript j and second subscript k will be denoted by

AJ,K or A[J,K]

• Two dimensional arrays are called matrices in mathematics and tables in business applications; hence two dimensional arrays are sometimes called matrix arrays.

Multidimensional Arrays (Contd.)

• There is a standard way of drawing a two-dimensional m×n array A where the elements of A form a rectangular array with m rows and n columns and where the element A[J, K] appears in row J and column K.

1 2 3 4

1 A[1, 1] A[1, 2] A[1, 3] A[1, 4]

2 A[2, 1] A[2, 2] A[2, 3] A[2, 4]

3 A[3, 1] A[3, 2] A[3, 3] A[3, 4]

Columns

Rows

Two-dimensional 3×4 array A

Multidimensional Arrays (Specification)

• Suppose A is a two-dimensional m×n array.

• The first dimension of A contains the index set 1, …….., m, with lower bound 1 and upper bound m; and the second dimension of A contains the index set 1, 2, …….., n, with lower bound 1 and upper bound n. the length of a dimension is the number of integers in its index set.

• The pair of lengths m×n (read “m by n”) is called the size of the array.

Multidimensional Arrays (Length)

• The length of a given dimension (i.e., the number of integers in its index set) can be obtained from the formula

• Length = upper bound – lower bound + 1

41

Representation of Two-Dimensional Arrays in memory

• Let A be a two-dimensional m×n array.

• Although A is pictured as a rectangular array of elements with m rows and n columns, the array will be represented in memory by a block of m.n sequential memory locations. Specifically, the programming language will store the array A either

• Column by column, what is called column major order,

or

• Row by row, in row major order.

• The figure shows these two ways when A is a two-dimensional 3×4 array. We emphasize that the particular representation used depends upon the programming language, not the user. 42

Representation of Two-Dimensional Arrays in memory

Subscript

(1, 1)

(2, 1) Col 1

(3, 1)

(1, 2)

(2, 2) Col 2

(3, 2)

(1, 3)

(2, 3) Col 3

(3, 3)

(1, 4)

(2, 4) Col 4

(3, 4)

A Subscript

(1, 1)

(1, 2) Row 1

(1, 3)

(1, 4)

(2, 2)

(2, 2) Row 2

(2, 3)

(2, 4)

(3, 1)

(3, 2) Row 3

(3, 2)

(3, 4)

A

a) Column major order b) Row major order

A A

43

2 Dimensional Arrays• Like linear array, 2-D array also keeps track of the address of

first element only i.e. the base address of the array.

• Using this base address, the computer computes the address of the element in the ith row & jth col. , i.e. loc(a[i][j]), using the following formula:

Column Major Order:

loc(a[i][j]) = base(a) + w[m(j – lbc)+ (i- lbr)] (in general)

loc(a[i][j]) = base(a) + w(m *j +i) ( in c/c++)

Row Major Order:

loc(a[i][j]) = base(a) + w[n(i – lbr)+ (j- lbc)] (in general)

loc(a[i][j]) = base(a) + w(n *i +j) ( in c/c++)

c program to transpose a matrix

#include<stdio.h>#include<conio.h> main(){ int m, n, c, d, matrix[10][10], transpose[10][10]; printf("Enter the number of rows and columns of matrix "); scanf("%d%d",&m,&n); printf("Enter the elements of matrix \n"); for( c = 0 ; c < m ; c++ ) { for( d = 0 ; d < n ; d++ ) { scanf("%d",&matrix[c][d]); } } for( c = 0 ; c < m ; c++ ) { for( d = 0 ; d < n ; d++ ) { transpose[d][c] = matrix[c][d]; } } printf("Transpose of entered matrix :-\n"); for( c = 0 ; c < n ; c++ ) { for( d = 0 ; d < m ; d++ ) { printf("%d\t",transpose[c][d]); } printf("\n"); } getch(); return 0;}

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