DataStructure - Arrays

8/12/2019 DataStructure - Arrays

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Data Structures & Algorithms

Lecture # 2

BS CS&IT Evening Program, FJ

Campus


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Today Topics

What are Arrays?

Arrays Data Structures

One Dimensional Array

Operations on Arrays Inserting Deleting

Traversing Searching Reversing


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What are Arrays?

Array can be defined as:

a group of data items of the same type

under one name orarray is a collection ofelements or items or components that havethe same data type.

For example array of 10 integers, arrays of

names, etcetera. It can also be defined asa list of two or more variables with thesame name.


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Why we use Array?

For example you want to find smallest numbersamongst four integer items and suppose that you donot know how to use array or you do not desire to useit. You may tackle the problem in the followingmanner:

Define four variables to hold four integers data items,i.e.,

int a=10, b=20, c=30; d=5;

After declaring these variables, you would be requiredto write a quite few statements to find the smallestnumber in them. Since there are four variables,therefore, you would need to use four compound ifstatements for the solution, such as:


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Why we use Array?

if (a


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Why we use Array?

Assuming if you would have to find smallest numberamongst 100 data items, off course you would berequired to write 100 compound ifstatements andeach compound condition made of 99 simpleconditions. Further, it would not be possible to do

other application such as sorting data, finding anyparticular number in large amount of data with theuse simple data structures. Therefore, the use ofarrays in programming languages is unavoidable.Processing on the data such as finding any number,calculating mean, average and sorting data are some

of the fascinating applications which can beimplemented very easily with the help of array.


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One dimensional array is most common form of the array used byprogrammers to solve different kinds of problems.

The general format of the one-dimensional array is as follows:data_type array_name[size];

wheredata_type: is any valid C++ data type (int, float, etc.)

array_name: is user defined identifier. User_definedname is followed by the square

brackets which makes difference between ordinary variable

and array type variable.

Size: shows the actual length of the array, i.e., number oflocations or no data items.


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Examples of One Dimensional Array

int a[10]; a is an integer array of10 elements

float f[10] f is a float array of 10elements

double d[10] d is a double arrayof 10 elements.


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Suppose you want to store identity-numbers of 20students, the following statement would be used todeclare a one dimensional integer array to hold data;

int student_id_no[20];

where int is type of the array, andstudent_id_no is name of the array (user defined).

Computer will allocate 20 locations forstudent_id_no.

C++ starts array locations from zero, hence, firstlocation is indicated as zero


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Diagrammatically Representation ofArray

0 1 2 3 4 5 6 7 8 9

Index

Number

101 103 105 107 109 111 113 115 117 119

MemoryAddress

LB UB


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Maximum number of elements in a One-DimensionalArray is computed as under:

Maximum number of elements = UB LB + 1UB Upper Bound LB Lower BoundIn pervious diagram UB = 9 and LB = 0So, Maximum number of elements = 9 0 +1 = 10

Each element of the array also has a unique memoryaddress

The starting address of array is called Base Address.

Elements of an array occupy a continuous block ofmemory


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The memory address of an element with index k iscomputed as:

L(X(k)) = L0+ C * (k 1)

L0is a Base Address, C memory size of each element in byte

Suppose that Base address is 101 and size of element is 2bytes, what will be the memory address for index 5(Previous diagram)

L0= 101, C = 2, k = 5

L(X(k)) = L0+ C * (k 1)L(X(5)) = 101 + 2 * (5 1)= 101 + 2 * 4 = 101 + 10 = 111


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Operations on 1-D Array

Traversal: Processing each element in the array

Search: Finding the location of an element with a givenvalue

Insertion: Adding a new element to an array

Deletion: Removing an element from an array

Sorting: Organizing the elements in some order

Merging: Combining two arrays into a single array

Reversing: Reversing the elements of an array


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Algorithms of Operations on 1-DArray

Traverse_Array

Algorithm for print the each element ofarray

1. Start

2. Repeat step-3 For i=1 to Max by 1

3. Print arr[i]

4. End


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Insert_Item(pos, num)

Algorithm for inserting an element atgiven position of the array

1. Start

2. Repeat step-3 For i=Max to Pos+1 by -1

3. arr[i] = arr[i-1]

4. Arr[Pos] = num5. End


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Delete_Item(pos)

Algorithm for delete an element from thegiven position of the array

1. Start

2. Repeat step-3 For i=Pos to Max-1 by 1

3. arr[i] = arr[i+1]

4. Arr[Max] = 05. End


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Search_Item(num)

Algorithm for search Pos of the given element in thearray

1. Start

2. Repeat step 3 For i=1 to Max by 1

3. If arr[i] = num then

Print I

Return

End If4. Print Element Not Found!

5. End


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Reverse_Array

Algorithm for reverse the entire array

1. Start

2. Repeat step 3 to 5 For i=1 to Max/2 by 1

3. temp = arr[i]

4. arr[i] = arr[Max-i+1]

5. arr[Max-i+1] = temp6. End


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Two Dimensional Array

A two-dimensional array is a collection of elements placedin m rows and n columns.

The syntax used to declare a 2-D array includes twosubscripts, of which one specifies the number of rows and

the other specifies the number of columns of an array.data_type array_name[rows][columns];where

data_type: is any valid C++ data type (int, float, etc.)

array_name: is user defined identifier. User defined

name is followed by the square

Rows: Specifies the number of rows of an arrayColumn: Specifies the number of columns of an array


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int a[3][4];a is an integer arraycontaining 3 rows and 4 columns

float f[3][4]; f is a float arraycontaining 3 rows and 4 columns

double d[3][4]; d is a double arraycontaining 3 rows and 4 columns


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Diagrammatically Representation ofArray

1 2 3 4

1 12 1 -9 23

2 14 7 11 121

3 6 78 15 34

Rows

Columns


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Two Dimensional Array

Maximum number of elements in a Two-DimensionalArray is computed as under:

Maximum number of elements = M x NM Number of rows N Number of columnsIn pervious diagram M = 3 and N = 4So, Maximum number of elements =3 x 4 = 12

Each element of the array also has a unique memoryaddress

The starting address of array is called Base Address.

Elements of an array occupy a continuous block ofmemory


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Two Dimensional arrays are represented in memory intwo ways. Row-major order

Representation of Two-DimensionalArrays in memory

12 1 -9 23 14 7 11 121 6 78 15 34

502 504 506 508 510 512 514 516 518 520 522 524MemoryAddress

Values

1stRow 2ndRow 3rdRow

12 14 6 1 7 78 -9 11 15 23 121 34

502 504 506 508 510 512 514 516 518 520 522 524MemoryAddress

Values

1stCol

Column-major order

Note: Each integer occupies two bytes.

2ndCol 3rdCol 4thCol


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Row-Major Order

The memory address of an element with index i (row) andj (Col) is computed as:

L(X[i][j]) = L0+ [(i 1) * N + (j 1)] * d

L0is a Base Address, d memory size of each element in byte

i is a row, j is a col and N is number of columns of an array

Suppose that Base address is 502 and size of element is 2bytes, what will be the memory address for position x[2][2](Previous diagram)

L0= 502, d = 2, i = 2, j = 2, N = 4

L(X[i][j]) = L0+ [(i 1) * N + (j 1)] * dL(X[2][2]) = 502 + [(2 1) * 4 + (2 1)] * 2

= 502 + [4 + 1] * 2 = 502 + 5 * 2 = 502 + 10 = 512


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Col-Major Order

The memory address of an element with index i (row) andj (Col) is computed as:

L(X[i][j]) = L0+ [(i 1) + (j 1) * M] * d

L0is a Base Address, d memory size of each element in byte

i is a row, j is a col and M is number of rows of an array

Suppose that Base address is 502 and size of element is 2bytes, what will be the memory address for position x[2][2](Previous diagram)

L0= 502, d = 2, i = 2, j = 2, M = 3

L(X[i][j]) = L0+ [(i 1) + (j 1) * M] * dL(X[2][2]) = 502 + [(2 1) + (2 1) * 3] * 2

= 502 + [1 + 3] * 2 = 502 + 4 * 2 = 502 + 8 = 510


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Operations on 2-D Array

Traversal: Processing each element in the array

Search: Finding the location of an element with a givenvalue

Insertion: Adding a new element to an array

Deletion: Removing an element from an array

Sorting: Organizing the elements in some order

Merging: Combining two arrays into a single array

Reversing: Reversing the elements of an array


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Traverse_Array

Algorithm for print the each element ofarray

1. Start

2. Repeat step-3 For i=1 to rMax by 1

3. Repeat step-4 For j=1 to cMax by 1

4. Print arr[i][j]5. End


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Insert_Item_Row_Major(r, c, num)Algorithm for inserting an element at given position of thearray

1. Start2. Repeat step-3 For i=rMax to r by -13. Repeat step-4 For j=cMax to 1 by -14. if i > r or j > c then

if j = 1 thenarr[i][j] = arr[i-1][cMax]

elsearr[i][j] = arr[i][j-1]

end if

end if5. Arr[r][c] = num6. End


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Delete_Item_Row_Major(r, c)Algorithm for delete an element from the given position of thearray

1. Start2. Repeat step-3 For i=r to rMax by 13. Repeat step-4 For j=1 to cMax by 14. if i r or j >= c then

if j = cMax thenarr[i][j] = arr[i+1][1]

elsearr[i][j] = arr[i][j+1]

end if

end if5. Arr[rMax][cMax] = 06. End


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Common Matrix Operations

Traverse

Addition

Multiplication

Transposition

Evaluation of the determinant of asquare matrix


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Algorithms for Matrix Addition

Matrix_Addition(m1, m2)

Algorithm for add two matrices

1. Start

2. Repeat step 3 For i=1 to rMax by 1

3. Repeat step 4 For j=1 to cMax by 1

4. Mat[i][j] = m1[i][j] + m2[i][j]

5. End


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Algorithms for Matrix Multiplication

Matrix_Multiplication(m1, m2)

Algorithm for multiply two matrices

1. Start



4. Repeat step 5 For k=1 to rMax

5. Mat[i][j] += m1[i][k] * m2[k][j]6. End


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Algorithms for Matrix Transposition

Matrix_Transpose(m)

Algorithm for transpose the matrix

1. Start



4. Mat[i][j] = m[j][i]

5. End


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Multi-Dimensional Array

A three-dimensional array can be thought of as an array ofarrays of arrays.

Following figure shows a 3-D array, which is a collection of

three 2-D arrays each containing 4 rows and 2 columnsint arr[3][4][2];

3 9

1 8

6 5

4 0

2 8

0 6

4 7

1 5

3 2

8 6

1 6

4 5

1st2-D Array 2nd2-D Array 3rd2-D Array


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Arrays and Polynomials

Polynomials like5X4+ 2X3+ 7X2+ 10X 8

Arithmetic operations like addition andmultiplication of polynomials are common

Need a way to represent the polynomial in anarray

Each element of the array should consist oftwo values, namely coefficient and exponent


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Arrays and Polynomials

To create a struct (User Defined Data Type) forPolynomials

struct Poly{

int coeff;

int exp;};

Poly P[Max];

P[0].coeff = c;

P[0].exp = e;


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Next Lecture

What are Stacks?

Representation of Stacks

Operations on Stacks

Push

Pop

Evaluation of Expressions

Infix to Postfix Conversion

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DataStructure - Arrays