U2.pptx Advanced Data Structures and Algorithms

Linear Data Structures Using
Sequential Organization

Sequential Organization
 Sequential organization allows storing data at a fixed
distance apart.
 If the ith element is stored at location X, then the next
sequential (i+1)th
element is stored at location X+C, where C is
constant.

ARRAYS
 If we intend to store a group of data together in a sequential manner in
computer’s memory, then arrays can be one of the possible data structures.
 An array is a finite ordered collection of homogeneous data elements which
provides direct access (or random access) to any of its elements.
 An array as a data structure is defined as a set of pairs (index, value) such
that with each index a value is associated.
• index — indicates the location of an element in an array.
• value - indicates the actual value of that data element.
• Declaration of an array in ‘C++’:
int A[20];

GENERIC DATA TYPE
 Generic Data type is a data type where the
operations are defined but the types of the
items being manipulated are not

ARRAYS AS AN ADT
 Formally ADT is a collection of domain, operations, and axioms (or rules)
 For defining an array as an ADT, we have to define its very basic operations or
functions that can be performed on it
 The basic operations of arrays are creation of an array, storing an element,
accessing an element, and traversing the array.
 Let us specify ADT Array in which we provide specifications with operations to
be performed.
 ADT ARRAY(Index, Value)
 declare CREATE( ) array
ACCESS (array, index) value
STORE (array, index, value) array
for all Array_A ε array, x Î value, and i, j ε index let
ACCESS (CREATE, i) = error.
ACCESS (STORE (Array_A, i, x), j) = x if EQUAL (i, j)
else ACCESS (Array_A, j)
end
end ARRAY.

MEMORY REPRESENTATION & ADDRESS
CALCULATION
A0
A1
.
.
.
Ai
.
An-1
a i
ai+1
ai+2
:
:
an-1
X(Base)
X+1
X+2
X+(n-1)
Array A
 The address of the ith
element is calculated by the following
formula
• (Base address) + (offset of the ith
element from base address)
• Here, base address is the address of the first element where
array storage starts.

 Arrays support various operations such as traversal, sorting,
searching, insertion, deletion, merging, block movement, etc.
 Insertion of an element into an array
 Deleting an element
 Memory Representation of Two-Dimensional Arrays
 Row-major Representation
 Column-major Representation

Columns
Col1 col2 .... coln
A11 A12 .... A1n
A11 A12 .... A1n
Am1 Am2 .... Amn
: :
:
m*n
Rows R1
R2
Rm
1 2 3 4
5 6 7 8
9 10 11 12
Matrix M =

ROW-MAJOR REPRESENTATION
 In row-major representation, the elements of
Matrix are stored row-wise, i.e., elements of 1st row,
2nd row, 3rd row, and so on till mth
row
(0,0) (0,1) (0,2) (0,3) (1,0) (1,1) (1,2) (1,3) (2,0) (2,1) (2,2) (2,3)
Row1 Row2 Row3
1 2 3 4 5 6 7 8 9 10 11 12

ROW MAJOR ARRANGEMENT
 Address of A[i][j] = base addr + (col_index * total no. of rows +
row_index) element size
Row 0
Row 1
Row m-1
Row 0
Row 1
Row
m-1
Memory Location
Row-major arrangement in memory , in row major representation

COLUMN-MAJOR ARRANGEMENT
 Address of A[i][j] = base addr + (row_index * total no. of columns +
col_index) *element size
col1 col2
Col
n-1
Col
0
Col
1
Col
2
Memory Location
…
Column-major arrangement in memory , in column major representation

(0,0) (1,0) (2,0) (0,1) (1,1) (2,1) (0,2) (1,2) (2,2) (0,3) (1,3) (2,3)
Col 1 Col 2 Col 3 Col 4
1 2 3 4 5 6 7 8 9 10 11 12
Column-Major Representation of 2-D array

U2.pptx Advanced Data Structures and Algorithms

Row-Major representation of 2D array

Three dimensions row-major
arrangement
(i*m2*m3) elements
A[0][m2][m3] A[1][m2][m3] A[i][m2][m3] A[m1-1][m2]

ARRAYS USING TEMPLATE
 The function is defined in similar way replacing int by T as
datatype of member of array
 In all member functions header, Array is replaced by Array
<T> :: now
 Following statements instantiate the template class Array to int
and float respectively. So P is array of ints and Q in array of
floats.
Array <int> P;
Array <float> Q;
 In similar we can also have array of any user defined data type

CONCEPT OF ORDERED LIST
 Ordered list is the most common and frequently used data object
 Def: it is set of elements where set may be empty or it can be written as
collection of elements such as (a1, a2, a3 ………, an)
 Linear elements of an ordered list are related with each other in a particular
order or sequence
 Following are some examples of the ordered list.
• 1, 3,5,7,9,11,13,15
• January, February, March, April, May, June, July, August, September,
• October, November, December
• Red, Blue, Green, Black, Yellow

 There are many basic operations that can be performed on the ordered list as
follows:
 Finding the length of the list
 Traverse the list from left to right or from right to left
 Access the ith element in the list
 Update (Overwrite) the value of the ith position
 Insert an element at the ith location
 Delete an element at the ith position

SINGLE VARIABLE POLYNOMIAL
 Representation Using Arrays
 Array of Structures
 Polynomial Evaluation
 Polynomial Addition
 Multiplication of Two Polynomials

 Polynomial as an ADT, the basic operations
are as follows:
 Creation of a polynomial
 Addition of two polynomials
 Subtraction of two polynomials
 Multiplication of two polynomials
 Polynomial evaluation

 STRUCTURE IS BETTER THAN ARRAY
FOR POLYNOMIAL:
 Such representation by an array is both time and space efficient when
polynomial is not a sparse one such as polynomial P(x) of degree 3 where P(x)=
3x3+x2–2x+5.
 But when polynomial is sparse such as in worst case a polynomial as A(x)= x99
+ 78 for degree of n =100, then only two locations out of 101 would be used.
 In such cases it is better to store polynomial as pairs of coefficient and exponent.
We may go for two different arrays for each or a structure having two members
as two arrays for each of coeff. and Exp or an array of structure that consists of
two data members coefficient and exponent.

POLYNOMIAL BY USING STRUCTURE
 Let us go for structure having two data
members coefficient and exponent and its
array.

AN ARRAY FOR FREQUENCY COUNT
We can use array to store the number of times a particular
element occurs in any sequence. Such occurrence of
particular element is known as frequency count.
void Frequency_Count ( int Freq[10 ], int A [ 100])
{
int i;
for ( i=0;i<10;i++)
Freq[i]=0;
for ( i=0;i<100;i++)
Freq[A[i] ++;
}

Frequency count of numbers ranging
between 0 to 9

SPARSE MATRIX
In many situations, matrix size is very large but out of
it, most of the elements are zeros (not necessarily
always zeros).
And only a small fraction of the matrix is actually
used. A matrix of such type is called a sparse matrix,

Sparse matrix and its representation

TRANSPOSE OF SPARSE MATRIX
 Simple Transpose
 Fast Transpose

Time complexity of manual technique is O (mn).

 Time complexity will be O (n . T)
= O (n . mn)
= O (mn2
)
which is worst than the conventional transpose with
time complexity O (mn)
Simple Sparse matrix transpose

FAST SPARSE MATRIX TRANSPOSE
 In worst case, i.e. T= m × n (non-zero elements) the
magnitude becomes O (n +mn) = O (mn) which is the
same as 2-D transpose
 However the constant factor associated with fast
transpose is quite high
 When T is sufficiently small, compared to its maximum of
m . n, fast transpose will work faster

STRING MANIPULATION
 Def: String is sequence of characters. The strings are defined with double quotes.
 The string is stored in the memory with terminating character ‘0’.
 There are various operations that can be performed on the string:
 To find the length of a string (strlen)
 To concatenate two strings (strcat)
 To copy a string (strcpy)
 To reverse a string (strrev)
 String compare (strcmp, strcmpi)
 Palindrome check
 To recognize a sub string. (strstr)

BASICALLY A STRING IS STORED AS A SEQUENCE
OF CHARACTERS IN ONE-DIMENSIONAL
CHARACTER ARRAY SAY A.
CHAR A[10] ="STRING" ;
EACH STRING IS TERMINATED BY A SPECIAL
CHARACTER THAT IS NULL CHARACTER ‘0’.
THIS NULL CHARACTER INDICATES THE END OR
TERMINATION OF EACH STRING.

CHARACTERISTICS OF ARRAY
 An array is a finite ordered collection of homogeneous data
elements.
 In array, successive elements of list are stored at a fixed distance
apart.
 Array is defined as set of pairs-( index and value).
 Array allows random access to any element
 In array, insertion and deletion of elements in between positions
requires data movement.
 Array provides static allocation, which means space allocation
done once during compile time, can not be changed run time.

ADVANTAGE OF ARRAY
DATA STRUCTURE
 Arrays permit efficient random access in constant time 0(1).
 Arrays are most appropriate for storing a fixed amount of data and also for high
frequency of data retrievals as data can be accessed directly.
 Wherever there is a direct mapping between the elements and there positions,
arrays are the most suitable data structures.
 Ordered lists such as polynomials are most efficiently handled using arrays.
 Arrays are useful to form the basis for several more complex data structures,
such as heaps, and hash tables and can be used to represent strings, stacks and
queues.

DISADVANTAGE OF
ARRAY DATA STRUCTURE
 Arrays provide static memory management. Hence
during execution the size can neither be grown nor
shrunk.
 Array is inefficient when often data is to inserted or
deleted as inserting and deleting an element in array
needs a lot of data movement.
 Hence array is inefficient for the applications, which
very often need insert and delete operations in
between.

APPLICATIONS OF ARRAYS
 Although useful in their own right, arrays also form the basis for
several more complex data structures, such as heaps, hash tables
and can be used to represent strings, stacks and queues.
 All these applications benefit from the compactness and direct
access benefits of arrays.
 Two-dimensional data when represented as Matrix and matrix
operations.

U2.pptx Advanced Data Structures and Algorithms

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