Data Structures Chapter-4

CHAPTER-4
ARRAYS, RECORDS & POINTERS

Data Structures
Linear
Linear relationship
between the
elements by means
of sequential
memory locations.
Linear relationship
between the
elements by means
of pointers or links
Non-Linear
Eg. Trees, Graphs
2 ways of representing linear structures
Arrays Linked List
M.Priyavani,MCA,DCHN,M.Phil, V.V.Vanniaperumal College
for Women, Virudhunagar

Operations performed on any Linear structure
 Traversal – Processing each element in the list.
 Search – Finding the location of the element with a given value or key.
 Insertion – Adding a new element to the list.
 Deletion – Removing an element from the list.
 Sorting – Arranging the elements in some type of order.
 Merging – Combining 2 lists into a single list.

When ARRAYS are preferred over Linked list?
 Since arrays are easy to Traverse, Search & Sort, they are frequently used
to store relatively permanent collections of data.
When ARRAYS may not be useful?
 If the size of the structure and the data in the structure are constantly
changing, then the array may not be as useful as the linked list.

Linear Arrays
 A Linear Arrays is a list of a finite number of n of homogenous data
elements, such that,
 The elements of the array are referenced respectively by an index set consisting
of n consecutive numbers.
 The elements of the array are stored respectively in successive memory
locations.
 The number of elements (n) is called the length or size of the array.
 The index set consists of the integers 1, 2, . . . N.

Linear Arrays – Continued . . .
 The length or the number of data elements of the array can be obtained from the index
set by the formula
Length = UB – LB + 1
where UB is the largest index, called the upper bound and
LB is the smallest index called the lower bound of the array.
 Index
 No.of Elements = 21 – 12 + 1 = 10
12 13 14 15 16 17 18 19 20 21

Linear Arrays
 The elements of the array A may be denoted by the subscript notation
 A1, A2, A3, . . . , An
 Or by the bracket notation
 A[1], A[2], A[3], . . . , A[n].
 The number K in A[K] is called the subscript or an index, and A[K] is called
a subscripted variable.
 Subscripts allow any element of A to be referenced by its relative position
in A.

Linear Arrays
 Where 0 is considered to be the lowest subscript, the highest subscript is
(size-1).
 Arrays can also be processed using FOR Loop.
 Among all the operators , the subscript of the array must have the highest
precedence.
 Any declaration of array must give the 3 items of information:
1. The name of the array
2. The data type of the array
3. The index set of the array

4.4 REPRESENTATION
OF
LINEAR ARRAYS IN
MEMORY

4.4 REPRESENTATION OF
LINEAR ARRAYS IN MEMORY
 Let X be a linear array in the memory of the computer.
 The memory of the computer is simply a sequence of addressed locations
as given below.
X[1]
X[2]
X[3]
X[4]
.
.
.
.
1000
1001
1002
1003
.
.
.
.
 Location( X[K] ) = Address of the element X[K].
 The elements of X are stored in successive memory
locations.
 The computer does not need to keep track of the
address of every element of X.
 But needs to keep track only of the address of the first
element of X, called the base address of X denoted by:
 Base(X)

4.4 REPRESENTATION OF
LINEAR ARRAYS IN MEMORY
 Using the base address, base(x), the computer calculates the address of
any element of X by the following formula:
 Location(X[k]) = Base(X) + w(K – lower_bound)
X[1]
X[2]
X[3]
X[4]
.
.
.
.
1000
1001
1002
1003
.
.
.
.
 Where w is the number of words per memory cell for the
array.
 the time to calculate X[k] is essentially the same for
any value of k.
 Given any subscript k, one can locate and access
the content of x[k] without scanning any other
element of x.

4.5 Traversing Linear Arrays
 Let A be a collection of data elements stored in the memory of the
computer.
 Suppose we want to print the contents of each element of A or we want
to count the number of elements with the given property.
 This can be accomplished by traversing A that is by accessing and
processing each element of A exactly once.
1. [ Initialize counter ] Set K := LB.
2. Repeat step-3 and 4 while K <= UB.
3. [ Visit Element ] Apply PROCESS to A[k].
4. [ Increase Counter ] Set K := K + 1.
[ End of Step 2 Loop ]
5. Exit

Inserting
 Inserting refers to a operation of adding another element to the collection A.
 Inserting an element at the “end” of a linear array can be easily done provided the
memory locations allocated for the array is large enough to accommodate the adding
element.
 Inserting an element in the middle of the array requires, part of (half of) the elements
must be moved downward to new locations to accommodate the new element and
keep the order of the other elements.

Inserting
NAME NAME NAME NAME
1 Brown 1 Brown 1 Brown 1 Brown
2 Davis 2 Davis 2 Davis 2 Davis
3 John 3 3 3 Ford
4 Smith 4 John 4 John 4 John
5 Wagner 5 Smith 5 Smith 5 Smith
6 6 Wagner 6 Wagner 6 Wagner
7 7 7 7
8 8 8 8

Algorithm for Inserting into a Linear Array
 Here LA is a linear array with N elements and K is a positive integer such
that K <= N.
 This algorithm inserts an element ITEM into the K-th position in LA.
1. [ Initialize Counter ] Set J := N.
2. Repeat Step 3 and 4 while J >= K.
3. [ Move j-th element downward] Set LA[ J+1 ] := LA[ J ]
4. [ Decrease counter ] Set J := J – 1
5. [ Insert Element ] Set LA[ K ] := Item
6. [ Reset N ] Set N := N +1
7. Exit

Deleting
 Deleting refers to the operation of removing one of the elements from A.
 Deleting an element at the “end” of an array presents no difficulties.
 Deleting an element in the middle of the array requires each subsequent elements
need to be moved one location upward in the order to “fill up” the array gap.

Deleting
NAME NAME NAME NAME
1 Brown 1 Brown 1 Brown 1 Brown
2 Davis 2 Davis 2 2 Ford
3 Ford 3 Ford 3 Ford 3 John
4 John 4 John 4 John 4 Smith
5 Smith 5 Smith 5 Smith 5 Wagner
6 Wagner 6 Wagner 6 Wagner 6
7 7 7 7
8 8 8 8

Algorithm for Deleting from a Linear Array
 Here LA is a linear array with N elements and K is a positive integer such
that K <= N.
 This algorithm deletes the K-th element from LA.
1. Set ITEM := LA[ k ].
2. Repeat for J := K to N-1.
3. [ Move j+1st element upward] Set LA[ J ] := LA[ J+1 ]
4. [ End of Loop ]
5. [ Reset N ] Set N := N -1
6. Exit

Multidimensional Arrays
 Arrays where elements are referenced by 2 and 3 subscripts are called as
two-dimensional and three-dimensional arrays.
 Some programming languages allow more than 2 subscripts also.
 These are called as multi-dimensional arrays.
 The length of the dimension is the number of integers in its index set.
 The pair of lengths m x n is called the size of the array.

Two-Dimensional Arrays
 A two0dimensional array is a collection pf m.n 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.
 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.
 A two-dimensional m x n array A, can be drawn as a rectangular array with m rows and n
columns and the element A[J,K] appears in row J and column K.
Columns 1 2 3 4
Row 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]

Storage Representations
Row-major Representation
 We can consider a two dimensional array since it has elements with a single dimension.
 As a result of this, a two-dimensional array can be assumed as a single column
with many rows and mapped sequentially.
 This is called a row-major representation.
A[1,1] A[1,2] A[1,3] A[1,4]
A[2,1] A[2,2] A[2,3] A[2,4]
A[3,1] A[3,2] A[3,3] A[3,4]
A[1,1]
A[1,2]
A[1,3]
A[1,4]
A[2,1]
A[2,2]
A[2,3]
A[2,4]
A[3,1]
A[3,2]
A[3,3]
A[3,4]M.Priyavani,MCA,DCHN,M.Phil, V.V.Vanniaperumal College

Row-major Representation
 We can calculate the address of the element of the m-th row and n-th column in a two dimensional
array using the formula shown below:
 addr[(a[m,n]) =
(Total number of rows present before the m-th row * size of a row )
+
(total number of elements present before the n-th element in the m-th row * size of an element)
 Addr(a[m,n]) = ((m-lb1)*(ub2-lb2+1)*size) + ((n-lb2)*size)
 Size of a row = Total no.of elements in a row * size of an element

Column-major Representation
 We can also represent a two-dimensional array as one single row of columns and map it
subsequently.
 This is called a column-major representation.
A[1,1] A[1,2] A[1,3] A[1,4]
A[2,1] A[2,2] A[2,3] A[2,4]
A[3,1] A[3,2] A[3,3] A[3,4]
A[1,1] A[2,1] A[3,1] A[1,2] A[2,2] A[3,2] A[1,3] A[2,3] A[3,3] A[1,4] A[2,4] A[3,4]

Representation of 2-dimensional Arrays in Memory
 The programming languages will store two dimensional arrays either
column-by-column (column major order) or row-by-row (row major order).
 The computer keeps track of Base(A) – the address of the first element
A[1,1] of A.
 It computes the address of A[J,K], where A is m x n array, using the formula
❖ LOC(A[J,K]) = Base(A) + w [ M(k-1) + (J-1) ], if column major order is used.
❖ LOC(A[J,K]) = Base(A) + w [ N(J-1) + (K-1) ], if row major order is followed.
 Here w denotes the number of words per memory location for the array A.

General Multi-dimensional Arrays
 An n dimensional m1 x m2 x . . . X mn array B is a collection of m1.m2. … mn
data elements in which each element is specified by a list of n integers
such as K1, K2, … , Kn called subscripts, with the property that 1<=K1<=m1,
1<=K2<=m2, … 1<=Kn<=mn.
 The element of B with subscripts K1, K2, … , Kn will be denoted by
Bk1,k2,…,kn
or
B[k1,k2,…,kn]

A Three Dimensional Array
A[1,1] A[1,2] A[1,3]
A[2,2]
4,
3,
1
4,
3,
2
4,
3,
3
4,
3,
4
4,
3,
5
Row-1
Row-2
Row-3
Row-4
Row-5
Row-6
Col-1 Col-2 Col-3 Col-4 Col-5

Representation of Polynomials using Arrays
 We need to evaluate polynomial expressions and perform basic arithmetic
operations like addition, multiplication on them.
 We have a method to represent a polynomial of degree ‘n’, by storing the
coefficient of (n+1) terms of a polynomial in a array.
 An expression x + 4 x2 – 7 x5 is a polynomial of degree 5.
 All the elements of an array has 2 values namely coefficient and
exponent.
 We also assume that the exponent of each successive term is less than
that of the previous term.
 A single dimensional array is used to store a single variable polynomial.

Polynomial representation using 1-d array
 3 x4 + 5 x3 + 6 x2 + 10 x – 14 is stored as below:
index Value Meaning
0 -14 – 14
1 10 10 x
2 6 6 x2
3 5 5 x3
4 3 3 x4

Polynomial addition
0 1 4 0 0 -7
-14 10 6 5 3
-7 x5 + 4 x2 + x
3 x4 + 5 x3 + 6 x2 + 10 x - 14
i
j
k
Set ‘i ‘ to point to first term in A
Set ‘j ‘ to point to first term in B
Set ‘k ‘ to point to first term in C

Polynomial addition
0 1 4 0 0 -7
-14 10 6 5 3
-14
-7 x5 + 4 x2 + x
3 x4 + 5 x3 + 6 x2 + 10 x - 14
i
j
k Add the terms pointed by I & j
into C

Polynomial addition
0 1 4 0 0 -7
-14 10 6 5 3
-14
-7 x5 + 4 x2 + x
3 x4 + 5 x3 + 6 x2 + 10 x - 14
i
j
k
Increment ‘i ‘ to point to next
term in A
Increment ‘j ‘ to point to nextt
term in B
Set ‘k ‘ to point to next term in C

Polynomial addition
0 1 4 0 0 -7
-14 10 6 5 3
-14 11
-7 x5 + 4 x2 + x
3 x4 + 5 x3 + 6 x2 + 10 x - 14
i
j
into C

Polynomial addition
0 1 4 0 0 -7
-14 10 6 5 3
-14 11
-7 x5 + 4 x2 + x
3 x4 + 5 x3 + 6 x2 + 10 x - 14
i
j
k
term in A
term in B

Polynomial addition
0 1 4 0 0 -7
-14 10 6 5 3
-14 11 10
-7 x5 + 4 x2 + x
3 x4 + 5 x3 + 6 x2 + 10 x - 14
i
j
into C

Polynomial addition
0 1 4 0 0 -7
-14 10 6 5 3
-14 11 10
-7 x5 + 4 x2 + x
3 x4 + 5 x3 + 6 x2 + 10 x - 14
i
j
k
term in A
term in B

Polynomial addition
0 1 4 0 0 -7
-14 10 6 5 3
-14 11 10 5
-7 x5 + 4 x2 + x
3 x4 + 5 x3 + 6 x2 + 10 x - 14
i
j
into C

Polynomial addition
0 1 4 0 0 -7
-14 10 6 5 3
-14 11 10 5
-7 x5 + 4 x2 + x
3 x4 + 5 x3 + 6 x2 + 10 x - 14
i
j
k
term in A
term in B

Polynomial addition
0 1 4 0 0 -7
-14 10 6 5 3
-14 11 10 5 3
-7 x5 + 4 x2 + x
3 x4 + 5 x3 + 6 x2 + 10 x - 14
i
j
into C
Polynomial B is over

Polynomial addition
0 1 4 0 0 -7
-14 10 6 5 3
-14 11 10 5 3 -7
-7 x5 + 4 x2 + x
3 x4 + 5 x3 + 6 x2 + 10 x - 14
i
j
k
Add the terms pointed by I & j
into C
Copy remaining A
terms into C

4.12 POINTERS & POINTER ARRAYS
 A variable P is called a pointer if P “points” to an element, i.e if P contains
the address of that element.
 An array PTR is called a pointer array if each element of PTR is a pointer.
2002 Data element
Address=2002P

Example
 Consider an organization which divides its members into 4 groups, where each group contains an
alphabetized list of members living in a certain area. There are 21 people totally, divided into 4 groups
of 4, 9, 2 and 6 people. Group-1 Group-2 Group-3 Group-4
Evans Conrad Davis Baker
Harris Felt Segal Cooper
Lewis Glass Ford
Shaw Hill Gray
King Jones
Penn Reed
Silver
Troy
WagnerM.Priyavani,MCA,DCHN,M.Phil, V.V.Vanniaperumal College

Example – continued …
 Here, the data can be stored into a 2-dimensional array either in row-major or column-major order.
 The drawback here is the wastage of memory locations. (show in green shades)
Group-1 Group-2 Group-3 Group-4
Evans Conrad Davis Baker
Harris Felt Segal Cooper
Lewis Glass Ford
Shaw Hill Gray
King Jones
Penn Reed
Silver
Troy
WagnerM.Priyavani,MCA,DCHN,M.Phil, V.V.Vanniaperumal College

Example – continued …
 Another way of storing without wasting space is
shown in the right side.
 Even this model has certain drawbacks.
 The list must be traversed from the beginning in
order to recognize a particular group.
 To correct these issues, we have the concept of
DYNAMIC MEMORY.
 Dynamic memory is a area in memory that is
managed automatically.
 We can request for and release storage as
required.
Evans
Harris
Lewis
Shaw
$$$
Conrad
…
Wagner
$$$
Davis
Segal
$$$
Baker
…
Reed
$$$
Group-1
Group-2
Group-3
Group-4

POINTER ARRAYS
 The space efficient data structure can
be easily modified so that the individual
groups can be indexed.
 This is done by using a pointer array,
which contains the locations of the
different groups, or say the locations of
the first element in that group.
Evans
Harris
Lewis
Shaw
Conrad
.
.
.
Wagner
Davis
Segal
Baker
.
.
.
Reed
$$$
1
5
14
16
22
Evans
Harris
Lewis
Shaw
Conrad
.
.
.
Wagner
Davis
Segal
Baker
.
.
.
Reed
$$$
GROUP

RECORDS; RECORD STRUCTURES
 Collection of data are frequently organized into a hierarchy of field, records and files.
 A record is a collection of related data items, each of which is called a field or attribute.
 A file is a collection of similar records.
 Each data item itself may be a group item composed of sub-items.
 The items which are indecomposable are called elementary items or atoms or scalars.
 The name given to various data items are called identifiers.
 Record differs from a linear array in the follow ways:
1. A record may be a collection of nonhomogeneous data.
2. The data items in a record are indexed by attribute names.

Example
 A hospital maintains a record
on each new born baby that
contains the following data
items: Name, Sex, Birthday
(Day, Month, Year), Father
(Name, Age), Mother (Name,
Age).
 The above structure can be
described as mentioned:
 The number in the left of
each identifier is called level
number.
 The subitems have greater
level number than the above
items.
1 New-born
2 Name
2 Sex
2 Birthday
3 Day
3 Month
3 Year
2 Father
3 Name
3 Age
2 Mother
3 Name
3 Age

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