arrays in data structure.pptx

Data Structure and
Algorithm
Mrs. K.KASTHURI, M.Sc., M.Phil., M.Tech.,
Assistant Professor,
Department of Information Technology,
V.V.Vanniaperumal College for Women,
1

Data Structure and Storage Structure
• Data Structure : The logical or mathematical model of a
particular organization of data .
• Storage Structure : Representation of a particular data
structure in the memory of a computer.
2

Data Structure and Storage Structure
 A data structure is a storage that is used to store and organize data.
 It is a way of arranging data on a computer
 so that it can be accessed and updated efficiently.
 A data structure is not only used for organizing the data.
 It is also used for processing, retrieving, and storing data.
 There are two types of Data Structures;
 They are Linear and Non-Linear.
 A Data structure is said to be linear if its elements from a sequence or in
other words form a linear list
 A Data structure is said to be Non-linear if its elements are not placed
sequentially or linearly.
3

Representation in Memory
 Two basic representation in memory
◦ Have a linear relationship between the elements represented
by means of sequential memory locations [ Arrays]
◦ Have the linear relationship between the elements
represented by means of pointer or links [ Linked List]
5

Operation on Linear Structure
 Traversal : Processing each element in the list
 Search : Finding the location of the element with a given
value or the record with a given key.
 Insertion: Adding a new element to the list
 Deletion: Removing an elements from the list
 Sorting : Arranging the elements in some type of order
 Merging : Combining two list into a single list
6

Linear Arrays
 A linear array is a list of a finite number of n
homogeneous data elements ( that is data elements of
the same type)
◦ The elements are of the arrays are referenced respectively
by an index set consisting of n consecutive numbers
◦ The elements of the arrays are stored respectively in
successive memory locations
7

Linear Arrays
 The number n of elements is called the
length or size of the array.
 The index set consists of the integer 1, 2, …
n
 Length or the number of data elements of
the array can be obtained from the index set
by
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 arrays
8

Linear Arrays
 Element of an array A may be denoted by
◦ Subscript notation A1, A2, , …. , An
◦ Parenthesis notation A(1), A(2), …. , A(n)
◦ Bracket notation A[1], A[2], ….. , A[n]
 The number K in A[K] is called subscript
or an index and A[K] is called a
subscripted variable
9

Representation of Linear Array in Memory
10

 Let LA be a linear array in the memory of the
computer
 LOC(LA[K]) = address of the element
LA[K] of the array LA
 The element of LA are stored in the
successive memory cells
 Computer does not need to keep track of the
address of every element of LA, but need to
track only the address of the first element of
the array denoted by Base(LA) called the
base address of LA
11

 LOC(LA[K]) = Base(LA) + w(K – lower
bound) where w is the number of words
per memory cell of the array LA
 w is a size of the data type.
12

Traversing Linear Arrays
 Traversing is accessing and processing
(visiting ) each element of the data
structure exactly ones
13
•••
Linear Array

(aka visiting ) each element of the data
14
•••
Linear Array

15
•••
Linear Array

16
•••
Linear Array

17
•••
Linear Array

18
•••
Linear Array

19
•••
Linear Array
1. Repeat for K = LB to UB
Apply PROCESS to LA[K]
[End of Loop]
2. Exit

Inserting and Deleting
 Insertion: Adding an element
◦ Beginning
◦ Middle
◦ End
 Deletion: Removing an element
◦ Beginning
◦ Middle
◦ End
20

Insertion
1 Brown
2 Davis
3 Johnson
4 Smith
5 Wagner
6
7
8
21
1 Brown
2 Davis
3 Johnson
4 Smith
5 Wagner
6 Ford
7
8
Insert Ford at the End of Array

Insertion
1 Brown
2 Davis
3 Johnson
4 Smith
5 Wagner
6
7
8
22
1 Brown
2 Davis
3 Johnson
4 Smith
5
6 Wagner
7
8
Insert Ford as the 3rd Element of Array
1 Brown
2 Davis
3 Johnson
4
5 Smith
6 Wagner
7
8
1 Brown
2 Davis
3
4 Johnson
5 Smith
6 Wagner
7
8
Ford
Insertion is not Possible without
loss of data if the array is FULL

Deletion
1 Brown
2 Davis
3 Ford
4 Johnson
5 Smith
6 Taylor
7 Wagner
8
23
1 Brown
2 Davis
3 Ford
4 Johnson
5 Smith
6 Taylor
7
8
Deletion of Wagner at the End of Array

Deletion
1 Brown
2 Davis
3 Ford
4 Johnson
5 Smith
6 Taylor
7 Wagner
8
24
1 Brown
2
3 Ford
4 Johnson
5 Smith
6 Taylor
7 Wagner
8
Deletion of Davis from the Array
1 Brown
2 Ford
3
4 Johnson
5 Smith
6 Taylor
7 Wagner
8
1 Brown
2 Ford
3 Johnson
4
5 Smith
6 Taylor
7 Wagner
8

Deletion
1 Brown
2 Ford
3 Johnson
4 Smith
5 Taylor
6 Wagner
7
8
25
No data item can be deleted from an
empty array

Insertion Algorithm
 INSERT (LA, N , K , ITEM) [LA is a linear
array with N elements and K is a positive integers
such that K ≤ N. This algorithm insert an element
ITEM into the Kth position in LA ]
1. [Initialize Counter] Set J := N
2. Repeat Steps 3 and 4 while J ≥ K
3. [Move the Jth 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
26

Deletion Algorithm
 DELETE (LA, N , K , ITEM) [LA is a linear
array with N elements and K is a positive integers
such that K ≤ N. This algorithm deletes Kth
element from LA ]
1. Set ITEM := LA[K]
2. Repeat for J = K to N -1:
[Move the J + 1st element upward] Set LA[J]
:= LA[J + 1]
3. [Reset the number N of elements] Set N :=
N - 1;
4. Exit
27

Multidimensional Array
 One-Dimensional Array
 Two-Dimensional Array
 Three-Dimensional Array
 Some programming Language allows as
many as 7 dimension
28

Two-Dimensional Array
 A Two-Dimensional m x n array A is a
collection of m . n data elements such
that each element is specified by a pair of
integer (such as J, K) called subscript with
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]
29

2D Arrays
The elements of a 2-dimensional array a is shown as below
a[0][0] a[0][1] a[0][2]
a[0][3]
a[1][0] a[1][1] a[1][2]
a[1][3]
a[2][0] a[2][1] a[2][2]
a[2][3]

Rows Of A 2D Array
a[0][0] a[0][1] a[0][2] a[0][3] row 0
a[1][0] a[1][1] a[1][2] a[1][3] row 1
a[2][0] a[2][1] a[2][2] a[2][3] row 2

Columns Of A 2D Array
a[0][0] a[0][1] a[0][2] a[0][3]
a[1][0] a[1][1] a[1][2] a[1][3]
a[2][0] a[2][1] a[2][2] a[2][3]
column 0 column 1 column 2 column 3

 Let A be a two-dimensional array m x n
 The array A will be represented in the
memory by a block of m x n sequential
memory location
 Programming language will store array A
either
◦ Column by Column (Called Column-Major
Order) Ex: Fortran, MATLAB
◦ Row by Row (Called Row-Major Order) Ex:
C, C++ , Java
33

2D Array in Memory
A Subscript
(1,1)
(2,1)
(3,1)
(1,2)
(2,2)
(3,2)
(1,3)
(2,3)
(3,3)
(1,4)
(2,4)
(3,4)
34
Column
1
Column
2
Column
3
Column
4
A Subscript
(1,1)
(1,2)
(1,3)
(1,4)
(2,1)
(2,2)
(2,3)
(2,4)
(3,1)
(3,2)
(3,3)
(3,4)
Row 1
Row 3
Row 2
Column-Major Order Row-Major Order

2D Array
 LOC(LA[K]) = Base(LA) + w(K -1)
 LOC(A[J,K]) of A[m,n]
Column-Major Order
LOC(A[J,K]) = Base(A) + w[m(K-1) + (J-1)]
Row-Major Order
LOC(A[J,K]) = Base(A) + w[n(J-1) + (K-1)]
35

 Let C be a n-dimensional array
 Length Li of dimension i of C is the
number of elements in the index set
Li = UB – LB + 1
 For a given subscript Ki, the effective
index Ei of Li is the number of indices
preceding Ki in the index set
Ei = Ki – LB
36

 Address LOC(C[K1,K2, …., Kn]) of an arbitrary
element of C can be obtained as
Column-Major Order
Base( C) + w[((( … (ENLN-1 + EN-1)LN-
2) + ….. +E3)L2+E2)L1+E1]
Row-Major Order
Base( C) + w[(… ((E1L2 + E2)L3 +
E3)L4 + ….. +EN-1)LN +EN]
37

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