Data strutcure and annalysis topic array

Unit-2
Linear Data
Structure
Array
Data Structures (DS)
GTU # 3130702

✓ Loopin
g
Outline
▪ Array
• Representation of arrays
• One dimensional array
• Two dimensional array
▪ Applications of arrays
• Symbol Manipulation (matrix representation of
polynomial equation)
• Sparse matrix
▪ Sparse matrix and its representation

One Dimensional Array
🞂 Simplest data structure that makes use of computed address to locate its
elements is the one-dimensional array or vector.
🞂 Number of memory locations is sequentially allocated to the vector.
🞂 A vector size is fixed and therefore requires a fixed number of memory
locations.
🞂 Vector A with subscript lower bound of “one” is represented as below….
• L0 is the address of the first word allocated to the first element of vector A
• C words is size of each element or node
• The address of element Ai is
Loc(Ai) = L0 + (C*(i-1))
• Let’s consider the more general case of a vector A with lower bound for it’s
subscript is given by some variable b.
• The address of element Ai is
Loc(Ai) = L0 + (C*(i-b))
A[i]
L
0
L0+(i-1)C
i-1

Two Dimensional Array
🞂 Two dimensional arrays are also called table or matrix
🞂 Two dimensional arrays have two subscripts
🞂 Column major order matrix: Two dimensional array in which elements are
stored column by column is called as column major matrix
🞂 Two dimensional array consisting of two rows and four columns is stored
sequentially by columns :
A[1,1],
A[2,1],
A[1,2],
A[2,2],
A[1,3],
A[2,3],
A[1,4],
A[2,4]
[1,1] [1,2] [1,3] [1,4]
[2,1] [2,2] [2,3] [2,4]
Row
1
Row
2
Col-
1
Col-
2
Col-
3
Col-
4

Column major order matrix
🞂 The address of element A [ i , j ] can be obtained by expression
Loc (A [ i , j ]) = L0 + (j-1)*2 + (i-1)
Loc (A [2, 3]) = L0 + (3-1)*2 + (2-1) = L0 + 5
🞂 In general for two dimensional array consisting of n rows and m columns the
address element A [ i , j ] is given by
Loc (A [ i , j ]) = L0 + (j-1)*n + (i – 1)
[1,1] [1,2] [1,3] [1,4]
[2,1] [2,2] [2,3] [2,4]
Row
1
Row
2
Col-
1
Col-
2
Col-
3
Col-
4
[1,1]
[2,1]
[1,2]
[2,2]
[1,3]

Applications of Array
1. Symbol Manipulation (matrix representation of polynomial equation)
2. Sparse Matrix
🞂 Matrix representation of polynomial equation
⮩ We can use array for different kind of operations in polynomial equation such as addition,
subtraction, division, differentiation etc…
⮩ We are interested in finding suitable representation for polynomial so that different
operations like addition, subtraction etc… can be performed in efficient manner.
⮩ Array can be used to represent Polynomial equation.

0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
X
X2
X3
X4
Y Y2
Y3 Y4
Representation of Polynomial equation
2X2
+ 5XY + Y2
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
X
X2
X3
X4
Y Y2
Y3 Y4
2
5
1
X2
+ 3XY + Y2
+Y-X
1
3
1
1
-1
Y Y2
Y3 Y4
X XY XY2
XY3
XY4
X2
X3
Y X2
Y2
X2
Y3
X2
Y4
X3
X3
Y X3
Y2
X3
Y3
X3
Y4
X4 X4
Y X4
Y2
X4
Y3
X4
Y4

Sparse matrix
🞂 An m x n matrix is said to be sparse if “many” of its elements are zero.
🞂 A matrix that is not sparse is called a dense matrix.
🞂 We can device a simple representation scheme whose space requirement
equals the size of the non-zero elements.
0 0 0 2 0 0 1 0
0 6 0 0 7 0 0 3
0 0 0 9 0 8 0 0
0 4 5 0 0 0 0 0
Terms
Row
Column
Value
4x8
Row - 1
Row - 2
Row - 3
Row - 4
Column
-
1
C
olumn
-
2
C
olumn
-
3
C
olumn
-
4
C
olumn
-
5
C
olumn
-
6
C
olumn
-
7
C
olumn
-
8
Linear Representation of given matrix
0
1
4
2
1
1
7
1
2
2
2
6
3
2
5
7
4
2
8
3
5
3
4
9
6
3
6
8
7
4
2
4
8
4
3
5

Sparse matrix Cont…
🞂 To construct matrix structure from liner representation we need to record.
⮩ Original row and columns of each non zero entries.
⮩ Number of rows and columns in the matrix.
🞂 So each element of the array into which the sparse matrix is mapped need to
have three fields: row, column and value

0 0 6 0 9 0 0
2 0 0 7 8 0 4
10 0 0 0 0 0 0
0 0 12 0 0 0 0
0 0 0 0 0 0 0
0 0 0 3 0 0 5
Row Colum
n
A
6x7
1
2
3
4
5
6
1 2 3 4 5 6 7
Memory Space required to
store
6x7 matrix
42 x 2 = 84 bytes
Memory Space required to
store
Linear Representation
30 x 2 = 60 bytes
Linear representation of
Matrix
A =
1 3 6
1 5 9
2 1 2
2 4 7
2 5 8
2 7 4
3 1 10
4 3 12
6 4 3
6 7 5
Space Saved = 84 – 60 = 24 bytes

Row
1
1
2
2
2
2
3
4
6
6
Colum
n
3
5
1
4
5
7
1
3
4
7
A
6
9
2
7
8
4
10
12
3
5
Linear Representation of
Matrix Colum
n
3
5
1
4
5
7
1
3
4
7
A
6
9
2
7
8
4
10
12
3
5
Linear Representation of
Matrix
Row
1
3
7
8
0
9
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
Memory Space required to store Liner Representation = 26 x 2 = 42 bytes

Data strutcure and annalysis topic array

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