1. many subjects (DFS, C++, SQL)
Array
Store marks of
1 subject (DFS)
for many (5) students.
Store marks of
many subjects (DFS, C++, SQL)
for many (5) students.
Store marks of
1 subject for 1 student.
Array
DFS
Array
C++
Array
SQL
Array
DFS 10 9 12 13 25 25 30 45 40 35 40 42 41 45 48
int 10 10 9 12 13 25
Array would be
preferred.
1 integer variable
is sufficient.
Array Marks
Array
DFS 2D
Array 10 9 12 13 25 25 30 45 40 35 40 42 41 45 48 10 9 12 13 25
Matrix

2. Array Array: To store marks of 1 subject for 1 student:
1 Integer variable is sufficient.
To store marks of 1 subject for many students:
Array would be preferred.
1 dimensional array is sufficient.
To store marks of many subjects of many students:
2-dimensional array / Matrix is required.

3. 2-D Array Index: A[ i ][ j ] C: int A[3][3] Position: APrPc
Fortran: int A[3][3]
Pascal: int A[-1...1][1...3]
Col (Pc)
Position 1 2 3 C
Col (j)
Row
(Pr)
Index 1 2
Row (i) 1 10 12 9
A[0][0]
A[0][2]
A[0][1]
A11
A13
A12 2 1 11 5 2
Rows
A[1][0]
A[1][1]
A[1][2]
A21
A22
A23 15 10 3 3 2
A[2][0]
A[2][1]
A[2][2]
Size
A31
A32
A33
3 x 3
m x n
Array A
Cols

4. 2-D Array Array A[-2...1][-3...-1] W = 2 Questions: Pc Pos 1 2 3
What is the value
of m, number of rows? j Pr
Index -3 -2 -1 i 4 1 -2
A[-2][-3]
A[-2][-2]
A[-2][-1]
What is the value
of n, number of cols?
A11
A12
A13 3 2 -1
A[-1][-3]
A[-1][-2]
A[-1][-1]
How many elements
are there in A?
A21
A23
A22
m x n
A[0][-3]
A[0][-2]
A[0][-1] 3
4 x 3
= 12
A31
A32
A33
What is the memory
occupied by entire array?
A[1][-3]
A[1][-2]
A[1][-1] 4 1
A41
12 x 2
= 24
A42
A43
4 x 3

5. 2-D Array Array A[-23...17][-37...11] W = 4 Pc Pos 1 2 . . . j Pr
Index
-37
-36
. . .
Questions: i 1
-23
How many elements
are there in A?
m x n 2
-22
What is the memory
occupied by entire array? 3
-21 . .
Difficult because the array is large.
Equations/Formulas are needed again.

6. 2-D Array Array A [-23...17] [-37...11] Size = U – L + 1 Row Array
Column Array
Index
-37
-36
-35
-34
-23
-22
-21
-20 .

7. 2-D Array Array A [-23...17] [-37...11] W = 2 Size = U – L + 1
Row Array
Column Array
Row Array
Col Array
Lr = -23
Lc = -37
Ur = 17
Uc = 11
Size(Ar) = Ur – Lr + 1
Size(Ac) = Uc – Lc + 1
Size(Ac) = 11 – (-37) + 1
Size(Ar) = 17 – (-23) + 1
Size(Ac) =
Size(Ar) =
Size(Ac) = 49
Size(Ar) = 41
n = 49
m = 41
Size(A) = m x n = 41 x 49 = 2009
Total Memory Occupied = Size x W = 2009 x 2 = 4018

8. 2-D Array Array A [-100...99] [-5...-1] W = 4 Size = U – L + 1
Find out the size and total memory occupied by
the array A[ ][ ] having word size of 4.
Question:
Array A [ ] [ ]
W = 4
Row Array
Column Array
Lr = -100
Lc = -5
Ur = 99
Uc = -1
Size(Ar) = Ur – Lr + 1
Size(Ac) = Uc – Lc + 1
Size(Ac) = -1 – (-5) + 1
Size(Ar) = 99 – (-100) + 1
Size(Ac) =
Size(Ar) =
Size(Ac) = 5
Size(Ar) = 200
n = 5
m = 200
Size(A) = m x n = 200 x 5 = 1000
Total Memory Occupied = Size x W = 1000 x 4 = 4000

9. 2-D Array Row by Row / Row-wise Column by Column / Column-wise 1
M = 1, W = 1
A11
A11 2
A12
A21 3
A13
A31
A11
A12
A13
A14 4
A14
A12 5
A21
A22 6
A21
A22
A23
A24
A22
A32 7
A23
A13 8
A24
A23
A31
A32
A33
A34 9
A31
A33 10
3 x 4
A32
A14 11
Will it be stored in memory?
Yes
A33
A24 12
On Continuous locations?
Yes
A34
A34
Row-major order
Column-major order

10. 2-D Array Memory representation of Matrix /
Storage mechanism of 2D array:
2 ways to store a 2D array / Matrix in memory:
Row-major order:
All elements are stored row by row / row-wise.
Example:
First row, Second row and so on.
Column-major order:
All elements are stored column by column / column-wise.
First column, Second column and so on.

11. 2-D Array Array A[-2...1][-1...1] M = 1000 W = 2 Row Major Order
Questions:
A11
1000
What is the address
location of A[0][0]?
A[-2][-1]
A11
A[-2][0]
A12
A[-2][1]
A13
A12
1002
1014
A13
1004
A21
1006
What is the address
location of A33?
A[-1][-1]
A21
A[-1][0]
A22
A[-1][1]
A23
A22
1008
1016
A23
1010
A31
1012
What is the address
location of A32?
A[0][-1]
A31
A[0][0]
A32
A[0][1]
A33
A32
1014
1014
A33
1016
A[1][-1]
A41
A[1][0]
A42
A[1][1]
A43
A41
1018
A42
1020
4 x 3
A43
1022

12. 2-D Array SPARSE MATRIX Sparse means ‘Not Dense’ / ‘Not Thick’ 10 - -9 - - - - - - - - - - - 2 - - - - - 1 12 - - - 11
Majority of the elements are NULL in this matrix.
Storing of NULL elements is a wastage of memory.
We need a technique by which only non-null elements are stored.

13. Classification of Sparse Matrix
Based on the availability of NON-NULL elements.
Sparse Matrix
Asymmetric Matrix
(Pattern)
Symmetric Matrix
Non-symmetric Matrix
Band
Triangular
Tridiagonal
Lower
Upper
Diagonal
Lower-
Left
Lower-
Right
Upper-
Left
Upper-
Right

14. Classification of Sparse Matrix
Lower-Left
Lower-Right
Upper-Left
Upper-Right
Null
* * * * *
* * * * *
* * * * *
* * * * *
Non-Null * * * * * * *
Diagonal * * * * * * * * * * * * * * * * * * * * * * * * * * *
Tri-Diagonal * * * * * * * * * * * * * * * * * * * * * * * *

15. Application of 2D Array / Matrix
Representation of Polynomials using Array
Polynomial in 2 variables
Co-efficient
2x2y2 + 3x2y + 5xy2 – 10xy – 15x + 20y - 100
-10 5
-100 20 - 3 2
-15
-10 5
Dimension? - 3 2
3 x 3
Axiyj ->
Store the co-efficient A in i th row and j th column.
Not Working
Axiyj -> Store the co-efficient A in (i+1) th row and (j+1) th column.
Processing?
Processing on Polynomials (Addition, Subtraction) can be easily done by
manipulating Matrices.

16. Application of 2D Array / Matrix
Representation of Polynomials using Array
Polynomial in 2 variables
10x4y3 + 5xy - 50
Axiyj -> Store the co-efficient A in (i+1) th row and (j+1) th column.
Sparse Matrix
-50 - - -
Lot of memory wastage. - 5 - - - - - - - - - - - - - 10
5 x 4