Row Major Order and Column Major Order

When it comes to organizing and accessing elements in a multi-dimensional array, two prevalent methods are Row Major Order and Column Major Order. These approaches define how elements are stored in memory and impact the efficiency of data access in computing.

Let us discuss them in detail one by one.

Row Major Order

Row major ordering assigns successive elements, moving across the rows and then down the next row, to successive memory locations. In simple language, the elements of an array are stored in a Row-Wise fashion.
To find the address of the element using row-major order uses the following formula:

Address of A[I][J] = B + W * ((I – LR) * N + (J – LC))   

I = Row Subset of an element whose address to be found, 
J = Column Subset of an element whose address to be found, 
B = Base address, 
W = Storage size of one element store in an array(in byte), 
LR = Lower Limit of row/start row index of the matrix(If not given assume it as zero), 
LC = Lower Limit of column/start column index of the matrix(If not given assume it as zero), 
N = Number of column given in the matrix.

How to find address using Row Major Order? 

Given an array, arr[1………10][1………15] with base value 100 and the size of each element is 1 Byte in memory. Find the address of arr[8][6] with the help of row-major order.

Solution:

Given:
Base address B = 100
Storage size of one element store in any array W = 1 Bytes
Row Subset of an element whose address to be found I = 8
Column Subset of an element whose address to be found J = 6
Lower Limit of row/start row index of matrix LR = 1 
Lower Limit of column/start column index of matrix = 1
Number of column given in the matrix N = Upper Bound – Lower Bound + 1
                                                                            = 15 – 1 + 1
                                                                            = 15

Formula:
Address of A[I][J] = B + W * ((I – LR) * N + (J – LC)) 

Solution:
Address of A[8][6] = 100 + 1 * ((8 – 1) * 15 + (6 – 1))
                                   = 100 + 1 * ((7) * 15 + (5))
                                  = 100 + 1 * (110)
Address of A[I][J] = 210

Column Major Order

If elements of an array are stored in a column-major fashion means moving across the column and then to the next column then it’s in column-major order. To find the address of the element using column-major order use the following formula:

Address of A[I][J] = B + W * ((J – LC) * M + (I – LR))  

I = Row Subset of an element whose address to be found, 
J = Column Subset of an element whose address to be found, 
B = Base address, 
W = Storage size of one element store in any array(in byte), 
LR = Lower Limit of row/start row index of matrix(If not given assume it as zero), 
LC = Lower Limit of column/start column index of matrix(If not given assume it as zero), 
M = Number of rows given in the matrix.

How to find address using Column Major Order?

Given an array arr[1………10][1………15] with a base value of 100 and the size of each element is 1 Byte in memory find the address of arr[8][6] with the help of column-major order.

Solution:

Given:
Base address B = 100
Storage size of one element store in any array W = 1 Bytes
Row Subset of an element whose address to be found I = 8
Column Subset of an element whose address to be found J = 6
Lower Limit of row/start row index of matrix LR = 1
Lower Limit of column/start column index of matrix = 1
Number of Rows given in the matrix M = Upper Bound – Lower Bound + 1
                                                                            = 10 – 1 + 1
                                                                           = 10

Formula: used
Address of A[I][J] = B + W * ((J – LC) * M + (I – LR))
Address of A[8][6] = 100 + 1 * ((6 – 1) * 10 + (8 – 1))
                                  = 100 + 1 * ((5) * 10 + (7))
                                 = 100 + 1 * (57)
Address of A[I][J] = 157 

From the above examples, it can be observed that for the same position two different address locations are obtained that’s because in row-major order movement is done across the rows and then down to the next row, and in column-major order, first move down to the first column and then next column. So both the answers are right.

Row Major Order vs Column Major Order

Aspect	Row Major Order	Column Major Order
Memory Organization	Elements are stored row by row in contiguous locations.	Elements are stored column by column in contiguous locations.
Memory Layout Example	For a 2D array A[m][n]: [A[0][0], A[0][1], …, A[m-1][n-1]]	For the same array: [A[0][0], A[1][0], …, A[m-1][n-1]]
Traversal Direction	Moves through the entire row before progressing to the next row.	Moves through the entire column before progressing to the next column.
Access Efficiency	Efficient for row-wise access, less efficient for column-wise access.	Efficient for column-wise access, less efficient for row-wise access.
Common Use Cases	Commonly used in languages like C and C++.	Commonly used in languages like Fortran.
Applications	Suitable for row-wise operations, e.g., image processing.	Suitable for column-wise operations, e.g., matrix multiplication.

So it’s all based on the position of the element whose address is to be found for some cases the same answers is also obtained with row-major order and column-major order and for some cases, different answers are obtained.