Mapping of multidimensional array to a one-dimensional array

Question:

A multidimensional array can be mapped to a one-dimensional array in two ways: In one method, called row-major mapping, we store the multidimensional array row by row. In the second method, called column-major mapping, we store it column by column . For example, consider the following multidimensional array:

11        12        13

21        22        23

Its row-major mapping is:

11        12        13        21        22        23

And column-major is:

11        21        12        22        13        23

Now, given the following array declaration in a pascal like language:

   int arr[l1..h1][l2..h2][l3..h3];

where l1,l2, & l3 are lower indices and h1,h2, & h3 are upper indices, inclusive. The memory is allocated in column-major fashion. Write the mapping formula for  arr[i][j][k] where l1 <= i <= h1, l2 <= j <= h2, & l3 <= k <= h3. Assume all indices are non-negative.

Further help: The first element can be referred by arr[l1][l2][l3] and the last by arr[h1][h2][h3]

SOLUTION:

An element at index position (i,j,k) is mapped to the following index of the 1-D array:

(k-l₃)*(h₂-l₂+1)*(h₁-l₁+1)

            +

(j-l₂)*(h₁-l₁+1)

            +

(i-l₁)