Lecture 87:- Formulas for nD Arrays

The formulas for calculating the linear index of an element in an n-dimensional (nD) array depend on the memory layout or storage order used. The two common storage orders are row-major and column-major. Here are the formulas for each:

1. Row-Major Formula:

   • Linear index = (index1 * size2 * size3 * ... * sizeN) + (index2 * size3 * ... * sizeN) + ... + indexN

   In this formula:

   • index1, index2, ..., indexN are the indices of the element along each dimension, starting from the outermost dimension (dimension 1) to the innermost dimension (dimension N).
   • size2, size3, ..., sizeN are the sizes (number of elements) along each dimension, excluding the outermost dimension.

2. Column-Major Formula:

   • Linear index = (indexN * size1 * size2 * ... * sizeN-1) + (indexN-1 * size1 * size2 * ... * sizeN-2) + ... + index1

   In this formula:

   • indexN, indexN-1, ..., index1 are the indices of the element along each dimension, starting from the innermost dimension (dimension N) to the outermost dimension (dimension 1).
   • size1, size2, ..., sizeN-1 are the sizes (number of elements) along each dimension, excluding the innermost dimension.

Note that these formulas assume 0-based indexing, where the indices start from 0.

The formulas can be extended for higher-dimensional arrays by adding more terms for each dimension. The principle remains the same: multiplying each index by the appropriate size and summing them to obtain the linear index.

It's important to use the correct formula based on the storage order used for the nD array. The choice of storage order depends on the programming language, library, or specific requirements of the application.