In row-major order, consecutive elements of the rows of the array are contiguously stored in memory; in column-major order, consecutive elements of the columns of the array are contiguously stored.
Array layout is critical for correctly passing arrays between programs written in different languages. It is also important for performance when traversing an array because accessing array elements that are contiguous in memory is usually faster than accessing elements which are not, due to caching. In some media such as tape or NAND flash memory, accessing sequentially is orders of magnitude faster than nonsequential access.
Explanation and example
would be stored as follows in the two orders:
To select the correct element one must know whether an array A[a][b] in a particular language is stored in a row-major (a subscript indexes rows, b subscript indexes columns) or in a column-major mode (a subscript indexes columns, b subscript indexes rows), eg. in C A would result in , while in Fortran it would result in .
Programming languages and libraries
Programming languages or their standard libraries that support multi-dimensional arrays typically have a native row-major or column-major storage order for these arrays.
Support for multi-dimensional arrays may also be provided by external libraries, which may even support arbitrary orderings, where each dimension has a stride value, and row-major or column-major are just two possible resulting interpretations.
As exchanging the indices of an array is the essence of array transposition, an array stored as row-major but read as column-major (or vice versa) will appear transposed. As actually performing this rearrangement in memory is typically an expensive operation, some systems provide options to specify individual matrices as being stored transposed.
Address calculation in general
The concept generalizes to arrays with more than two dimensions.
For a d-dimensional array with dimensions Nk (k=1...d), a given element of this array is specified by a tuple of d (zero-based) indices .
In row-major order, the last dimension is contiguous, so that the memory-offset of this element is given by:
In column-major order, the first dimension is contiguous, so that the memory-offset of this element is given by:
For a given order, the stride in dimension k is given by the multiplication value in parentheses before index nk in the right hand-side summations above.
More generally, there are d! possible orders for a given array, one for each permutation of dimensions (with row-major and column-order just 2 special cases), although the lists of stride values are not necessarily permutations of each other, e.g., in the 2-by-3 example above, the strides are (3,1) for row-major and (1,2) for column-major.
- Matrix representation
- Vectorization (mathematics), the equivalent of turning a matrix into the corresponding column-major vector
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