The difference is simply this: in row-major order, rows of the array are contiguous in memory; in column-major order, the columns are contiguous.
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 flash memory, accessing sequentially is orders of magnitude faster than nonsequential access.
Explanation and example
Following conventional matrix notation, rows are numbered by the first index of a two-dimensional array and columns by the second index, i.e., a1,2 is the second element of the first row, counting downwards and rightwards. (Note this is the opposite of Cartesian conventions.)
The difference between row-major and column-major order is simply that the order of the dimensions is reversed. Equivalently, in row-major order the rightmost indices vary faster as one steps through consecutive memory locations, while in column-major order the leftmost indices vary faster.
Would be stored as follows in the two orders:
Programming languages which support multi-dimensional arrays have a native storage order for these arrays.
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 trivially 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:
- Matrix representation
- Vectorization (mathematics), the equivalent of turning a matrix into the corresponding column-major vector.
- MATLAB documentation, MATLAB Data Storage (retrieved from Mathworks.co.uk, January 2014).
- Spiegelhalter et al. (2003, p. 17): Spiegelhalter, David; Thomas, Andrew; Best, Nicky; Lunn, Dave (January 2003), "Formatting of data: S-Plus format", WinBUGS User Manual (Version 1.4 ed.), Robinson Way, Cambridge CB2 2SR, UK: MRC Biostatistics Unit, Institute of Public Health, PDF document
- An Introduction to R, Section 5.1: Arrays (retrieved March 2010).
- Donald E. Knuth, The Art of Computer Programming Volume 1: Fundamental Algorithms, third edition, section 2.2.6 (Addison-Wesley: New York, 1997).