Mesh Data Structure:

Data structure in distributed environment depends upon the part of matrix ${\bf X}$ that is residing on a node at the beginning of the computation. In a column-major structure $P_{in}$ would be equals to $P(MN,M)$. Therefore, implementation of (a) would consist of matrix transposes after a Fourier transform computations on each dimension. Implementation of (b) puts the transposes prior to the Fourier transform computations. In a row-major structure $P_{in}$ would be an $M \times N$ identity matrix, and so is the ${P_{in}}^{-1}$.