The communication pattern of a set of MPI processes can be represented by a graph. The nodes represent MPI processes, and the edges connect MPI processes that communicate with each other. MPI provides message-passing between any pair of MPI processes in a group. There is no requirement for opening a channel explicitly. Therefore, a ``missing link'' in the user-defined graph of MPI processes does not prevent the corresponding MPI processes from exchanging messages. It means rather that this connection is neglected in the virtual topology. This strategy implies that the virtual topology gives no convenient way of naming this pathway of communication. Another possible consequence is that an automatic mapping tool (if one exists for the runtime environment) will not take account of this edge when mapping.
Specifying the virtual topology in terms of a graph is sufficient for all applications. However, in many applications the graph structure is regular, and the detailed set-up of the graph would be inconvenient for the user and might be less efficient at run time. A large fraction of all parallel applications use MPI process topologies like rings, two- or higher-dimensional grids, or tori. These structures are completely defined by the number of dimensions and the numbers of MPI processes in each coordinate direction. Also, the mapping of grids and tori is generally an easier problem than that of general graphs. Thus, it is desirable to address these cases explicitly.
The coordinates of MPI processes in a Cartesian structure begin their numbering at 0. Row-major numbering is always used for the MPI processes in a Cartesian structure. This means that, for example, for four MPI processes in a (2 × 2) grid, the relationship between their ranks in the group and their coordinates in the virtual topology is as follows:
| coord (0,0): | rank 0 |
| coord (0,1): | rank 1 |
| coord (1,0): | rank 2 |
| coord (1,1): | rank 3 |