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# MPI Virtual topologies
**Virtual topologies** register grid structures so `MPI_Cart_shift` computes neighbor ranks instead of requiring manual arithmetic.

```c
int row = rank / Q,  col = rank % Q;
int north = ((row - 1 + P) % P) * Q + col;
int south = ((row + 1)     % P) * Q + col;
```

This is rather fragile. Changing the process count or grid shape breaks the formula. `MPI_Cart_create` registers a Cartesian grid topology with MPI so that `MPI_Cart_shift` can compute neighbor ranks for you, including periodic wraparound.

```c
int dims[2] = {4, 4};
int periods[2] = {1, 1};   // periodic in both dimensions
MPI_Comm cart;
MPI_Cart_create(MPI_COMM_WORLD, 2, dims, periods, 1, &cart);

int north, south;
MPI_Cart_shift(cart, 0, 1, &north, &south);  // neighbors along row dimension
```

Topologies do not change the underlying communication; they are a convenience layer for structured grids and stencil codes where the natural pattern is nearest-neighbor exchange. `MPI_Cart_coords` converts a rank to its grid coordinates and `MPI_Cart_rank` does the reverse.