The problem of finding data decompositions
and computation decompositions
can be broken down
into three distinct components using the affine model.
The partition determines the computation and data
that are allocated to the same processor.
Mathematically, data and computation partitions are described by the
nullspace of the matrices D and C from Definitions 2.1
and 2.2.
The orientation, represented by the matrices D and C,
describes the mapping between the axes of the array elements and
loop iterations, and the processors.
Lastly, the displacement gives the offset
of the starting position of the data and computation, and corresponds
to the constant vectors 
and 
.
We illustrate the partition, orientation and displacement by
developing a communication-free decomposition for the sample program in
Figure 1.