Polytope model


The polyhedral model is a mathematical framework for programs that perform large numbers of operations -- too large to be explicitly enumerated -- thereby requiring a compact representation. Nested loop programs are the typical, but not the only example, and the most common use of the model is for loop nest optimization in program optimization. The polyhedral method treats each loop iteration within nested loops as lattice points inside mathematical objects called polyhedra, performs affine transformations or more general non-affine transformations such as tiling on the polytopes, and then converts the transformed polytopes into equivalent, but optimized, loop nests through polyhedra scanning.

Simple example

Consider the following example written in C:

const int n = 100;
int i, j, a;
for

The essential problem with this code is that each iteration of the inner loop on a requires that the previous iteration's result, a, be available already. Therefore, this code cannot be parallelized or pipelined as it is currently written.
An application of the polytope model, with the affine transformation and the appropriate change in the boundaries, will transform the nested loops above into:

a = a + a;

In this case, no iteration of the inner loop depends on the previous iteration's results; the entire inner loop can be executed in parallel.

Detailed example

The following C code implements a form of error-distribution dithering similar to Floyd–Steinberg dithering, but modified for pedagogical reasons. The two-dimensional array src contains h rows of w pixels, each pixel having a grayscale value between 0 and 255 inclusive. After the routine has finished, the output array dst will contain only pixels with value 0 or value 255. During the computation, each pixel's dithering error is collected by adding it back into the src array.
Each iteration of the inner loop modifies the values in src based on the values of src, src, and src. We can illustrate the dependencies of src graphically, as in the diagram on the right.
Performing the affine transformation on the original dependency diagram gives us a new diagram, which is shown in the next image. We can then rewrite the code to loop on p and t instead of i and j, obtaining the following "skewed" routine.