Fusing a triangle loop for parallelization, calculating sub-indices

Question

A common technique in parallelization is to fuse nested for loops like this

for(int i=0; i

to

Answer 1

Considering that you're trying to fuse a triangle with the intent of parallelizing, the non-obvious solution is to choose a non-trivial mapping of x to (i,j):

j |\ i ->
  | \             ____
| |  \    =>    |\\   |
V |___\         |_\\__|

After all, you're not processing them in any special order, so the exact mapping is a don't care.

So calculate x->i,j as you'd do for a rectangle, but if i > j then { i=N-i, j = N-j } (mirror Y axis, then mirror X axis).

   ____
 |\\   |      |\           |\
 |_\\__|  ==> |_\  __  =>  | \
                  / |      |  \
                 /__|      |___\