Similar to strip mining (Strip mining works for single loops)
Loop tiling is used for nested loops
Tiling boundaries are parallel to the iteration space axes and not to iteration space boundaries
The eventual goal is to interchange tile loops outward and element loops inward
Tiling is characterized by tile size ts and a tile offset to (0 ≤ to < ts)
Each tile starts an iteration i such that i mod ts = to
Each tile iterates from tn-ts+to to (tn+1)-ts+to-1 where tn is tile number
The compiler must determine the minimum and maximum tile numbers
The compiler must ensure that element loop does not execute outside its original iteration space
The general formula for tiling for a loop such as
for I = lo, hi
is
for it = floor((lo-to)/ts)*ts+to, floor((hi-to)/ts)*ts+to, ts
for I = max(lo,it), min(hi,it+ts-1)
Tile following loops with a tile
size of 20 and an offset of 5
For I = 1, 50
for j = i, 60
A[I,j] = A[I,j]+1
endfor
endfor
just applying the formula produces following loop
For It = -15, 45, 20
for i = max(1,it), min(50,it+19)
for jt=floor((i-5)/20)*20+5, 45, 20
for j=max(I,jt), min(60, jt+19)
A[I,j] = A[I,j]+1
endfor
endfor
endfor
endfor
