Compile-time Loop Splitting for Distributed Memory ... - Stanford AI Lab

aref (aref (A, div-a-i), rem-a-i);
aref (aref (B, div-b-i + (i-n * div-b-j)),
rem-b-i + (i-spread * rem-b-j));
aref (aref (C, div-c-i + i-proc * (div-c-j + (j-proc * div-c-k))),
rem-c-i + i-spread * (rem-c-j + j-spread * rem-c-k));
Figure 2-8: The same references with all the divisions replaced by interval invariants and
all modulos replaced by an incrementing counter.
aref (aref (B, pid-B-ij), rem-b-i + (i-spread * rem-b-j));
aref (aref (C, pid-C-ijk), rem-c-i + i-spread * (rem-c-j + j-spread * rem-c-k);
Figure 2-9: The 2-D and 3-D reference expressions after further compiler optimization.
each reference.
An optimizing compiler could even further reduce the calculation by replacing the
multiplication by � ����—�in the second reference (� ����—�in the third reference) with an
addition of � ����—�(� ����—�) on each iteration. This type of optimization, called strength
reduction, is described in Section 3.3.2.
In the context of rectangular partitioning, “sufficiently small” interval values are
those that keep the loop nest occupied with a single processor. If this condition is met, the
processor ID is constant, and the offset into the processor’s memory can be determined
with monotonically increasing or decreasing counters. Thus, appropriate intervals can
Number of Operations Per Array Reference
Reference Before Optimization After Optimization
mod 4 2 C mod 4 2 C
1-D 1 1 0 0 0 0 0 0
2-D 2 2 2 2 0 0 1 1
3-D 3 3 4 4 0 0 2 2
Table 2.3: Reduction in operations for array references by an optimizing compiler.
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