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

Chapter 3
Overview of Loop Splitting
Now that array referencing on multiprocessors has been presented, this section continues
the background presentation by defining the term loop splitting. This includes describing
the loop and code transformations comprising loop splitting as well as its application to
simplifying array reference expressions.
3.1 Introduction
As supported by Section 2.5, array references can require involved calculations. For
example, in a simple matrix transpose (2-D loop with body A[i][j] = B[j][i]), the assignment
requires 4 each of modulos, divisions, multiplications, and additions. In matrix addition
(A[i][j] = B[i][j] + C[i][j]), of the 6 modulos, 6 divisions, 6 multiplications, and 7 additions,
only one addition was specified by the programmer. All of this array reference overhead
significantly inhibits a loop nest’s execution performance.
However, loop splitting is an attempt to minimize this overhead by altering the loop
structure. By modifying the intervals of the loop nest and introducing another nesting
level, not only are the array reference operations fewer in number, but they are also no
longer performed on every iteration. This reduction in calculation leads to faster program
execution time.
This section presents sufficient descriptions of the components of loop splitting.
Section 3.2 describes the two loop transformations composing loop splitting, and Section
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