A Static Data Dependence Analysis Approach for Software Pipelining

Abstract

This paper introduces a new static data dependence constraint, called dependence difference inequality, which can deal with coupled subscripts for multi-dimensional array references. Unlike direction vectors, dependence difference inequalities are related to not only the iteration space for a loop program but also the operation distance between two operations. They are more strict than other methods, and can act as additional constraints to each variable in a linear system on their own or with others. As a result, the solution space for a linear system can be compressed heavily. So long as dependence difference inequalities do not satisfy simultaneously, the loop can be software-pipelined with any initiation interval even if there exists a data dependence between two operations. Meanwhile, by replacing direction vectors with dependence difference inequalities some conservative estimations made by other traditional data dependence analysis approaches can be eliminated.

A Dynamic Data Dependence Analysis Approach for Software Pipelining

“…op6 (3) op7 (2) op1 (5) op5 (4) op8 (1) RTPAD(op5 (5) , op8 (2) ) RTPAD(op5 (5) , op8 (3) ) RTPAD(op5 (5) , op8 (4) ) 6…”

“…op6 (6) op7 (5) op1 (8) op9 (3) op5 (7) op8 (4) RTPAD(op5 (8) , op8 (5) ) RTPAD(op5 (8) , op8 (6) ) RTPAD(op5 (8) , op8 (7) ) 9 op6 (7) op7 (6) op1 (9) op9 (4) op5 (8) op8 (5) RTPAD(op5 (9) , op8 (6) ) RTPAD(op5 (9) , op8 (7) ) RTPAD(op5 (9) , op8 (8) ) Table 2. An address conflict between op 5 (6) and op 8 (4) is detected…”

Coping with Data Dependencies of Multi-dimensional Array References

“…Mathematically the problem can be reduced to that of checking whether or not a system of m linear equations with 2n unknown variables has a simultaneous integer solution, which satisfies the constraints for each variable in the system. It has been proved that a loop can be software-pipelined with any value of initiation interval if the dependence difference inequalities do not satisfy simultaneously [3]. That is to say, dependence difference inequalities can act as additional constraints to each variable in the system of m linear equations on their own or with other constraints, such as direction vectors.…”

“…The paper is the ongoing work of [3]. We focuses on applying dependence difference inequalities for analyzing data dependence.…”

“…(2) and Eq. (5) presented in[3]. Given a line in R m corresponding to an equation vλ T = 0, if F λ = 0 intersects V in R m for any fixed point λ ≠ 0 in the line, such as its canonical solution, then for every λ in the line F λ = 0 also intersects V.Theorem 1 shows that there are at most 2n ψ-planes and n φ-planes, and there are determined by these hyperplanes.…”