Circular-arc graph coloring: On chords and circuits in the meeting graph

On the effectiveness of register moves to minimise post-pass unrolling in software pipelined loops

“…Some state-of-art work [6], [9] studied the circular lifetime intervals as an alternative representation for register allocation.…”

“…To address this problem, we represent in Figure 1(c) an alternative representation called circular lifetime intervals (CLI) [6], [13], [9]. The x-axis represents the different clock cycles and the y-axis represents the live ranges of the different variables.…”

“…Many heuristics, exploiting the information provided in such representation are proposed in the litterature [6], [9]. The number of minimum registers needed for a circular interval is related to the width of the circular lifetime intervals.…”

“…This can have a dramatic impact by causing unnessary instruction cache misses when the code size of the loop happens to be larger than the size of the instruction cache. For simplicity, we did not expand the code to assign registers for b and c. In addition, brute force searching for the best solution using loop unrolling has a prohibitive cost, existing solutions may either sacrifice the register optimality [9], [12], [16] or incur large unrolling overhead [6], [17].…”

“…In this case, known methods exist for computing unroll factors: (1) Modulo Variable Expansion (MVE) [11], [12] which computes a minimal unroll factor but may introduce spill (since MVE may need more than MAXLIVE registers without proving an appropriate upper-bound); (2) Hendren's heuristic [9] which computes a sufficient unroll factor without introducing spill, but with no guarantee in terms of minimal register usage or unroll factor; (3) the Meeting Graph framework [6] whose unroll factor is formally proven to minimize register usage (MAXLIVE), but which does not guarantee a minimal unroll factor.…”