Non-uniform dependences partitioned by recurrence chains

Nonuniform distance loop dependences are a known obstacle to find parallel iterations. To find the outermost loop parallelism in these "irregular" loops, a novel method is presented based on recurrence chains. The scheme organizes nonuniformly dependent iterations into lexicographically ordered monotonic chains. While the initial and final iterations of monotonic chains form two parallel sets, the remaining iterations form an intermediate set that can be partitioned further. When there is only one pair of coupled array references, the nonuniform dependences are represented by a single recurrence equation. In that case, the chains in the intermediate set do not bifurcate and each can be executed as a WHILE loop. The independent and the initial iterations of monotonic dependence chains constitute the outermost parallelism. The proposed approach compares favorably with other treatments of nonuniform dependences in the literature. When there are multiple recurrence equations, a dataflow parallel execution can be scheduled using the technique to find maximum loop parallelism.