Nested Circular Intervals: A Model for Barrier Placement in Single-Program, Multiple-Data Codes with Nested Loops.

Abstract

(eng) We want to perform compile-time analysis of an SPMD program and place barriers in it to synchronize it correctly, minimizing the runtime cost of the synchronization. This is the barrier minimization problem. No full solution to the problem has been given previously. Here we model the problem with a new combinatorial structure, a nested family of sets of circular intervals. We show that barrier minimization is equivalent to finding a hierarchy of minimum cardinality point sets that cut all intervals. For a single loop, modeled as a simple family of circular intervals, a linear-time algorithm is known. We extend this result, finding a linear-time solution for nested circular intervals families. This result solves the barrier minimization problem for general nested loops.