On the Alignment Problem

This paper deals with the problem of aligning data and computations when mapping uniform or affine loop nests onto SPMD distributed memory parallel computers. For affine loop nests we formulate the problem by introducing the communication graph, which can be viewed as the counterpart for the mapping problem of the dependence graph for scheduling. We illustrate the approach with several examples to show the difficulty of the problem. In the simplest case, that of perfect loop nests with uniform dependences, we show that minimizing the number of communications is NP-complete, although we are able to derive a good alignment heuristic in most practical cases.