New Complexity Results on Array Contraction and Related Problems

Array contraction is an optimization that transforms array variables into scalar variables within a loop. While the opposite transformation, scalar expansion, is used for enabling parallelism (with a penalty in memory size), array contraction is used to save memory by removing temporary arrays and to increase locality. Several heuristics have already been proposed to perform array contraction through loop fusion and/or loop shifting. But so far, the complexity of the problem was unknown, and no exact approach was available (and even more, only a sufficient condition for array contraction was used). In this report, we focus on the theoretical aspects of the problem. We prove several NP-complete results that characterize precisely its complexity and we provide an integer linear programming formulation to solve the problem exactly. Our study also proves the NP-completeness of similar problems whose complexity was not established so far.