Optimal weighted loop fusion for parallel programs

Abstract

Much of the computation involved in parallel programs occurs within loops, either nested loops as in parallel scientific applications or collections of loops as in stream-based applications. Loop fusion is a well-known program transformation that has shown to be effective in improving data locality in parallel programs by reducing inter-processor communication and improving register and cache locality. Weighted loop fusion is the problem of finding a legal partition of loop nests into fusible clusters so as to minimize the total inter-cluster weights. The loop nests may contain parallel or sequential loops; care is taken to ensure that a parallel loop does not get serialized after fusion. It has been shown in past work that the weighted loop fusion problem is NP-hard. Despite the NP-hardness property, we show how optimal solutions can be found efficiently (i.e., within the compile-time constraints of a product-quality optimizing compiler) for weighted loop fusion problem sizes that occur in practice. In this paper, we present an integer programming formulation for weighted loop fusion with size (number of variables and constraints) that is linearly proportional to the size of the input weighted loop fusion problem. The linear-sized formulation is key to making the execution time small enough for use in a product-quality optimizing compiler, since the natural integer programming formulation for this problem has cubic size for which the execution time would be too large to be practical. The linear-sized integer programming formulation can be solved efficiently using any standard optimization package but we also present a custom branch-and-bound algorithm that can be used if greater efficiency is desired. A prototype implementation of this approach has been completed, and preliminary compile-time measurements are included in the paper as validation of the practicality of this approach.