Access normalization: loop restructuring for NUMA computers
ACM Transactions on Computer Systems (TOCS)
Probabilistic Loop Scheduling for Applications with Uncertain Execution Time
IEEE Transactions on Computers
Properties and Algorithms for Unfolding of Probabilistic Data-Flow Graphs
Journal of VLSI Signal Processing Systems
Loop Transformation Using Nonunimodular Matrices
IEEE Transactions on Parallel and Distributed Systems
Mapping normalization technique on the HPF compiler fhpf
ISHPC'05/ALPS'06 Proceedings of the 6th international symposium on high-performance computing and 1st international conference on Advanced low power systems
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In this paper, we discuss a loop transformation framework that is based on integer non-singular matrices. The transformations included in this framework are called $\Lambda$-transformations and include permutation, skewing and reversal, as well as a transformation called loop scaling. This framework is more general than the existing ones; however, it is also more difficult to generate code in our framework. This paper shows how integer lattice theory can be used to generate efficient code. An added advantage of our framework over existing ones is that there is a simple completion algorithm which, given a partial transformation matrix, produces a full transformation matrix that satisfies all dependences. This completion procedure has applications in parallelization and in the generation of code for NUMA machines.