Graphs and algorithms
Software pipelining: an effective scheduling technique for VLIW machines
PLDI '88 Proceedings of the ACM SIGPLAN 1988 conference on Programming Language design and Implementation
On optimal parallelization of arbitrary loops
Journal of Parallel and Distributed Computing
Circular scheduling: a new technique to perform software pipelining
PLDI '91 Proceedings of the ACM SIGPLAN 1991 conference on Programming language design and implementation
An efficient resource-constrained global scheduling technique for superscalar and VLIW processors
MICRO 25 Proceedings of the 25th annual international symposium on Microarchitecture
Lifetime-sensitive modulo scheduling
PLDI '93 Proceedings of the ACM SIGPLAN 1993 conference on Programming language design and implementation
Rotation scheduling: a loop pipelining algorithm
DAC '93 Proceedings of the 30th international Design Automation Conference
Minimum register requirements for a modulo schedule
MICRO 27 Proceedings of the 27th annual international symposium on Microarchitecture
Specification of software pipelining using Petri nets
International Journal of Parallel Programming
Decomposed software pipelining: a new perspective and a new approach
International Journal of Parallel Programming
ACM Computing Surveys (CSUR)
Circuit Retiming Applied to Decomposed Software Pipelining
IEEE Transactions on Parallel and Distributed Systems
Computer architecture (2nd ed.): a quantitative approach
Computer architecture (2nd ed.): a quantitative approach
The IA-64 Architecture at Work
Computer
Perfect Pipelining: A New Loop Parallelization Technique
ESOP '88 Proceedings of the 2nd European Symposium on Programming
MICRO 14 Proceedings of the 14th annual workshop on Microprogramming
Swing Modulo Scheduling: A Lifetime-Sensitive Approach
PACT '96 Proceedings of the 1996 Conference on Parallel Architectures and Compilation Techniques
Tiling and Memory Reuse for Sequences of Nested Loops
Euro-Par '02 Proceedings of the 8th International Euro-Par Conference on Parallel Processing
Journal of VLSI Signal Processing Systems
Improving the parallelism of iterative methods by aggressive loop fusion
The Journal of Supercomputing
Software Pipelining in Nested Loops with Prolog-Epilog Merging
HiPEAC '09 Proceedings of the 4th International Conference on High Performance Embedded Architectures and Compilers
Unrolling and retiming of stream applications onto embedded multicore processors
Proceedings of the 49th Annual Design Automation Conference
Sub-polyhedral scheduling using (unit-)two-variable-per-inequality polyhedra
POPL '13 Proceedings of the 40th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages
The resource-constrained modulo scheduling problem: an experimental study
Computational Optimization and Applications
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The idea of decomposed software pipelining is to decouple the software pipelining problem into a cyclic scheduling problem without resource constraints and an acyclic scheduling problem with resource constraints. In terms of loop transformation and code motion, the technique can be formulated as a combination of loop shifting and loop compaction. Loop shifting amounts to moving statements between iterations thereby changing some loop independent dependences into loop carried dependences and vice versa. Then, loop compaction schedules the body of the loop considering only loop independent dependences, but taking into account the details of the target architecture. In this paper, we show how loop shifting can be optimized so as to minimize both the length of the critical path and the number of dependences for loop compaction. The first problem is well-known and can be solved by an algorithm due to Leiserson and Saxe. We show that the second optimization (and the combination with the first one) is also polynomially solvable with a fast graph algorithm, variant of minimum-cost flow algorithms. Finally, we analyze the improvements obtained on loop compaction by experiments on random graphs.