A Framework for Resource-Constrained Rate-Optimal Software Pipelining
The rapid advances in high-performance computer architecture and compilation techniques provide both challenges and opportunities to exploit the rich solution space of software pipelined loop schedules. In this paper, we develop a framework to construct a software pipelined loop schedule which runs on the given architecture (with a fixed number of processor resources) at the maximum possible iteration rate (a la rate-optimal) while minimizing the number of buffers-a close approximation to minimizing the number of registers. The main contributions of this paper are: First, we demonstrate that such problem can be described by a simple mathematical formulation with precise optimization objectives under a periodic linear scheduling framework. The mathematical formulation provides a clear picture which permits one to visualize the overall solution space (for rate-optimal schedules) under different sets of constraints. Secondly, we show that a precise mathematical formulation and its solution does make a significant performance difference. We evaluated the performance of our method against three leading contemporary heuristic methods. Experimental results show that the method described in this paper performed significantly better than these methods. The techniques proposed in this paper are useful in two different ways: 1) As a compiler option which can be used in generating faster schedules for performance-critical loops (if the interested users are willing to trade the cost of longer compile time with faster runtime). 2) As a framework for compiler writers to evaluate and improve other heuristics-based approaches by providing quantitative information as to where and how much their heuristic methods could be further improved.

title: A theory for software-hardware co-scheduling for ASIPs and embedded processors
abstract: Exploiting instruction-level parallelism (ILP) is extremely important for achieving high performance in application specific instruction set processors (ASIPs) and embedded processors. Existing techniques deal with either scheduling hardware pipelines to obtain higher throughput or software pipeline-an instruction scheduling technique for iterative computation-loops for exploiting greater ILP. We integrate these techniques to co-schedule hardware and software pipelines to achieve greater instruction throughput. In this paper, we develop the underlying theory of co-scheduling, called the Modulo-Scheduled Pipeline (or MS-Pipeline) theory. More specifically, we establish the necessary and sufficient condition for achieving the maximum throughput in a given pipeline operating under module scheduling. Further, we establish a sufficient condition to achieve a specified throughput, based on which we also develop a methodology for designing the hardware pipelines that achieve such a throughput.

title: A Framework for Resource-Constrained Rate-Optimal Software Pipelining
abstract: The rapid advances in high-performance computer architecture and compilation techniques provide both challenges and opportunities to exploit the rich solution space of software pipelined loop schedules. In this paper, we develop a framework to construct a software pipelined loop schedule which runs on the given architecture (with a fixed number of processor resources) at the maximum possible iteration rate (a la rate-optimal) while minimizing the number of buffers-a close approximation to minimizing the number of registers. The main contributions of this paper are: First, we demonstrate that such problem can be described by a simple mathematical formulation with precise optimization objectives under a periodic linear scheduling framework. The mathematical formulation provides a clear picture which permits one to visualize the overall solution space (for rate-optimal schedules) under different sets of constraints. Secondly, we show that a precise mathematical formulation and its solution does make a significant performance difference. We evaluated the performance of our method against three leading contemporary heuristic methods. Experimental results show that the method described in this paper performed significantly better than these methods. The techniques proposed in this paper are useful in two different ways: 1) As a compiler option which can be used in generating faster schedules for performance-critical loops (if the interested users are willing to trade the cost of longer compile time with faster runtime). 2) As a framework for compiler writers to evaluate and improve other heuristics-based approaches by providing quantitative information as to where and how much their heuristic methods could be further improved.

title: Co-scheduling hardware and software pipelines
abstract: In this paper we propose co-scheduling, a framework for simultaneous design of hardware pipelines structures and software-pipelined schedules. Two important components of the co-scheduling framework are: (1) An extension to the analysis of hardware pipeline design that meets the needs of periodic (or software pipelined) schedules. Reservation tables, forbidden latencies, collision vectors, and state diagrams from classical pipeline theory are revisited and extended to solve the new problems. (2) An efficient method, based on the above extension of pipeline analysis, to perform (a) software pipeline scheduling and (b) hardware pipeline reconfiguration which are mutually "compatible". The proposed method has been implemented and preliminary experimental results for 1008 kernel loops are reported. Co-scheduling successfully obtains a schedule for 95% of these loops. The median time to obtain these schedules is 0.25 seconds on a Sparc-20.

title: Star-extremal graphs and the lexicographic product
abstract: The star-chromatic number and the fractional-chromatic number are two generalizations of the ordinary chromatic number of a graph. We say a graph G is star-extremal if its star-chromatic number is equal to its fractional-chromatic number. We prove that star-extremal graphs G have the following interesting property: For an arbitrary graph H the star-chromatic number χ ⋆ (G[H]) of the lexicographic product G [ H ] is equal to the product of χ ⋆ ( G ) and χ( H ). Then we show that several classes of circulant graphs are star-extremal. Thus for these circulant graphs G and arbitrary graphs H , if χ ⋆ ( G ) and χ( H ) are known then we can easily determine the star-chromatic number (hence the ordinary chromatic number) of the lexicographic product G [ H ]. For these star-extremal circulant graphs, we also derive polynomial-time anti-clique-finding and coloring algorithms.