On synthesizing systolic arrays from recurrence equations with linear dependencies

Journal ArticleWe present a technique for synthesizing systolic architectures from Recurrence Equations. A class of such equations (Recurrence Equations with Linear Dependencies) is defined and and the problem of mapping such equations onto a two dimensional architecture is studied. We show that such a mapping is provided by means of a linear allocation and timing function. An important result is that under such a mapping the dependencies remain linear. After obtaining a two-dimensional architecture by applying such a mapping, a systolic array can be derived if t h e communication can be spatially and temporally localized. We show that a simple test consisting of finding the zeroes of a matrix is sufficient to determine whether this localization can be achieved by pipelining and give a construction that generates the array when such a pipelining is possible. The technique is illustrated by automatically deriving a well known systolic array for factoring a band matrix into lower and upper triangular factors

Systolic array synthesis by static analysis of program dependencies

Journal ArticleWe present a technique for mapping recurrence equations to systolic arrays. While this problem has been studied in fairly great detail, the recurrence equations that are analysed here are a generalization of those studied previously. In a n earlier paper (14] we have showed how systolic arrays can be synthesized from such generalized recurrence equations by a combination of affine transformations and explicit pipelining. This paper extends the results in two directions. Firstly, a multistage pipelining technique is proposed, which permits the synthesis of systolic arrays with irregular data flow. Secondly we develop analysis techniques for the synthesis of systolic arrays whose computation is governed by control signals in a systematic manner which is amenable to mechanization. The full paper also discusses how these techniques can be applied to the mapping problem for more general architectures

Parallelization of dynamic programming recurrences in computational biology

The rapid growth of biosequence databases over the last decade has led to a performance bottleneck in the applications analyzing them. In particular, over the last five years DNA sequencing capacity of next-generation sequencers has been doubling every six months as costs have plummeted. The data produced by these sequencers is overwhelming traditional compute systems. We believe that in the future compute performance, not sequencing, will become the bottleneck in advancing genome science. In this work, we investigate novel computing platforms to accelerate dynamic programming algorithms, which are popular in bioinformatics workloads. We study algorithm-specific hardware architectures that exploit fine-grained parallelism in dynamic programming kernels using field-programmable gate arrays: FPGAs). We advocate a high-level synthesis approach, using the recurrence equation abstraction to represent dynamic programming and polyhedral analysis to exploit parallelism. We suggest a novel technique within the polyhedral model to optimize for throughput by pipelining independent computations on an array. This design technique improves on the state of the art, which builds latency-optimal arrays. We also suggest a method to dynamically switch between a family of designs using FPGA reconfiguration to achieve a significant performance boost. We have used polyhedral methods to parallelize the Nussinov RNA folding algorithm to build a family of accelerators that can trade resources for parallelism and are between 15-130x faster than a modern dual core CPU implementation. A Zuker RNA folding accelerator we built on a single workstation with four Xilinx Virtex 4 FPGAs outperforms 198 3 GHz Intel Core 2 Duo processors. Furthermore, our design running on a single FPGA is an order of magnitude faster than competing implementations on similar-generation FPGAs and graphics processors. Our work is a step toward the goal of automated synthesis of hardware accelerators for dynamic programming algorithms

N–Dimensional Orthogonal Tile Sizing Problem

AMS subject classification: 68Q22, 90C90We discuss in this paper the problem of generating highly efficient code when a n + 1-dimensional nested loop program is executed on a n-dimensional torus/grid of distributed-memory general-purpose machines. We focus on a class of uniform recurrences with non-negative components of the dependency matrix. Using tiling the iteration space strategy we show that minimizing the total running time reduces to solving a non-trivial non-linear integer optimization problem. For the later we present a mathematical framework that enables us to derive an O(n log n) algorithm for finding a good approximate solution. The theoretical evaluations and the experimental results show that the obtained solution approximates the original minimum sufficiently well in the context of the considered problem. Such algorithm is realtime usable for very large values of n and can be used as optimization techniques in parallelizing compilers as well as in performance tuning of parallel codes by hand

Time-Optimal and Conflict-Free Mappings of Uniform Dependence Algorithms into Lower Dimensional Processor Arrays

Most existing methods of mapping algorithms into processor arrays are restricted to the case where n-dimensional algorithms or algorithms with n nested loops are mapped into (n—l)-dimensional arrays. However, in practice, it is interesting to map n-dimensional algorithms into (k —l)-dimensional arrays where k\u3c.n. For example, many algorithms at bit-level are at least 4-dimensional (matrix multiplication, convolution, LU decomposition, etc.) and most existing bit level processor arrays are 2-dimensional. A computational conflict occurs if two or more computations of an algorithm are mapped into the same processor and the same execution time. In this paper, necessary and sufficient conditions are derived to identify all mappings without computational conflicts, based on the Hermite normal form of the mapping matrix. These conditions are used to propose methods of mapping any n-dimensional algorithm into (k— l)-dimensional arrays, kn—3, optimality of the mapping is guaranteed