Yevgen Voronenko

Multiplierless multiple constant multiplication

A variable can be multiplied by a given set of fixed-point constants using a multiplier block that consists exclusively of additions, subtractions, and shifts. The generation of a multiplier block from the set of constants is known as the multiple constant multiplication (MCM) problem. Finding the optimal solution, namely, the one with the fewest number of additions and subtractions, is known to be NP-complete. We propose a new algorithm for the MCM problem, which produces solutions that require up to 20% less additions and subtractions than the best previously known algorithm. At the same time our algorithm, in contrast to the closest competing algorithm, is not limited by the constant bitwidths. We present our algorithm using a unifying formal framework for the best, graph-based MCM algorithms and provide a detailed runtime analysis and experimental evaluation. We show that our algorithm can handle problem sizes as large as 100 32-bit constants in a time acceptable for most applications. The implementation of the new algorithm is available at www.spiral.net.

Discrete fourier transform on multicore

This article gives an overview on the techniques needed to implement the discrete Fourier transform (DFT) efficiently on current multicore systems. The focus is on Intel-compatible multicores, but we also discuss the IBM Cell and, briefly, graphics processing units (GPUs). The performance optimization is broken down into three key challenges: parallelization, vectorization, and memory hierarchy optimization. In each case, we use the Kronecker product formalism to formally derive the necessary algorithmic transformations based on a few hardware parameters. Further code-level optimizations are discussed. The rigorous nature of this framework enables the complete automation of the implementation task as shown by the program generator Spiral. Finally, we show and analyze DFT benchmarks of the fastest libraries available for the considered platforms.

FFT program generation for shared memory: SMP and multicore

The chip makers response to the approaching end of CPU frequency scaling are multicore systems, which offer the same programming paradigm as traditional shared memory platforms but have different performance characteristics. This situation considerably increases the burden on library developers and strengthens the case for automatic performance tuning frameworks like Spiral, a program generator and optimizer for linear transforms such as the discrete Fourier transform (DFT). We present a shared memory extension of Spiral. The extension within Spiral consists of a rewriting system that manipulates the structure of transform algorithms to achieve load balancing and avoids false sharing, and of a backend to generate multithreaded code. Application to the DFT produces a novel class of algorithms suitable for multicore systems as validated by experimental results: we demonstrate a parallelization speed-up already for sizes that fit into L1 cache and compare favorably to other DFT libraries across all small and midsize DFTs and considered platforms

Algebraic Signal Processing Theory: Cooley–Tukey Type Algorithms for DCTs and DSTs

In this paper, we systematically derive a large class of fast general-radix algorithms for various types of real discrete Fourier transforms (real DFTs) including the discrete Hartley transform (DHT) based on the algebraic signal processing theory. This means that instead of manipulating the transform definition, we derive algorithms by manipulating the polynomial algebras underlying the transforms using one general method. The same method yields the well-known Cooley-Tukey fast Fourier transform (FFT) as well as general radix discrete cosine and sine transform algorithms. The algebraic approach makes the derivation concise, unifies and classifies many existing algorithms, yields new variants, enables structural optimization, and naturally produces a human-readable structural algorithm representation based on the Kronecker product formalism. We show, for the first time, that the general-radix Cooley-Tukey and the lesser known Bruun algorithms are instances of the same generic algorithm. Further, we show that this generic algorithm can be instantiated for all four types of the real DFT and the DHT.

Bandit-based optimization on graphs with application to library performance tuning

The problem of choosing fast implementations for a class of recursive algorithms such as the fast Fourier transforms can be formulated as an optimization problem over the language generated by a suitably defined grammar. We propose a novel algorithm that solves this problem by reducing it to maximizing an objective function over the sinks of a directed acyclic graph. This algorithm valuates nodes using Monte-Carlo and grows a subgraph in the most promising directions by considering local maximum k-armed bandits. When used inside an adaptive linear transform library, it cuts down the search time by an order of magnitude compared to the existing algorithm. In some cases, the performance of the implementations found is also increased by up to 10% which is of considerable practical importance since it consequently improves the performance of all applications using the library.

Formal loop merging for signal transforms

A critical optimization in the domain of linear signal transforms, such as the discrete Fourier transform (DFT), is loop merging, which increases data locality and reuse and thus performance. In particular, this includes the conversion of shuffle operations into array reindexings. To date, loop merging is well understood only for the DFT, and only for Cooley-Tukey FFT based algorithms, which excludes DFT sizes divisible by large primes. In this paper, we present a formal loop merging framework for general signal transforms and its implementation within the SPIRAL code generator. The framework consists of Ε-SPL, a mathematical language to express loops and index mappings; a rewriting system to merge loops in Ε-SPL and a compiler that translates Ε-SPL into code. We apply the framework to DFT sizes that cannot be handled using only the Cooley-Tukey FFT and compare our method to FFTW 3.0.1 and the vendor library Intel MKL 7.2.1. Compared to FFTW our generated code is a factor of 2--4 faster under equal implementation conditions (same algorithms, same unrolling threshold). For some sizes we show a speed-up of a factor of 9 using Bluesteins algorithm. Further, we give a detailed comparison against the Intel vendor library MKL; our generated code is between 2 times faster and 4.5 times slower.

Computer Generation of General Size Linear Transform Libraries

The development of high-performance libraries has become extraordinarily difficult due to multiple processor cores, vector instruction sets, and deep memory hierarchies. Often, the library has to be reimplemented and reoptimized, when a new platform is released. In this paper we show how to automatically generate general input-size libraries for the domain of linear transforms. The input to our generator is a formal specification of the transform and the recursive algorithms the library should use; the output is a library that supports general input size, is vectorized and multithreaded, provides an adaptation mechanism for the memory hierarchy, and has excellent performance, comparable to or better than the best human-written libraries. Further, we show that our library generator enables various customizations; one example is the generation of Java libraries.

Automatic Tuning of Discrete Fourier Transforms Driven by Analytical Modeling

Analytical models have been used to estimate optimal values for parameters such as tile sizes in the context of loop nests. However, important algorithms such as fast Fourier transforms (FFTs) present a far more complex search space consisting of many thousands of different implementations with very different complex access patterns and nesting and code structures. As a results, some of the best available FFT implementations use heuristic search based on runtime measurements. In this paper we present the first analytical model that can successfully replace the measurement in this search on modern platforms. The model includes many details of the platforms memory system including the TLBs, and, for the first time, physically addressed caches and hardware prefetching. The effect, as we show, is a dramatically reduced search time to find the best FFT without significant loss in performance. Even though our model is adapted to the FFT in this paper, its underlying structure should be applicable for a much larger set of code structures and hence is a candidate for iterative compilation.

Offline library adaptation using automatically generated heuristics

Automatic tuning has emerged as a solution to provide high-performance libraries for fast changing, increasingly complex computer architectures. We distinguish offline adaptation (e.g., in ATLAS) that is performed during installation without the full problem description from online adaptation (e.g., in FFTW) that is performed at runtime. Offline adaptive libraries are simpler to use, but, unfortunately, writing the adaptation heuristics that power them is a daunting task. The overhead of online adaptive libraries, on the other hand, makes them unsuitable for a number of applications. In this paper, we propose to automatically generate heuristics in the form of decision trees using a statistical classifier, effectively converting an online adaptive library into an offline one. As testbed we use Spiral-generated adaptive transform libraries for current multicores with vector extensions. We show that replacing the online search with generated decision trees maintains a performance competitive with vendor libraries while allowing for a simpler interface and reduced computation overhead.

Automatic generation of implementations for DSP transforms on fused multiply-add architectures

Many modern computer architectures feature fused multiply-add (FMA) instructions, which offer potentially faster performance for numerical applications. For DSP transforms, compilers can only generate FMA code to a very limited extent because optimal use of FMAs requires modifying the chosen algorithm. In this paper, we present a framework for automatically generating FMA code for every linear DSP transform, which we implemented as an extension to the SPIRAL code generation system. We show that for many transforms and transform sizes, our generated FMA code matches the best-known hand-derived FMA algorithms in terms of arithmetic cost. Further, we present actual runtime results that show the speed-up obtained by using FMA instructions.