CSR5: An Efficient Storage Format for Cross-Platform Sparse Matrix-Vector Multiplication

Sparse matrix-vector multiplication (SpMV) is a fundamental building block for numerous applications. In this paper, we propose CSR5 (Compressed Sparse Row 5), a new storage format, which offers high-throughput SpMV on various platforms including CPUs, GPUs and Xeon Phi. First, the CSR5 format is insensitive to the sparsity structure of the input matrix. Thus the single format can support an SpMV algorithm that is efficient both for regular matrices and for irregular matrices. Furthermore, we show that the overhead of the format conversion from the CSR to the CSR5 can be as low as the cost of a few SpMV operations. We compare the CSR5-based SpMV algorithm with 11 state-of-the-art formats and algorithms on four mainstream processors using 14 regular and 10 irregular matrices as a benchmark suite. For the 14 regular matrices in the suite, we achieve comparable or better performance over the previous work. For the 10 irregular matrices, the CSR5 obtains average performance improvement of 17.6\%, 28.5\%, 173.0\% and 293.3\% (up to 213.3\%, 153.6\%, 405.1\% and 943.3\%) over the best existing work on dual-socket Intel CPUs, an nVidia GPU, an AMD GPU and an Intel Xeon Phi, respectively. For real-world applications such as a solver with only tens of iterations, the CSR5 format can be more practical because of its low-overhead for format conversion. The source code of this work is downloadable at https://github.com/bhSPARSE/Benchmark_SpMV_using_CSR5Comment: 12 pages, 10 figures, In Proceedings of the 29th ACM International Conference on Supercomputing (ICS '15

GraphR: Accelerating Graph Processing Using ReRAM

This paper presents GRAPHR, the first ReRAM-based graph processing accelerator. GRAPHR follows the principle of near-data processing and explores the opportunity of performing massive parallel analog operations with low hardware and energy cost. The analog computation is suit- able for graph processing because: 1) The algorithms are iterative and could inherently tolerate the imprecision; 2) Both probability calculation (e.g., PageRank and Collaborative Filtering) and typical graph algorithms involving integers (e.g., BFS/SSSP) are resilient to errors. The key insight of GRAPHR is that if a vertex program of a graph algorithm can be expressed in sparse matrix vector multiplication (SpMV), it can be efficiently performed by ReRAM crossbar. We show that this assumption is generally true for a large set of graph algorithms. GRAPHR is a novel accelerator architecture consisting of two components: memory ReRAM and graph engine (GE). The core graph computations are performed in sparse matrix format in GEs (ReRAM crossbars). The vector/matrix-based graph computation is not new, but ReRAM offers the unique opportunity to realize the massive parallelism with unprecedented energy efficiency and low hardware cost. With small subgraphs processed by GEs, the gain of performing parallel operations overshadows the wastes due to sparsity. The experiment results show that GRAPHR achieves a 16.01x (up to 132.67x) speedup and a 33.82x energy saving on geometric mean compared to a CPU baseline system. Com- pared to GPU, GRAPHR achieves 1.69x to 2.19x speedup and consumes 4.77x to 8.91x less energy. GRAPHR gains a speedup of 1.16x to 4.12x, and is 3.67x to 10.96x more energy efficiency compared to PIM-based architecture.Comment: Accepted to HPCA 201

GraphBLAST: A High-Performance Linear Algebra-based Graph Framework on the GPU

High-performance implementations of graph algorithms are challenging to implement on new parallel hardware such as GPUs because of three challenges: (1) the difficulty of coming up with graph building blocks, (2) load imbalance on parallel hardware, and (3) graph problems having low arithmetic intensity. To address some of these challenges, GraphBLAS is an innovative, on-going effort by the graph analytics community to propose building blocks based on sparse linear algebra, which will allow graph algorithms to be expressed in a performant, succinct, composable and portable manner. In this paper, we examine the performance challenges of a linear-algebra-based approach to building graph frameworks and describe new design principles for overcoming these bottlenecks. Among the new design principles is exploiting input sparsity, which allows users to write graph algorithms without specifying push and pull direction. Exploiting output sparsity allows users to tell the backend which values of the output in a single vectorized computation they do not want computed. Load-balancing is an important feature for balancing work amongst parallel workers. We describe the important load-balancing features for handling graphs with different characteristics. The design principles described in this paper have been implemented in "GraphBLAST", the first high-performance linear algebra-based graph framework on NVIDIA GPUs that is open-source. The results show that on a single GPU, GraphBLAST has on average at least an order of magnitude speedup over previous GraphBLAS implementations SuiteSparse and GBTL, comparable performance to the fastest GPU hardwired primitives and shared-memory graph frameworks Ligra and Gunrock, and better performance than any other GPU graph framework, while offering a simpler and more concise programming model.Comment: 50 pages, 14 figures, 14 table

Performance Analysis and Optimization of Sparse Matrix-Vector Multiplication on Modern Multi- and Many-Core Processors

This paper presents a low-overhead optimizer for the ubiquitous sparse matrix-vector multiplication (SpMV) kernel. Architectural diversity among different processors together with structural diversity among different sparse matrices lead to bottleneck diversity. This justifies an SpMV optimizer that is both matrix- and architecture-adaptive through runtime specialization. To this direction, we present an approach that first identifies the performance bottlenecks of SpMV for a given sparse matrix on the target platform either through profiling or by matrix property inspection, and then selects suitable optimizations to tackle those bottlenecks. Our optimization pool is based on the widely used Compressed Sparse Row (CSR) sparse matrix storage format and has low preprocessing overheads, making our overall approach practical even in cases where fast decision making and optimization setup is required. We evaluate our optimizer on three x86-based computing platforms and demonstrate that it is able to distinguish and appropriately optimize SpMV for the majority of matrices in a representative test suite, leading to significant speedups over the CSR and Inspector-Executor CSR SpMV kernels available in the latest release of the Intel MKL library.Comment: 10 pages, 7 figures, ICPP 201

AlSub: Fully Parallel and Modular Subdivision

In recent years, mesh subdivision---the process of forging smooth free-form surfaces from coarse polygonal meshes---has become an indispensable production instrument. Although subdivision performance is crucial during simulation, animation and rendering, state-of-the-art approaches still rely on serial implementations for complex parts of the subdivision process. Therefore, they often fail to harness the power of modern parallel devices, like the graphics processing unit (GPU), for large parts of the algorithm and must resort to time-consuming serial preprocessing. In this paper, we show that a complete parallelization of the subdivision process for modern architectures is possible. Building on sparse matrix linear algebra, we show how to structure the complete subdivision process into a sequence of algebra operations. By restructuring and grouping these operations, we adapt the process for different use cases, such as regular subdivision of dynamic meshes, uniform subdivision for immutable topology, and feature-adaptive subdivision for efficient rendering of animated models. As the same machinery is used for all use cases, identical subdivision results are achieved in all parts of the production pipeline. As a second contribution, we show how these linear algebra formulations can effectively be translated into efficient GPU kernels. Applying our strategies to $\sqrt{3}$, Loop and Catmull-Clark subdivision shows significant speedups of our approach compared to state-of-the-art solutions, while we completely avoid serial preprocessing.Comment: Changed structure Added content Improved description