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Sparse data structures are ubiquitous in modern computing, and numerous formats have been designed to represent them. These formats may exploit specific sparsity patterns, aiming to achieve higher performance for key numerical computations than more general-purpose formats such as CSR and COO. In this work we present UZP, a new sparse format based on polyhedral sets of integer points. UZP is a fexible format that subsumes CSR, COO, DIA, BCSR, etc., by raising them to a common mathematical abstraction: a union of integer polyhedra, each intersected with an ane lattice. We present a modular approach to building and optimizing UZP: it captures equivalence classes for the sparse structure, enabling the tuning of the representation for target-specifc and application-specifc performance considerations. UZP is built from any input sparse structure using integer coordinates, and is interoperable with existing software using CSR and COO data layout. We provide detailed performance evaluation of UZP on 200+ matrices from SuiteSparse, demonstrating how simple and mostly unoptimized generic executors for UZP can already achieve solid performance by exploiting Z-polyhedra structures. 

A formal, high-level representation of programs is typically needed for static and dynamic analyses performed by compilers. However, the source code of target applications is not always available in an analyzable form, e.g., to protect intellectual property. To reason on such applications, it becomes necessary to build models from observations of its execution. This paper details an algebraic approach which, taking as input the trace of memory addresses accessed by a single memory reference, synthesizes an affine loop with a single perfectly nested reference that generates the original trace. This approach is extended to support the synthesis of unions of affine loops, useful for minimally modeling traces generated by automatic transformations of polyhedral programs, such as tiling. The resulting system is capable of processing hundreds of gigabytes of trace data in minutes, minimally reconstructing 100% of the static control parts in PolyBench/C applications and 99.99% in the Pluto-tiled versions of these benchmarks. As an application example of the trace modeling method, trace compression is explored. The affine representations built for the memory traces of PolyBench/C codes achieve compression factors of the order of 106 and 103 with respect to gzip for the original and tiled versions of the traces, respectively. 

Python has become one of the most used and taught languages nowadays. Its expressiveness, cross-compatibility and ease of use have made it popular in areas as diverse as finance, bioinformatics or machine learning. However, Python programs are often significantly slower to execute than an equivalent native C implementation, especially for computation-intensive numerical kernels. This work presents PolyBench/Python, implementing the 30 kernels in PolyBench/C, one of the standard benchmark suites for polyhedral optimization, in Python. In addition to the benchmark kernels, a functional wrapper including mechanisms for performance measurement, testing, and execution configuration has been developed. The framework includes support for different ways to translate C-array codes into Python, offering insight into the tradeoffs of Python lists and NumPy arrays. The benchmark performance is thoroughly evaluated on different Python interpreters, and compared against its PolyBench/C counterpart to highlight the profitability (or lack thereof) of using Python for regular numerical codes. 

Modern polyhedral compilers excel at aggressively optimizing codes with static control parts, but the state-of-practice to find high-performance polyhedral transformations especially for different hardware targets still largely involves auto-tuning. In this work we propose a novel customizable polyhedral scheduling technique, with the aim of delivering high performance for several hardware targets. We design constraints and objectives that model several crucial aspects of performance such as stride optimization or the trade-off between parallelism and reuse, while considering important architectural features of the target machine. We evaluate our work using the PolyBench/C benchmark suite and experimentally validate it against large optimization spaces generated with the Pluto compiler on 3 representative architectures: an IBM Power9, an Intel Xeon Phi and an Intel Core-i9. Our results show we can achieve comparable or superior performance to Pluto on the majority of benchmarks, without implementing tiling in the source code nor using experimental autotuning. 

We present a novel framework to automatically derive highly efficient parametric multi-way recursive divide-&-conquer algorithms for a class of dynamic programming (DP) problems. Standard two-way or any fixed R-way recursive divide-&-conquer algorithms may not fully exploit many-core processors. To run efficiently on a given machine, the value of R may need to be different for every level of recursion based on the number of processors available and the sizes of memory/caches at different levels of the memory hierarchy. The set of R values that work well on a given machine may not work efficiently on another machine with a different set of machine parameters. To improve portability and efficiency, Multi-way Autogen generates parametric multi-way recursive divide-&-conquer algorithms where the value of R can be changed on the fly for every level of recursion. We present experimental results demonstrating the performance and scalability of the parallel programs produced by our framework.