On the parallelization of triangular decompositions

Asadi, Mohammadali; Brandt, Alexander; Moir, Robert H. C.; Maza, Marc Moreno; Xie, Y.

doi:10.1145/3373207.3404065

Cited by 6 publications

(5 citation statements)

References 13 publications

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“…These parallel algorithms are implemented using generic support for task parallelism, thread pools, and asynchronous generators, also provided in the BPAS library. The details of this parallel support are discussed in [3] and [4].…”

Section: Experimentation and Discussionmentioning

confidence: 99%

“…Within numerical linear algebra, parallel pipelines have already been employed in parallel implementations of singular value decomposition [9], LU decomposition, and Gaussian elimination [12]. Meanwhile, to the best of our knowledge, the only use of parallel pipeline in symbolic computation is [3], which examines a parallel implementation of triangular decomposition of polynomial systems.…”

Section: Introductionmentioning

confidence: 99%

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On the Complexity and Parallel Implementation of Hensel's Lemma and Weierstrass Preparation

Brandt¹,

Maza²

2021

Preprint

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Hensel's lemma, combined with repeated applications of Weierstrass preparation theorem, allows for the factorization of polynomials with multivariate power series coefficients. We present a complexity analysis for this method and leverage those results to guide the load-balancing of a parallel implementation to concurrently update all factors. In particular, the factorization creates a pipeline where the terms of degree k of the first factor are computed simultaneously with the terms of degree k − 1 of the second factor, etc. An implementation challenge is the inherent irregularity of computational work between factors, as our complexity analysis reveals. Additional resource utilization and load-balancing is achieved through the parallelization of Weierstrass preparation. Experimental results show the efficacy of this mixed parallel scheme, achieving up to 9× speedup on 12 cores.

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Section: Experimentation and Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the Complexity and Parallel Implementation of Hensel's Lemma and Weierstrass Preparation

Brandt¹,

Maza²

2021

Preprint

Self Cite

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“…The complexity of algorithms computing characteristic sets or regular chains are double-exponential. It remains a very active research field and some practical improvements were recently proposed [1,19].…”

Section: Wu's Methods In 3dmentioning

confidence: 99%

“…As far as we know, this proof has not be mechanized into a formal geometric framework yet. We propose below two new proofs of this theorem 1 .The first one follows an algebraic approach popularized by the late Prof. Wu. Since this approach requires using coordinates and polynomials, only the direction "from Pappus to Dandelin-Gallucci" can be achieved.…”

Section: Dandelin-gallucci's Theoremmentioning

confidence: 99%

Two New Ways to Formally Prove Dandelin-Gallucci's Theorem

Braun

Magaud

Schreck

2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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Mechanizing proofs of geometric theorems in 3D is significantly more challenging than in 2D. As a first noteworthy case study, we consider an iconic theorem of 3D geometry: Dandelin-Gallucci's theorem. We work in the very simple but powerful framework of projective incidence geometry, where only incidence relationships are considered. We study and compare two new and very different approaches to prove this theorem. First, we propose a new proof based on the well-known Wu's method. Second, we use an original method based on matroid theory to generate a proof script which is then checked by the Coq proof assistant. For each method, we point out which parts of the proof we manage to carry out automatically and which parts are more difficult to automate and require human interaction. We hope these first developments will lead to formally proving more 3D theorems automatically and that it will be used to formally verify some key properties of computational geometry algorithms in 3D. CCS CONCEPTS• Theory of computation → Automated reasoning; • Computing methodologies → Theorem proving algorithms.

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“…It is realized a preliminary implementations on SMPs and gets a promising speedups. Asadi et al 40 develop this algorithm by combining the fork‐join model and producer‐consumer patterns in order to better capture component‐level parallelization. This idea is reported to the publicly Basic Polynomial Algebra Subprograms (BPAS) library.…”

Section: Related Workmentioning

confidence: 99%

Solving Boolean polynomial systems by parallelizing characteristic set method for cyber‐physical systems

Zhao

Zhu

et al. 2020

Softw Pract Exp

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Summary Many cyber‐attach schemes and coding models established by algebra tools are build to address the problem of security of cyber‐pysical systems (CPS). As an important field of algebra computing, Boolean Polynomial System Solving (PoSSo) problem plays a very important role in many algebra applications. In this article, we propose an efficient Parallel Boolean Characteristic Set method (PBCS) under the high‐performance computing environment to improve the efficiency of solving Boolean polynomial systems. The PBCS is implemented based on the state‐of‐the‐art Boolean Characteristic Set method (BCS). It adopts a master‐slave parallel pattern, and distributes tasks based on the polynomial sets after initial zero decomposition. We design a strategy of dynamically reallocating tasks to ameliorate load imbalance, which is caused by dynamical zero decomposition of polynomials. Furthermore, we improve its performance by optimizing the parameter settings of PBCS, including the maximum number of polynomial branches that trigger the dynamic allocation policy and the scheduling time. Experimental results with solving several Boolean polynomial systems confirm that PBCS is efficient and scalable, especially for the equations generating from stream ciphers that have block triangular structure. Moreover, the method also has good scalability. It shows a stable speedup as well even extending to the size of thousands of CPU cores.

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On the parallelization of triangular decompositions

Cited by 6 publications

References 13 publications

On the Complexity and Parallel Implementation of Hensel's Lemma and Weierstrass Preparation

On the Complexity and Parallel Implementation of Hensel's Lemma and Weierstrass Preparation

Two New Ways to Formally Prove Dandelin-Gallucci's Theorem

Solving Boolean polynomial systems by parallelizing characteristic set method for cyber‐physical systems

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