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Speeding Up the GVW Algorithm via a Substituting Method
Li Ting1,2; Sun Yao1; Huang Zhenyu1; Wang Dingkang3,4; Lin Dongdai1
AbstractThe GVW algorithm is an efficient signature-based algorithm for computing Grobner bases. In this paper, the authors consider the implementation of the GVW algorithm by using linear algebra, and speed up GVW via a substituting method. As it is well known that, most of the computing time of a Grobner basis is spent on reductions of polynomials. Thus, linear algebraic techniques, such as matrix operations, have been used extensively to speed up the implementations. Particularly, one-direction (also called signature-safe) reduction is used in signature-based algorithms, because polynomials (or rows in matrices) with larger signatures can only be reduced by polynomials (rows) with smaller signatures. The authors propose a new method to construct sparser matrices for signature-based algorithms via a substituting method. Specifically, instead of only storing the original polynomials in GVW, the authors also record many equivalent but sparser polynomials at the same time. In matrix construction, denser polynomials are substituted by sparser equivalent ones. As the matrices get sparser, they can be eliminated more efficiently. Two specifical algorithms, Block-GVW and LMGVW, are presented, and their combination is the Sub-GVW algorithm. The correctness of the new proposed method is proved, and the experimental results demonstrate the efficiency of this new method.
KeywordGrobner basis GVW signature-based algorithm time-memory tradeoff
Funding ProjectNational Nature Science Foundation of China[61877058] ; National Nature Science Foundation of China[61872359] ; Strategy Cooperation Project[AQ-1701] ; CAS[QYZDJ-SSW-SYS022]
WOS Research AreaMathematics
WOS SubjectMathematics, Interdisciplinary Applications
WOS IDWOS:000458795500012
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Document Type期刊论文
Corresponding AuthorSun Yao
Affiliation1.Chinese Acad Sci, Inst Informat Engn, SKLOIS, Beijing 100093, Peoples R China
2.Univ Chinese Acad Sci, Sch Cyber Secur, Beijing 100190, Peoples R China
3.Chinese Acad Sci, Acad Math & Syst Sci, KLMM, Beijing 100090, Peoples R China
4.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100190, Peoples R China
Recommended Citation
GB/T 7714
Li Ting,Sun Yao,Huang Zhenyu,et al. Speeding Up the GVW Algorithm via a Substituting Method[J]. JOURNAL OF SYSTEMS SCIENCE & COMPLEXITY,2019,32(1):205-233.
APA Li Ting,Sun Yao,Huang Zhenyu,Wang Dingkang,&Lin Dongdai.(2019).Speeding Up the GVW Algorithm via a Substituting Method.JOURNAL OF SYSTEMS SCIENCE & COMPLEXITY,32(1),205-233.
MLA Li Ting,et al."Speeding Up the GVW Algorithm via a Substituting Method".JOURNAL OF SYSTEMS SCIENCE & COMPLEXITY 32.1(2019):205-233.
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