Speeding Up the GVW Algorithm via a Substituting Method

doi:10.1007/s11424-019-8345-3

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	Speeding Up the GVW Algorithm via a Substituting Method
	Li Ting 1,2; Sun Yao 1; Huang Zhenyu 1; Wang Dingkang3,4 ; Lin Dongdai 1
	2019-02-01
发表期刊	JOURNAL OF SYSTEMS SCIENCE & COMPLEXITY
ISSN	1009-6124
卷号	32 期号:1 页码:205-233
摘要	The 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.
关键词	Grobner basis GVW signature-based algorithm time-memory tradeoff
DOI	10.1007/s11424-019-8345-3
语种	英语
资助项目	National Nature Science Foundation of China[61877058] ; National Nature Science Foundation of China[61872359] ; Strategy Cooperation Project[AQ-1701] ; CAS[QYZDJ-SSW-SYS022]
WOS研究方向	Mathematics
WOS类目	Mathematics, Interdisciplinary Applications
WOS记录号	WOS:000458795500012
出版者	SPRINGER HEIDELBERG
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/32682
专题	系统科学研究所
通讯作者	Sun Yao
作者单位	1.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
推荐引用方式 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.