Linear and sublinear time algorithms for the basis of abelian groups

It is well known that every finite abelian group G can be represented as a direct product of cyclic groups: G ≅ G 1 × G 2 × ⋯ × G t , where each G i is a cyclic group of order p j for some prime p and integer j ≥ 1 . If a i generates the cyclic group of G i , i = 1 , 2 , … , t , then the elements a...

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Bibliographic Details
Published in:Theoretical computer science Vol. 412; no. 32; pp. 4110 - 4122
Main Authors: Chen, Li, Fu, Bin
Format: Journal Article Conference Proceeding
Language:English
Published: Oxford Elsevier B.V 22.07.2011
Elsevier
Subjects:
Abelian group
Algorithmics. Computability. Computer arithmetics
Algorithms
Applied sciences
Basis of Abelian group
Computer science; control theory; systems
Decomposition
Exact sciences and technology
Factorization
Generators
Group theory
Group theory and generalizations
Integers
Mathematical analysis
Mathematics
Miscellaneous
Randomization
Sciences and techniques of general use
Theoretical computing
Basis of Abelian group
Randomization
Decomposition
Abelian group
Cyclic group
Computer theory
Decomposition method
Product
Computing
Factorization
Integer
Input
Finite group
Ordered group
Deterministic algorithms
Generator
ISSN:0304-3975, 1879-2294
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Summary:It is well known that every finite abelian group G can be represented as a direct product of cyclic groups: G ≅ G 1 × G 2 × ⋯ × G t , where each G i is a cyclic group of order p j for some prime p and integer j ≥ 1 . If a i generates the cyclic group of G i , i = 1 , 2 , … , t , then the elements a 1 , a 2 , … , a t are called a basis of G . We show a randomized algorithm such that given a set of generators M = { x 1 , … , x k } for an abelian group G and the prime factorization of order ord ( x i ) ( i = 1 , … , k ) , it computes a basis of G in O ( | M | ( log n ) 2 + ∑ i = 1 t n i p i n i / 2 ) time, where n = | G | has prime factorization p 1 n 1 p 2 n 2 ⋯ p t n t (which is not a part of input). This generalizes Buchmann and Schmidt’s algorithm that takes O ( | M | | G | ) time. In another model, all elements in an abelian group are put into a list as a part of input. We obtain an O ( n ) time deterministic algorithm and a sublinear time randomized algorithm for computing a basis of an abelian group.
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ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2010.06.011