Linear Time Algorithms for the Basis of Abelian Groups

It is well known that every finite Abelian group G can be represented as a product of cyclic groups: G = G1×G2×· · ·Gt, where each Gi is a cyclic group of size p j for some prime p and integer j ≥ 1. If ai is the generator of the cyclic group of Gi, i = 1, 2, · · · , t, then the elements a1, a2, · · · , at are the basis of G. In this paper, we first obtain an O(n)-time deterministic algorithm for computing the basis of an Abelian group with n elements. This improves the previous O(n) time algorithm found by Chen [1]. We then derive an O(( ∑t i=1 p ni−1 i n 2 i log pi)(log n)(log logn))-time randomized algorithm to compute the basis of Abelian group G of size n with factorization n = p1 1 · · · p nt k , which is also a part of the input. This shows that for a large number of cases, the basis of a finite Abelian group can be computed in sublinear time. For example, it implies an O(n 1 d (logn) log log n)-time randomized algorithm to compute the basis of an Abelian group G of size n = p1 1 · · · p nt t , where d = max{ni|i = 1, · · · , t}. It is a sublinear time algorithm if max{ni|i = 1, · · · , t} is bounded by a constant. It also implies that if n is an integer in [1,m]−G(m, c), then the basis of an Abelian group of size n can be computed in O((log n) log log n)time, where c is any positive constant and G(m, c) is a subset of the small fraction of integers in [1,m] with |G(m,c)| m = O( 1 (logm)c/2 ) for every integer m.