1 research outputs found
Property Testing for Cyclic Groups and Beyond
This paper studies the problem of testing if an input (Gamma,*), where Gamma
is a finite set of unknown size and * is a binary operation over Gamma given as
an oracle, is close to a specified class of groups. Friedl et al. [Efficient
testing of groups, STOC'05] have constructed an efficient tester using
poly(log|Gamma|) queries for the class of abelian groups. We focus in this
paper on subclasses of abelian groups, and show that these problems are much
harder: Omega(|Gamma|^{1/6}) queries are necessary to test if the input is
close to a cyclic group, and Omega(|Gamma|^c) queries for some constant c are
necessary to test more generally if the input is close to an abelian group
generated by k elements, for any fixed integer k>0. We also show that knowledge
of the size of the ground set Gamma helps only for k=1, in which case we
construct an efficient tester using poly(log|Gamma|) queries; for any other
value k>1 the query complexity remains Omega(|Gamma|^c). All our upper and
lower bounds hold for both the edit distance and the Hamming distance. These
are, to the best of our knowledge, the first nontrivial lower bounds for such
group-theoretic problems in the property testing model and, in particular, they
imply the first exponential separations between the classical and quantum query
complexities of testing closeness to classes of groups.Comment: 15 pages, full version of a paper to appear in the proceedings of
COCOON'11. v2: Ref. [14] added and a few modifications to Appendix A don