One-Sided Error Testing of Monomials and Affine Subspaces

We study the query complexity of testing monomials (resp., affine and linear subspaces) with one-sided error. Actually, we consider three versions of each of these properties, and obtain the following results regarding the query complexity of testing them with one-sided error. The general version, in which the arity of the monomial (resp., co-dimension of the subspace) is not specified, has query complexity O~(1/ϵ), where ϵ denotes the proximity parameter.The bounded version, in which the arity of the monomial (resp., co-dimension of the subspace) is upper bounded by a fixed parameter, has query complexity O~(1/ϵ).The exact version, in which the arity of the monomial (resp., co-dimension of the subspace) is required to equal a fixed parameter (e.g., equals 2), has query complexity Ω~(logn), where n denotes the length of the argument for the tested function. The general version, in which the arity of the monomial (resp., co-dimension of the subspace) is not specified, has query complexity O~(1/ϵ), where ϵ denotes the proximity parameter. The bounded version, in which the arity of the monomial (resp., co-dimension of the subspace) is upper bounded by a fixed parameter, has query complexity O~(1/ϵ). The exact version, in which the arity of the monomial (resp., co-dimension of the subspace) is required to equal a fixed parameter (e.g., equals 2), has query complexity Ω~(logn), where n denotes the length of the argument for the tested function. The running time of the testers in the positive results is linear in their query complexity.