Testing Pattern-Freeness.

We consider the problem of testing pattern-freeness (PF): given a string $I$ and a fixed pattern $J$ of length $k$ over a finite alphabet $\Sigma$, decide whether $I$ is $J$-free (has no occurrence of $J$) or alternatively, one has to modify $I$ in at least an $\epsilon$-fraction of its locations to obtain a string that is $J$-free. The 2D analog where one is given a 2D-image and a fixed pattern of dimension $k$ is studied as well. We show that other than a small number of specific patterns, for both 1D and 2D cases, there are simple one-sided testers for this problem whose query complexity is $O(1/\epsilon)$. The testers work for any finite alphabet, with complexity that does not depend on the template dimension $k$, which might depend on $n$. For the 1D case, testing PF is a specific case of the problem of testing regular languages. Our algorithm improves upon the query complexity of the known tester of Alon et. al., removing a polynomial dependency on $k$ as well as a polylogarithmic factor in $1/\epsilon$. For the 2D case, it is the first testing algorithm we are aware of. The PF property belongs to the more general class of matrix properties, studied by Fisher and Newman. They provide a 2-sided tester, doubly exponential in $1/\epsilon$, for matrix properties that can be characterized by a finite set of forbidden induced submatrices and pose the problem of testing such properties for tight submatrices (with consecutive rows and columns) as an open problem. Our algorithm provides a strong tester for this class of properties, in which the forbidden set is of size 1. The dependence of our testers on $\epsilon$ is tight up to constant factors since any tester erring with probability at most $1/3$ must make $\Omega(1/\epsilon)$ queries to $I$. The analysis of our testers builds upon novel combinatorial properties of strings and images, which may be of independent interest.