On the problem of deciding if a polyomino tiles the plane by translation

The words that tile the plane by translation are characterized by the Beauquier-Nivat condition. By using the constant time algorithms for computing the longest common extensions in two words, we provide a linear time algorithm in the case of pseudo-square polyominoes, improving the previous quadratic algorithm of Gambini and Vuillon. For pseudo-hexagon polyominoes not containing arbitrarily large square factors we also have a linear algorithm.