In our previous work, we introduced the concept of semilocal string comparison, and developed for it an efficient method called the seaweed algorithm. In the current paper, we introduce its extension, called the periodic seaweed algorithm. The new algorithm allows efficient exploitation of the periodic structure in one of the input strings. By application of the periodic seaweed algorithm, we obtain new algorithms for the tandem LCS problem and the tandem cyclic alignment problem, improving on existing algorithms in running time. © 2009 Springer Berlin Heidelberg.