Identifying periodic occurrences of a template with applications to protein structure

We consider a string matching problem where the pattern is a template that matches many different strings with various degrees of perfection. The quality of a match is given by a penalty matrix that assigns each pair of characters a score that characterizes how well the characters match. Superfluous characters in the text and superfluous characters in the pattern may also occur and the respective penalties for such gaps in the alignment are also given by the penalty matrix. For a text T of length n, and a template P of length m, we wish to find the best alignment of T with Pⁿ, which is the concatenation of n copies of P, (m will typically be much smaller than n). Such an alignment can simply be obtained by solving a dynamic programming problem of size O(n²m), and ignoring the periodic character of Pn. We show that the structure of Pⁿ can be exploited and the problem reduced to essentially solving a dynamic programming of size O(mn). If the complexity of computing gap penalties is O(1), (which is frequently the case), our algorithm runs in O(mn) time. The problem was motivated by a protein structure problem.