A perfect single tandem repeat is defined as a nonempty string that can be divided into two identical substrings, e.g. abcabc. An approximate single tandem repeat is one in which the substrings are similar, but not identical, e.g. abcdaacd.

There are many solutions to the string matching problem which are strictly linear in the input size and independent of alphabet size. Furthermore, the model of computation for these algorithms is very weak: they allow only simple arithmetic and comparisons of equality between characters of the input. In contrast, algorithm for two dimensional matching have needed stronger models of computation, most notably assuming a totally ordered alphabet. The fastest algorithms for two dimensional matching have therefore had a logarithmic dependence on the alphabet size. In the worst case, this gives an algorithm that runs in O(n log m) with O(m log m) preprocessing.

This paper discusses how to count and generate strings that are "distinct" in two senses: p-distinct and b-distinct. Two strings x on alphabet A and x 0 on alphabet A 0 are said to be p-distinct iff they represent distinct "patterns"; that is, iff there exists no one-one mapping from A to A 0 that transforms x into x 0 . Thus aab and baa are p-distinct while aab and ddc are p-equivalent. On the other hand, x and x 0 are said to be b-distinct iff they give rise to distinct border (failure function) arrays: thus aab with border array 010 is b-distinct from aba with border array 001. The number of p-distinct (respectively, b-distinct) strings of length n formed using exactly k different letters is the [k; n] entry in an infinite p 0 (respectively, b 0 ) array. Column sums p[n] and b[n] in these arrays give the number of distinct strings of length n. We present algorithms to compute, in constant time per string, all p-distinct (respectively, b-distinct) strings of length n ...

We propose an algorithm for finding in a word all pairs of occurrences of the same subword with a given distance r between them. The obtained complexity is O(n log r + S), where S is the size of the output. We also show how the algorithm can be modified in order to find all such pairs of occurrences separated by a given word. The solution uses an algorithm for finding all quasi-squares in two strings, a problem that generalizes the known problem of searching for squares.