Two-Pattern Strings — Computing Repetitions & Near-Repetitions

In a recent paper we introduced infinite two-pattern strings on the alphabet {a, b} as a generalization of Sturmian strings, and we posed three questions about them: • Given a finite string x, can we in linear time O(|x|) recognize whether or not x is a prefix/substring of some infinite two-pattern string? • If recognized as two-pattern, can all the repetitions in x be computed in linear time? • Given an integer ℓ, how many of these " two-pattern " strings x of length ℓ are there? In the previous paper we were able to answer the first of these questions in the affirmative, at least for " complete " two-pattern strings x. Here we show that, once a complete two-pattern string x has 1 been recognized, its repetitions can all be computed in linear time using an iterative algorithm that in addition computes all the " near-repetitions " in x. The third question is dealt with in a subsequent paper.