On the Monotonicity of the String Correction Factor for Words with Mismatches

The string correction factor is the term by which the probability of a word w needs to be multiplied in order to account for character changes or “errors” occurring in at most k arbitrary positions in that word. The behavior of this factor, as a function of k and of the word length, has implications on the number of candidates that need to be considered and weighted when looking for subwords of a sequence that present unusually recurrent replicas within some bounded number of mismatches. Specifically, it is seen that over intervals of mono- or bi- tonicity for the correction factor, only some of the candidates need be considered. This mitigates the computation and leads to tables of over- represented words that are more compact to represent and inspect. In recent work, expectation and score monotonicity has been established for a number of cases of interest, under i.i.d. probabilistic assumptions. The present paper reviews the cases of bi-tonic behavior for the correction factor, concentrating on the instance in which the question is still open.