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

Abstract

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 {em 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.