Monotone scoring of patterns with mismatches

We study the problem of extracting, from given source x and error threshold k, substrings of x that occur unusually often in x within k substitutions or mismatches. Specifically, we assume that the input textstring x of n characters is produced by an i.i.d. source, and design efficient methods for computing the probability and expected number of occurrences for substrings of x with (either exactly or up to) k mismatches. Two related schemes are presented. In the first one, an O(nk) time preprocessing of x is developed that supports the following subsequent queries: for any substring w of x arbitrarily specified as input, the probability of occurrence of w in x within (either exactly or up to) k mismatches is reported in O(k(2)) time. In the second scheme, a length or length range is arbitrarily specified, and the above probabilities are computed for all substrings of x having length in that range, in overall O(nk) time. Further, monotonicity conditions are introduced and studied for probabilities and expected occurrences of a substring under unit increases in its length, allowed number of errors, or both. Over intervals of constant frequency count, these monotonicities translate to some of the scores in use, thereby reducing the size of tables at the outset and enhancing the process of discovery. These latter derivations extend to patterns with mismatches an analysis previously devoted to exact patterns.