On Indeterminate Strings Matching

Given two indeterminate equal-length strings p and t with a set of characters per position in both strings, we obtain a determinate string pw from p and a determinate string tw from t by choosing one character per position. Then, we say that p and t match when pw and tw match for some choice of the characters. While the most standard notion of a match for determinate strings is that they are simply identical, in certain applications it is more appropriate to use other definitions, with the prime examples being parameterized matching, order-preserving matching, and the recently introduced Cartesian tree matching. We provide a systematic study of the complexity of string matching for indeterminate equal-length strings, for different notions of matching. We use n to denote the length of both strings, and r to be an upper-bound on the number of uncertain characters per position. First, we provide the first polynomial time algorithm for the Cartesian tree version that runs in deterministic O(n log2 n) and expected O(n log n log log n) time using O(n log n) space, for constant r. Second, we establish NP-hardness of the order-preserving version for r = 2, thus solving a question explicitly stated by Henriques et al. [CPM 2018], who showed hardness for r = 3. Third, we establish NP-hardness of the parameterized version for r = 2. As both parameterized and order-preserving indeterminate matching reduce to the standard determinate matching for r = 1, this provides a complete classification for these three variants. 2012 ACM Subject Classification Theory of computation ! Pattern matching.