Set intersection and sequence matching

In the classical pattern matching problem, one is given a text and a pattern, both of which are sequences of letters, and is required to find all occurrences of the pattern in the text. We study two modifications of the classical problem, where each letter in the text and pattern is a set (Set Intersection Matching problem) or a sequence (Sequence Matching problem). Two "letters" are considered to be match if the intersection of the two corresponding sets is not empty, or if the two sequences have a common element in the same index. We show the first known non-trivial and efficient algorithms for these problems, for the case the maximum set/sequence size is small. The first, randomized, that takes Θ (2^dn ln n log m) time, where d is the maximum set/sequence size, and can also fit, with slight modifications, for the case one is also interested in up to k mismatches. The second is deterministic and takes Θ (4^dn log m). The third algorithm, also deterministic, is able to count the number of matches at each index of the text in total running time Θ (∑_i=1^d ( _i^|∑|)n log m).