Transposition Invariant Pattern Matching for Multi-Track Strings

We consider the problem of multi-track string matching. The task is to find the occurrences of a pattern across parallel strings. Given an alphabet Σ of natural numbers and a set S over Σ of h strings si = s1 · · · s i n for i = 1, . . . , h, a pattern p = p1 · · · pm has such an occurrence at position j of S if p1 = s i1 j , p2 = s i2 j+1, . . . , pm = s im j+m−1 holds for i1, . . . , im ∈ {1, . . . , h}. An application of the problem is music retrieval where occurrences of a monophonic query pattern are searched in a polyphonic music database. In music retrieval it is even more pertinent to allow invariance for pitch level transpositions, i.e., the task is to find whether there are occurrences of p in S such that the formulation above becomes p1 = s i1 j + c, p2 = s i2 j+1 + c, . . . , pm = s im j+m−1 + c for some constant c. We present several algorithms solving the problem. Our main contribution, the MP algorithm, is a transposition-invariant bit-parallel filtering algorithm for static databases. After an O(nhe) time preprocessing, it finds candidates for transposition invariant occurrences in time O(n⌈m/w⌉+m + d) where w, e, and d denote the size of the machine word in bits and two factors dependent on the size of the alphabet, respectively. A straightforward algorithm is used to check whether the candidates are proper occurrences. The algorithm needs time O(hm) per candidate. ACM CCS