Relations Between δ-Matching and Matching with Don’t Care Symbols: δ-distinguishing Morphisms (Preliminary Version)

The δ-matching problem is a special version of approximate pattern-matching, motivated by applications in musical information retrieval, where the alphabet Σ is an interval of integers. The δ-match is an occurrence of the pattern such that |a − b| ≤ δ for each pair of letters (integers) whose positions match in this occurrence. We investigate relations between δ-matching and pattern-matching with don’t care symbol ∗ (a symbol matching every symbol, including itself). We show a close correspondence between pattern-matching with don’t cares and δ-matching. The δ-matching is reducible to k instances of pattern-matching with don’t cares. We investigate how the numbers δ and k are related by introducing δ-distinguishing families H of morphisms. The size of H corresponds to k. We show that for minimal families H we have |H| = Θ(δ). ⋆ This work was supported in part by NSF grant CCR-9800085. ⋆⋆ The work was partially supported by a NATO grant PST.CLG.977017, a Marie Curie Fellowship, a Royal Society and a Wellcome Foundation grant. ⋆ ⋆ ⋆ The work of this author was partially supported by a NATO grant PST.CLG.977017. † Partially supported by a Royal Society grant.