Pattern Matching under Polynomial Transformation

Abstract

We consider a class of pattern matching problems where a normalizing polynomial transformation can be applied at every alignment of the pattern and text. Normalized pattern matching plays a key role in fields as diverse as image processing and musical information processing, where application specific transformations are often applied to the input. By considering a wide range of such transformations, we provide fast algorithms and the first lower bounds for both new and old problems. Given a pattern of length $m$ and a longer text of length $n$, where both are assumed to contain integer values only, we first show $O(n\log m)$ time algorithms for pattern matching under linear transformations even when wildcard symbols can occur in the input. We then show how to extend the technique to polynomial transformations of arbitrary degree. Next we consider the problem of finding the minimum Hamming distance under polynomial transformation. We show that, for any $\varepsilon>0$, there cannot exist an $O(nm^{1-\varepsilon})$ time algorithm for additive and linear transformations conditional on the hardness of the classic 3Sum problem. Finally, we consider a version of the Hamming distance problem under additive transformations with a bound $k$ on the maximum distance that needs to be reported. We give a deterministic $O(nk\log k)$ time solution, which we then improve by careful use of randomization to $O(n\sqrt{k\log k}\log n)$ time for sufficiently small $k$. Our randomized solution outputs the correct answer at every position with high probability.