Pattern Matching under Polynomial Transformation

TL;DR

This paper develops fast algorithms and establishes lower bounds for pattern matching under polynomial transformations, addressing both linear and higher-degree cases, with applications in image and music processing.

Contribution

It introduces the first efficient algorithms and lower bounds for pattern matching under polynomial transformations, extending techniques to arbitrary degrees and analyzing Hamming distance problems.

Findings

O(n log m) time algorithms for linear transformations with wildcards

Extension of algorithms to polynomial transformations of arbitrary degree

Conditional lower bounds based on 3SUM hardness for certain problems

Abstract

We consider a class of pattern matching problems where a normalising transformation is applied at every alignment. Normalised 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 the class of polynomial transformations of the input, 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 epsilon>0, there cannot exist an O(n m^(1-epsilon)) 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 need be reported. We give a deterministic O(nk log k) time solution which we then improve by careful use of randomisation to O(n sqrt(k log k) log n) time for sufficiently small k. Our randomised solution outputs the correct answer at every position with high probability.