Fast Multi-dimensional Approximate Pattern Matching

Abstract

We address the problem of approximate string matching in d dimensions, that is, to find a pattern of size m ^d in a text of size n ^d with at most k < m ^d errors (substitutions, insertions and deletions along any dimension). We use a novel and very flexible error model, for which there exists only an algorithm to evaluate the similarity between two elements in two dimensions at O(m ⁴) time. We extend the algorithm to d dimensions, at O(d!m ^2d) time and O(d!m ^2d-1) space.We also give the first search algorithm for such model, which is O(d!m ^d n ^d) time and O(d!m ^d n ^d-1) space. We show how to reduce the space cost to O(d!3^d m ^2d-1) with little time penalty. Finally, we present the first sublinear-time (on average) searching algorithm (i.e. not all text cells are inspected), which is O(kn ^d/m ^d-1) for k < (m/(d(logσ m- logσ d)))^d-1, where σ is the alphabet size. After that error level the filter still remains better than dynamic programming for k ≤ m ^d-1/(d(logσ m -logσ d))^(d-1)/d. These are the first search algorithms for the problem. As side-effects we extend to d dimensions an already proposed algorithm for two-dimensional exact string matching, and we obtain a sublinear-time filter to search in d dimensions allowing k mismatches.

Supported in part by Fondecyt grant 1-990627.