Approximated pattern matching with the L1, L2 and L∞ metrics

Abstract

Given an alphabet ∑ = {1,2,...,|∑|} text string T ∈ ∑n and a pattern string P ∈ ∑ m , for each i = 1,2,...,n - m + 1 define L d (i) as the d-norm distance when the pattern is aligned below the text and starts at position i of the text. The problem of pattern matching with L p distance is to compute L p (i) for every i = 1,2,...,n - m + 1. We discuss the problem for d = 1, ∞. First, in the case of L 1 matching (pattern matching with an L 1 distance) we present an algorithm that approximates the L 1 matching up to a factor of 1 + ε, which has an run time. Second, we provide an algorithm that approximates the L ∞ matching up to a factor of 1 + ε with a run time of . We also generalize the problem of String Matching with mismatches to have weighted mismatches and present an O(nlog4 m) algorithm that approximates the results of this problem up to a factor of O(logm) in the case that the weight function is a metric. © 2009 Springer Berlin Heidelberg.