Approximated Pattern Matching with the L ₁, L ₂ and L _∞ Metrics

Abstract

Given an alphabet Σ = {1,2,...,|Σ|} text string T ∈ Σ ⁿ and a pattern string P ∈ Σ ^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 $O(\frac{1}{\varepsilon^2} n\log mlog |\Sigma|)$ run time. Second, we provide an algorithm that approximates the L _∞ matching up to a factor of 1 + ε with a run time of $O(\frac{1}{\varepsilon} n\log mlog |\Sigma|)$. 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.