Approximate pattern matching with the L ₁, L ₂ and L _∞ metrics

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 _p (i) as the p-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=1,2,∞. First, in the case of L ₁ matching (pattern matching with an L ₁ distance) we show a reduction of the string matching with mismatches problem to the L ₁ matching problem and we present an algorithm that approximates the L ₁ matching up to a factor of 1+ε, which has an O(1/ε ²n log m log|Σ|) run time. Then, the L ₂ matching problem (pattern matching with an L ₂ distance) is solved with a simple O(nlog∈m) time algorithm. Finally, we provide an algorithm that approximates the L _∞ matching up to a factor of 1+ε with a run time of O(1/εnlog mlog|Σ|). We also generalize the problem of String Matching with mismatches to have weighted mismatches and present an O(nlog∈ ⁴ m) algorithm that approximates the results of this problem up to a factor of O(log∈m) in the case that the weight function is a metric.