L ₁ Pattern Matching Lower Bound

Abstract

Let a text string T = t ₀,...,t _{n − − 1} and a pattern string P = p ₀,..., p _{m − − 1} t _i , p _j  ∈ IN be given. In The Approximate Pattern Matching in the L ₁ metric problem (L ₁-matching for short) the output is, for every text location i, the L ₁ distance between the pattern and the length m substring of the text starting at i, i.e. Σ$_{j=0}^{m-1}|{\it t}_{i+{\it j}}$ – p _j | . The Less Than Matching problem is that of finding all locations i of T where t $_{i+{\it j}}$ ≥ p _j j = 0,..., m–1 . The String Matching with Mismatches problem is that of finding the number of mismatches between the pattern and every length m substring of the text. For the three above problems, the fastest known deterministic solution is $O(n\sqrt{m{\rm log}m})$ time.

In this paper we show that the latter two problems can be linearly reduced to the problem of L ₁-matching.

Partially supported by GIF Young Scientists Program grant 2055-1168.6/2002.