L1 pattern matching lower bound

Authors:
Ohad Lipsky;Ely Porat
Affiliations:
Department of Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel;Department of Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel
Venue:
Information Processing Letters
Year:
2008

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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@?N be given. In the Approximate Pattern Matching in theL"1metric 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. @?"j"="0^m^-^1|t"i"+"j-p"j|. The Less Than Matching problem is that of finding all locations i of T where t"i"+"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(nmlogm) time. In this paper we show that the latter two problems can be linearly reduced to the problem of L"1-matching.