Space lower bounds for online pattern matching

Authors:
Raphaël Clifford;Markus Jalsenius;Ely Porat;Benjamin Sach
Affiliations:
University of Bristol, Department of Computer Science, Bristol, UK;University of Bristol, Department of Computer Science, Bristol, UK;Bar-Ilan University, Department of Computer Science, Ramat-Gan, Israel;University of Warwick, Department of Computer Science, Coventry, UK
Venue:
Theoretical Computer Science
Year:
2013

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Abstract

We present space lower bounds for online pattern matching under a number of different distance measures. Given a pattern of length m and a text that arrives one character at a time, the online pattern matching problem is to report the distance between the pattern and a sliding window of the text as soon as the new character arrives. We require that the correct answer is given at each position with constant probability. We give @W(m) bit space lower bounds for L"1, L"2, L"~, Hamming, edit and swap distances as well as for any algorithm that computes the cross-correlation/convolution. We then show a dichotomy between distance functions that have wildcard-like properties and those that do not. In the former case which includes, as an example, pattern matching with character classes, we give @W(m) bit space lower bounds. For other distance functions, we show that there exist space bounds of @W(logm) and O(log^2m) bits. Finally we discuss space lower bounds for non-binary inputs and show how in some cases they can be improved.