Space Lower Bounds for Online Pattern Matching

TL;DR

This paper establishes fundamental space lower bounds for online pattern matching across various distance measures, revealing a dichotomy based on the properties of the distance functions and extending to non-binary inputs.

Contribution

It provides the first tight space lower bounds for multiple distance measures in online pattern matching and characterizes the complexity based on distance function properties.

Findings

Omega(m) space lower bounds for several distance measures

A dichotomy between wildcard-like and non-wildcard-like distance functions

Potential for improved bounds in non-binary input scenarios

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 Omega(m) bit space lower bounds for L_1, L_2, L_\infty, 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 Omega(m) bit space lower bounds. For other distance functions, we show that there exist space bounds of Omega(log m) and O(log^2 m) bits. Finally we discuss space lower bounds for non-binary inputs and show how in some cases they can be improved.