  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.
