Cell-probe bounds for online edit distance and other pattern matching problems

We give cell-probe bounds for the computation of edit distance, Hamming distance, convolution and longest common subsequence in a stream. In this model, a fixed string of n symbols is given and one δ-bit symbol arrives at a time in a stream. After each symbol arrives, the distance between the fixed string and a suffix of most recent symbols of the stream is reported. The cell-probe model is perhaps the strongest model of computation for showing data structure lower bounds, subsuming in particular the popular word-RAM model. • We first give an Ω ( (δ log n)/(w + log log n) ) lower bound for the time to give each output for both online Hamming distance and convolution, where w is the word size. This bound relies on a new encoding scheme and for the first time holds even when w is as small as a single bit. • We then consider the online edit distance and longest common subsequence problems in the bit-probe model (w = 1) with a constant sized input alphabet. We give a lower bound of Ω( √ log n/(log log n)) which applies for both problems. This second set of results relies both on our new encoding scheme as well as a carefully constructed hard distribution. • Finally, for the online edit distance problem we show that there is an O((log n)/w) upper bound in the cell-probe model. This bound gives a contrast to our new lower bound and also establishes an exponential gap between the known cell-probe and RAM model complexities.