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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([EQUATION]log n/(log log n)3/2) which applies for both problems. This second set of results relies both on our new encoding scheme as well as a carefully constructed hard input distribution.  • Finally, for the online edit distance problem we show that there is an O((log2n)/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.

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