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 $\delta$-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 $\mathrm{\Omega }((\delta \log n)\mathrm{/}(w+\log \log n))$ 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 $\mathrm{\Omega }(\sqrt{\log n}\mathrm{/}(\log \log n{)}^{3\mathrm{/}2})$ which applies for both problems. This second set of results relies both on our new encoding scheme as well as a carefully constructed hard distribution.
• Finally, for the online edit distance problem we show that there is an $O((\log n{)}^2\mathrm{/}w)$ 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.