# Cell-Probe Bounds for Online Edit Distance and Other Pattern Matching Problems

Raphael Clifford, Markus Jalsenius, Benjamin Sach

Research output: Chapter in Book/Report/Conference proceedingConference Contribution (Conference Proceeding)

6 Citations (Scopus)

## Abstract

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 $\Omega((\delta \log n)/(w+\log\log n))$ 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 $\Omega(\sqrt{\log n}/(\log\log n)^{3/2})$ which applies for both problems. This second set of results relies both on our new encoding scheme as well as a carefully constructed hard distribution. * Finally, for the online edit distance problem we show that there is an $O((\log n)^2/w)$ 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.
Original language English ACM-SIAM Symposium on Discrete Algorithms Published - 4 Jan 2015 ACM-SIAM Symposium on Discrete Algorithms (2015) - San Diego, United StatesDuration: 4 Jan 2015 → 6 Jan 2015Conference number: 26

### Conference

Conference ACM-SIAM Symposium on Discrete Algorithms (2015) SODA15 United States San Diego 4/01/15 → 6/01/15

### Bibliographical note

Accepted: 9th September 2014

• cs.DS
• F.2.2; F.1.2

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