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

TL;DR

This paper establishes strong cell-probe lower bounds for online pattern matching problems like edit distance and Hamming distance, revealing fundamental limits of data structures in the streaming model, and contrasts these with upper bounds.

Contribution

It introduces new cell-probe lower bounds for online edit distance, Hamming distance, convolution, and LCS, using novel encoding schemes and hard distributions, and compares these with upper bounds.

Findings

Lower bounds for online Hamming distance and convolution depend on word size.

Lower bounds for online edit distance and LCS are established for bit-probe model.

An upper bound for online edit distance shows an exponential gap with the lower bounds.

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.