Lower Bounds and Improved Algorithms for Asymmetric Streaming Edit Distance and Longest Common Subsequence

TL;DR

This paper establishes new space lower bounds for asymmetric streaming edit distance and LCS, and introduces improved algorithms that significantly reduce space complexity for these problems, advancing understanding in streaming models.

Contribution

The paper provides the first explicit lower bounds for ED and LCS in the asymmetric streaming model and presents improved algorithms with reduced space complexity.

Findings

Lower bounds for ED and LCS in asymmetric streaming model.

Exponential separation between edit distance and Hamming distance.

Improved algorithms with reduced space complexity for ED and LCS.

Abstract

In this paper, we study edit distance (ED) and longest common subsequence (LCS) in the asymmetric streaming model, introduced by Saks and Seshadhri [SS13]. As an intermediate model between the random access model and the streaming model, this model allows one to have streaming access to one string and random access to the other string. Our first main contribution is a systematic study of space lower bounds for ED and LCS in the asymmetric streaming model. Previously, there are no explicitly stated results in this context, although some lower bounds about LCS can be inferred from the lower bounds for longest increasing subsequence (LIS) in [SW07][GG10][EJ08]. Yet these bounds only work for large alphabet size. In this paper, we develop several new techniques to handle ED in general and LCS for small alphabet size, thus establishing strong lower bounds for both problems. In particular, our lower bound for ED provides an exponential separation between edit distance and Hamming distance in the asymmetric streaming model. Our lower bounds also extend to LIS and longest non-decreasing sequence (LNS) in the standard streaming model. Together with previous results, our bounds provide an almost complete picture for these two problems. As our second main contribution, we give improved algorithms for ED and LCS in the asymmetric streaming model. For ED, we improve the space complexity of the constant factor approximation algorithms in [FHRS20][CJLZ20] from $\tilde{O}(\frac{n^\delta}{\delta})$ to $O(\frac{d^\delta}{\delta}\;\mathsf{polylog}(n))$, where $n$ is the length of each string and $d$ is the edit distance between the two strings. For LCS, we give the first $1/2+\epsilon$ approximation algorithm with space $n^{\delta}$ for any constant $\delta>0$, over a binary alphabet.