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Streaming and Small Space Approximation Algorithms for Edit Distance and Longest Common Subsequence

Authors Kuan Cheng, Alireza Farhadi, MohammadTaghi Hajiaghayi, Zhengzhong Jin, Xin Li, Aviad Rubinstein, Saeed Seddighin, Yu Zheng

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Author Details

Kuan Cheng
  • Peking University, Beijing, China
Alireza Farhadi
  • University of Maryland, College Park, MD, USA
MohammadTaghi Hajiaghayi
  • University of Maryland, College Park, MD, USA
Zhengzhong Jin
  • Johns Hopkins University, Baltimore, MD, USA
Xin Li
  • Johns Hopkins University, Baltimore, MD, USA
Aviad Rubinstein
  • Stanford University, CA, USA
Saeed Seddighin
  • Toyota Technological Institute, Chicago, IL, USA
Yu Zheng
  • Johns Hopkins University, Baltimore, MD, USA

Funding

  • Cheng, Kuan: supported by a start-up funding of Peking University.
  • Farhadi, Alireza: supported in part by Facebook Fellowship.
  • Hajiaghayi, MohammadTaghi: supported in part by NSF BIGDATA grant IIS-1546108, and NSF SPX grant CCF-1822738.
  • Jin, Zhengzhong: supported by NSF Award CCF-1617713 and NSF CAREER Award CCF-1845349.
  • Li, Xin: supported by NSF Award CCF-1617713 and NSF CAREER Award CCF-1845349.
  • Rubinstein, Aviad: supported in part by a David and Lucile Packard Fellowship.
  • Seddighin, Saeed: supported by an Adobe Research Award and a Google Research Gift.
  • Zheng, Yu: supported by NSF Award CCF-1617713 and NSF CAREER Award CCF-1845349.

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Kuan Cheng, Alireza Farhadi, MohammadTaghi Hajiaghayi, Zhengzhong Jin, Xin Li, Aviad Rubinstein, Saeed Seddighin, and Yu Zheng. Streaming and Small Space Approximation Algorithms for Edit Distance and Longest Common Subsequence. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 54:1-54:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ICALP.2021.54

Abstract

The edit distance (ED) and longest common subsequence (LCS) are two fundamental problems which quantify how similar two strings are to one another. In this paper, we first consider these problems in the asymmetric streaming model introduced by Andoni, Krauthgamer and Onak [Andoni et al., 2010] (FOCS'10) and Saks and Seshadhri [Saks and Seshadhri, 2013] (SODA'13). In this model we have random access to one string and streaming access the other one. Our main contribution is a constant factor approximation algorithm for ED with memory Õ(n^δ) for any constant δ > 0. In addition to this, we present an upper bound of Õ _ε(√n) on the memory needed to approximate ED or LCS within a factor 1±ε. All our algorithms are deterministic and run in polynomial time in a single pass.
We further study small-space approximation algorithms for ED, LCS, and longest increasing sequence (LIS) in the non-streaming setting. Here, we design algorithms that achieve 1 ± ε approximation for all three problems, where ε > 0 can be any constant and even slightly sub-constant. Our algorithms only use poly-logarithmic space while maintaining a polynomial running time. This significantly improves previous results in terms of space complexity, where all known results need to use space at least Ω(√n). Our algorithms make novel use of triangle inequality and carefully designed recursions to save space, which can be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Edit Distance
  • Longest Common Subsequence
  • Longest Increasing Subsequence
  • Space Efficient Algorithm
  • Approximation Algorithm

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