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On Estimating Edit Distance: Alignment, Dimension Reduction, and Embeddings

Authors Moses Charikar, Ofir Geri, Michael P. Kim, William Kuszmaul

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Author Details

Moses Charikar
  • Department of Computer Science, Stanford University, Stanford, CA, USA
Ofir Geri
  • Department of Computer Science, Stanford University, Stanford, CA, USA
Michael P. Kim
  • Department of Computer Science, Stanford University, Stanford, CA, USA
William Kuszmaul
  • Department of Computer Science, Stanford University, Stanford, CA, USA

Funding

  • Charikar, Moses: Supported by NSF grant CCF-1617577 and a Simons Investigator Award.
  • Geri, Ofir: Supported by NSF grant CCF-1617577, a Simons Investigator Award, and the Google Graduate Fellowship in Computer Science in the School of Engineering at Stanford University.
  • Kim, Michael P.: Supported by NSF grant CCF-1763299.

Cite AsGet BibTex

Moses Charikar, Ofir Geri, Michael P. Kim, and William Kuszmaul. On Estimating Edit Distance: Alignment, Dimension Reduction, and Embeddings. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ICALP.2018.34

Abstract

Edit distance is a fundamental measure of distance between strings and has been widely studied in computer science. While the problem of estimating edit distance has been studied extensively, the equally important question of actually producing an alignment (i.e., the sequence of edits) has received far less attention. Somewhat surprisingly, we show that any algorithm to estimate edit distance can be used in a black-box fashion to produce an approximate alignment of strings, with modest loss in approximation factor and small loss in run time. Plugging in the result of Andoni, Krauthgamer, and Onak, we obtain an alignment that is a (log n)^{O(1/epsilon^2)} approximation in time O~(n^{1 + epsilon}). Closely related to the study of approximation algorithms is the study of metric embeddings for edit distance. We show that min-hash techniques can be useful in designing edit distance embeddings through three results: (1) An embedding from Ulam distance (edit distance over permutations) to Hamming space that matches the best known distortion of O(log n) and also implicitly encodes a sequence of edits between the strings; (2) In the case where the edit distance between the input strings is known to have an upper bound K, we show that embeddings of edit distance into Hamming space with distortion f(n) can be modified in a black-box fashion to give distortion O(f(poly(K))) for a class of periodic-free strings; (3) A randomized dimension-reduction map with contraction c and asymptotically optimal expected distortion O(c), improving on the previous O~(c^{1 + 2 / log log log n}) distortion result of Batu, Ergun, and Sahinalp.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • edit distance
  • alignment
  • approximation algorithms
  • embedding
  • dimension reduction

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