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 blackbox 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 minhash 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 blackbox fashion to give distortion O(f(poly(K))) for a class of periodicfree strings; (3) A randomized dimensionreduction 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.
BibTeX  Entry
@InProceedings{charikar_et_al:LIPIcs:2018:9038,
author = {Moses Charikar and Ofir Geri and Michael P. Kim and William Kuszmaul},
title = {{On Estimating Edit Distance: Alignment, Dimension Reduction, and Embeddings}},
booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
pages = {34:134:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770767},
ISSN = {18688969},
year = {2018},
volume = {107},
editor = {Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9038},
URN = {urn:nbn:de:0030drops90383},
doi = {10.4230/LIPIcs.ICALP.2018.34},
annote = {Keywords: edit distance, alignment, approximation algorithms, embedding, dimension reduction}
}
Keywords: 

edit distance, alignment, approximation algorithms, embedding, dimension reduction 
Collection: 

45th International Colloquium on Automata, Languages, and Programming (ICALP 2018) 
Issue Date: 

2018 
Date of publication: 

04.07.2018 