The Computational Hardness of Estimating Edit Distance
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Andoni-2010-THE COMPUTATIONAL HA.pdf
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Author(s)
Andoni, Alexandr
Krauthgamer, Robert
Alternative Title
THE COMPUTATIONAL HARDNESS OF ESTIMATING EDIT DISTANCE
Date Issued
April 2010
Journal
SIAM Journal of Computing
Publisher
Society for Industrial and Applied Mathematics
Citation
Andoni, Alexandr, and Robert Krauthgamer. “The Computational Hardness of Estimating Edit Distance.” SIAM Journal on Computing 39.6 (2010): 2398-2429. ©2010 Society for Industrial and Applied Mathematics.
Version
Final published version
Abstract
We prove the first nontrivial communication complexity lower bound for the problem of estimating the edit distance (aka Levenshtein distance) between two strings. To the best of our knowledge, this is the first computational setting in which the complexity of estimating the edit distance is provably larger than that of Hamming distance. Our lower bound exhibits a trade-off between approximation and communication, asserting, for example, that protocols with $O(1)$ bits of communication can obtain only approximation $\alpha\geq\Omega(\log d/\log\log d)$, where $d$ is the length of the input strings. This case of $O(1)$ communication is of particular importance since it captures constant-size sketches as well as embeddings into spaces like $l_1$ and squared-$l_2$, two prevailing algorithmic approaches for dealing with edit distance. Indeed, the known nontrivial communication upper bounds are all derived from embeddings into $l_1$. By excluding low-communication protocols for edit distance, we rule out a strictly richer class of algorithms than previous results. Furthermore, our lower bound holds not only for strings over a binary alphabet but also for strings that are permutations (aka the Ulam metric). For this case, our bound nearly matches an upper bound known via embedding the Ulam metric into $l_1$. Our proof uses a new technique that relies on Fourier analysis in a rather elementary way.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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DOI of Published Version
https://doi.org/10.1137/080716530
