Randomized algorithms
Approximate nearest neighbors: towards removing the curse of dimensionality
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Bounds for the String Editing Problem
Journal of the ACM (JACM)
Approximate nearest neighbors and sequence comparison with block operations
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Communication complexity of document exchange
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Space lower bounds for distance approximation in the data stream model
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Efficient Search for Approximate Nearest Neighbor in High Dimensional Spaces
SIAM Journal on Computing
Lower bounds for embedding edit distance into normed spaces
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
A sublinear algorithm for weakly approximating edit distance
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Algorithmic Applications of Low-Distortion Geometric Embeddings
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Optimal space lower bounds for all frequency moments
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Approximate Nearest Neighbor under edit distance via product metrics
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
An information statistics approach to data stream and communication complexity
Journal of Computer and System Sciences - Special issue on FOCS 2002
Approximating Edit Distance Efficiently
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Oblivious string embeddings and edit distance approximations
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Improved lower bounds for embeddings into L1
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
The string edit distance matching problem with moves
ACM Transactions on Algorithms (TALG)
Estimating the sortedness of a data stream
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Low distortion embeddings for edit distance
Journal of the ACM (JACM)
Earth mover distance over high-dimensional spaces
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Efficient and private distance approximation in the communication and streaming models
Efficient and private distance approximation in the communication and streaming models
The influence of variables on Boolean functions
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Fast and compact regular expression matching
Theoretical Computer Science
Overcoming the l1 non-embeddability barrier: algorithms for product metrics
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Approximating edit distance in near-linear time
Proceedings of the forty-first annual ACM symposium on Theory of computing
Lower bounds for edit distance and product metrics via Poincaré-type inequalities
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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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.