LCS approximation via embedding into locally non-repetitive strings
Information and Computation
SIAM Journal on Discrete Mathematics
The smoothed complexity of edit distance
ACM Transactions on Algorithms (TALG)
Efficient communication protocols for deciding edit distance
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Faster algorithm for computing the edit distance between SLP-Compressed strings
SPIRE'12 Proceedings of the 19th international conference on String Processing and Information Retrieval
Homomorphic fingerprints under misalignments: sketching edit and shift distances
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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We present a near-linear time algorithm that approximates the edit distance between two strings within a polylogarithmic factor. For strings of length $n$ and every fixed $\eps0$, the algorithm computes a $(\log n)^{O(1/\eps)}$ approximation in $n^{1+\eps}$ time. This is an {\em exponential} improvement over the previously known approximation factor, $2^{\tilde O(\sqrt{\log n})}$, with a comparable running time [Ostrovsky and Rabani, J. ACM 2007, Andoni and Onak, STOC 2009]. This result arises naturally in the study of a new \emph{asymmetric query} model. In this model, the input consists of two strings $x$ and $y$, and an algorithm can access $y$ in an unrestricted manner, while being charged for querying every symbol of $x$. Indeed, we obtain our main result by designing an algorithm that makes a small number of queries in this model. We then provide a nearly-matching lower bound on the number of queries. Our lower bound is the first to expose hardness of edit distance stemming from the input strings being ``repetitive'', which means that many of their substrings are approximately identical. Consequently, our lower bound provides the first rigorous separation between edit distance and Ulam distance.