Approximate Similarity Search Under Edit Distance Using Locality-Sensitive Hashing

Edit distance similarity search, also called approximate pattern matching, is a fundamental problem with widespread database applications. The goal of the problem is to preprocess $n$ strings of length $d$, to quickly answer queries $q$ of the form: if there is a database string within edit distance $r$ of $q$, return a database string within edit distance $cr$ of $q$. Previous approaches to this problem either rely on very large (superconstant) approximation ratios $c$, or very small search radii $r$. Outside of a narrow parameter range, these solutions are not competitive with trivially searching through all $n$ strings. In this work give a simple and easy-to-implement hash function that can quickly answer queries for a wide range of parameters. Specifically, our strategy can answer queries in time $\tilde{O}(d3^rn^{1/c})$. The best known practical results require $c \gg r$ to achieve any correctness guarantee; meanwhile, the best known theoretical results are very involved and difficult to implement, and require query time at least $24^r$. Our results significantly broaden the range of parameters for which we can achieve nontrivial bounds, while retaining the practicality of a locality-sensitive hash function. We also show how to apply our ideas to the closely-related Approximate Nearest Neighbor problem for edit distance, obtaining similar time bounds.