We study a fundamental problem in data analytics: similarity search under edit distance (or, edit similarity search for short). In this problem we try to build an index on a set of n strings S = s1, ..., sn, with the goal of answering the following two types of queries: (1) the threshold query: given a query string t and a threshold K, output all si ∈ S such that the edit distance between si and t is at most K; (2) the top-k query: given a query string t, output the k strings in S that are closest to t in terms of edit distance. Edit similarity search has numerous applications in bioinformatics, databases, data mining, information retrieval, etc., and has been studied extensively in the literature. In this paper we propose a novel algorithm for edit similarity search named MinSearch. The algorithm is randomized, and we can show mathematically that it outputs the correct answer with high probability for both types of queries. We have conducted an extensive set of experiments on MinSearch, and compared it with the best existing algorithms for edit similarity search. Our experiments show that MinSearch has a clear advantage (often in orders of magnitudes) against the best previous algorithms in query time, and MinSearch is always one of the best among all competitors in the indexing time and space usage. Finally, MinSearch achieves perfect accuracy for both types of queries on all datasets that we have tested.