Near-Linear Time Insertion-Deletion Codes And (1+Epsilon)-Approximating Edit Distance Via Indexing

We introduce fast-decodable indexing schemes for edit distance which can be used to speed up edit distance computations to near-linear time if one of the strings is indexed by an indexing string I. In particular, for every length n and every epsilon > 0, one can in near linear time construct a string I is an element of Sigma'(n) with vertical bar Sigma'vertical bar = O-epsilon (1), such that, indexing any string S epsilon Sigma(n), symbol-by-symbol, with I results in a string S' is an element of Sigma ''(n) where Sigma ''' = Sigma x Sigma' for which edit distance computations are easy, i.e., one can compute a (1 + epsilon)-approximation of the edit distance between S' and any other string in O(npoly(logn)) time. Our indexing schemes can be used to improve the decoding complexity of state-of-the-art error correcting codes for insertions and deletions. In particular, they lead to near-linear time decoding algorithms for the insertion-deletion codes of [Haeupler, Shahrasbi; STOC '17] and faster decoding algorithms for list-decodable insertion-deletion codes of [Haeupler, Shahrasbi, Sudan; ICALP '18]. Interestingly, the latter codes are a crucial ingredient in the construction of fast-decodable indexing schemes.