Near-linear time insertion-deletion codes and (1+ε)-approximating edit distance via indexing

Abstract

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 ε > 0, one can in near linear time construct a string I ∈ Σ′n with |Σ′| = Oε (1), such that, indexing any string S ∈ Σn, symbol-by-symbol, with I results in a string S′ ∈ Σ′′n where Σ′′ = Σ × Σ′ for which edit distance computations are easy, i.e., one can compute a (1 + ε)-approximation of the edit distance between S′ and any other string in O(npoly(log n)) 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.