Near-Linear Time Insertion-Deletion Codes and (1+ε)-Approximating Edit Distance via Indexing
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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'| = O_ε(1), such that, indexing any string S ∈Σ^n, 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(n poly( n)) time. Our indexing schemes can be used to improve the decoding complexity of state-of-the-art error correcting codes for insertion and deletions. In particular, they lead to near-linear time decoding algorithms for the insertion-deletion codes of [Haeupler, Shahrasbi; STOC `17] and 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.
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