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(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