  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 $\varepsilon >0$, one can in near linear time construct a string
$I \in \Sigma'^n$ with $|\Sigma'| = O_{\varepsilon}(1)$, such that, indexing
any string $S \in \Sigma^n$, symbol-by-symbol, with $I$ results in a string $S'
\in \Sigma''^n$ where $\Sigma'' = \Sigma \times \Sigma'$ for which edit
distance computations are easy, i.e., one can compute a
$(1+\varepsilon)$-approximation of the edit distance between $S'$ and any other
string in $O(n \text{poly}(\log n))$ 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.
