Conference: The 41st ACM Symposium on Theory of Computing (STOC 2009).

Journal: SIAM Journal on Computing, 41(6), 2012 (special issue on STOC 2009).

Abstract: We show how to compute the edit distance between two strings of length
$n$ up to a factor of $2^{\tilde O(\sqrt{\log n})}$ in $n^{1+o(1)}$
time. This is the first sub-polynomial approximation algorithm for this problem
that runs in near-linear time, improving on the state-of-the-art
$n^{1/3+o(1)}$ approximation. Previously, approximation of
$2^{\tilde O(\sqrt{\log n})}$ was known only for *embedding* edit
distance into $\ell_1$, and it is not known if that
embedding can be computed in less than quadratic time.

Update: Our new paper *Polylogarithmic Approximation for Edit Distance and the Asymmetric Query Complexity* gives a polylogarithmic approximation algorithm that runs in strongly subquadratic time.