A (1+epsilon)-approximation of edit distance in almost-linear time

Main theorem: one randomized algorithm, uniform in a rational accuracy input $\varepsilon\in(0,1)$, returns for explicitly stored strings with polynomially bounded integer symbols an estimate within factor $1+\varepsilon$ of unit-cost edit distance with probability at least 2/3, in worst-case expected time $N^{1+o(1)}$ on a logarithmic-word RAM; equal strings give 0 deterministically. The bound is pointwise in fixed $\varepsilon$: no polynomial dependence on $1/\varepsilon$ is claimed, and exact computation is used below very large accuracy-dependent thresholds, so it is an asymptotic scheme. Not shown: an alignment or edit script, weighted costs, a deterministic algorithm, or bounds when $\varepsilon$ shrinks with $N$.

Main theorem: one randomized algorithm, uniform in a rational accuracy input $\varepsilon\in(0,1)$, returns for explicitly stored strings with polynomially bounded integer symbols an estimate within factor $1+\varepsilon$ of unit-cost edit distance with probability at least 2/3, in worst-case expected time $N^{1+o(1)}$ on a logarithmic-word RAM; equal strings give 0 deterministically. The bound is pointwise in fixed $\varepsilon$: no polynomial dependence on $1/\varepsilon$ is claimed, and exact computation is used below very large accuracy-dependent thresholds, so it is an asymptotic scheme. Not shown: an alignment or edit script, weighted costs, a deterministic algorithm, or bounds when $\varepsilon$ shrinks with $N$.