The least distortion of embedding edit distance into l1

Theorem 1.1: there are absolute $c,C>0$ and $d_0$ with $\exp(c\sqrt{\log d\log\log d})\le E_\Sigma(d)\le\exp(C\sqrt{\log d\log\log d})$ for all $d\ge d_0$ and all finite $|\Sigma|\ge2$, so the Ostrovsky-Rabani scale is sharp up to constants in the exponent. The upper bound is Ostrovsky-Rabani's method made uniform over alphabets and lengths; the new content is the matching lower bound, given by three constructions across the family. The constants $c,C$ are not determined, and nothing is said about computational efficiency.

Theorem 1.1: there are absolute $c,C>0$ and $d_0$ with $\exp(c\sqrt{\log d\log\log d})\le E_\Sigma(d)\le\exp(C\sqrt{\log d\log\log d})$ for all $d\ge d_0$ and all finite $|\Sigma|\ge2$, so the Ostrovsky-Rabani scale is sharp up to constants in the exponent. The upper bound is Ostrovsky-Rabani's method made uniform over alphabets and lengths; the new content is the matching lower bound, given by three constructions across the family. The constants $c,C$ are not determined, and nothing is said about computational efficiency.