The sharp exponential scale of edit-distance distortion

We determine the exponential scale of the least ℓ_(1) distortion of unit-cost edit distance on all strings of length at most d. For every sufficiently large d, uniformly over finite alphabets of size at least two, the distortion lies between exp(c\sqrt{log dloglog d}) and exp(C\sqrt{log dloglog d}) for absolute constants c,C> 0. The lower bound already holds on binary strings of one common length. Thus the order of logarithmic distortion is sharp up to absolute constants.

We give two finite-circle constructions of binary strings whose least ℓ_(1) distortion is exp(\Omega(\sqrt{log dloglog d})), where d bounds their length. Both constructions supply words of one common length for every sufficiently large cap. Two direct binary coding arguments transfer the constructions with absolute distortion and logarithmic block width. We also develop the overlapping-substring method of Ostrovsky and Rabani into a complete finite histogram embedding at the same exponential scale, uniformly over all finite alphabets and all words of length at most d, including the empty word.

We give two independent constructions of binary words of one length at most d whose ordinary edit-distance metrics require ℓ_(1) distortion exp(\Omega(\sqrt{log dloglog d})) for every sufficiently large d. We also prove a constant-distortion binary conversion for one prescribed input length. Together with the companion upper embedding theorem, these lower bounds determine the order of logarithmic distortion uniformly over finite alphabets with at least two symbols.