The sharp exponential scale of edit-distance distortion
The following describes the scope of the Lean formalization related to the following accompanying paper(s):
- Edit Distance in $\ell_1$: Matching Bounds up to Constants in the Exponent
- Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance
- Tree Constructions for the $\ell_1$ Distortion of Binary Edit Distance
Scope
The formalization determines the exponential scale of the least $\ell_1$ distortion of unit-cost edit distance on strings of length at most $d$. For every sufficiently large $d$, uniformly over finite alphabets with at least two symbols, the distortion lies between $\exp(c\sqrt{\log d,\log\log d})$ and $\exp(C\sqrt{\log d,\log\log d})$ for absolute constants $c,C>0$. The lower bound has a witness consisting of binary strings of one common length, and the same scale controls the supremum over finite alphabets.
The formalization gives two finite-circle constructions of binary strings whose least $\ell_1$ distortion is at least $\exp(c\sqrt{\log d,\log\log d})$ for all sufficiently large length caps $d$. The witnesses use strings of one common length and include the binary coding transfers with their stated uniform bounds.
It also gives a finite histogram embedding with distortion at most $\exp(C\sqrt{\log d,\log\log d})$ for all strings of length at most $d$, uniformly over finite alphabets, including the empty string. Thus the lower and upper exponential scales agree up to absolute constants.
The formalization gives two lower-bound constructions for the $\ell_1$ distortion of ordinary edit distance on binary words. For every sufficiently large length cap $d$, each construction supplies a finite set of at least two binary words of one common length at most $d$ whose least $\ell_1$ distortion is at least $\exp(c\sqrt{\log d,\log\log d})$ for an absolute $c>0$.
The selected statements are the binary lower bounds. The paper's constant-distortion binary conversion and the companion upper embedding theorem are outside them.
Comparator links
| Result | Comparator statement |
|---|---|
| Matching exponential-scale bounds for edit-distance distortion | EditDistance.lean |
| Finite-circle obstructions and histogram embeddings for edit distance | FiniteCircle.lean |
| Binary edit-distance distortion lower bound | BinaryEditLower.lean |
| Tree-based binary edit-distance distortion lower bound | TreeEdit.lean |