# Problem 59: Low Expansion Encoding of Edit Distance

Let $T = \bigcup_{i=1}^{n} \{0,1\}^i$. For a pair of strings $(x,y) \in T \times T$, let $\operatorname{ed}(x,y)$ denote the edit distance between $x$ and $y$, which is defined as the minimum number of character insertion, deletion, and substitution needed for converting $x$ into $y$.
Question: Is there a mapping $f:T \rightarrow \{0,1\}^{m}$ satisfying the following conditions
• $f$ is injective, i.e. it does not map different inputs to the same point.
• $m=O(n^c)$ for some constant $c \geq 1$.
• For strings with $\operatorname{ed}(x,y)=1$ we have $\mathcal{H}(f(x),f(y)) \le C$ for $C=o(\log n)$.
The same question holds for randomized mappings as long as they map different $x$ and $y$ to different points with high probability. Currently the best upper bound on $C$ is $O(\log n\log^*n)$ achieved through a randomized mapping that deploys the Locally Consistent Parsing method [CormodePSV-00]. For non-repetitive strings (the Ulam distance) there is a deterministic mapping with $C\leq 6$ and $c=2$. Preferably we would like to have mappings that are efficiently computable and are equipped with polynomial time decoding algorithms ($x$ can be obtained from $f(x)$ efficiently). See [Jowhari-12] for motivations on the problem.