Optimal Document Exchange and New Codes for Small Number of Insertions and Deletions

This paper gives a communication-optimal document exchange protocol and an efficient near optimal derandomization. This also implies drastically improved error correcting codes for small number of adversarial insertions and deletions. For any $n$ and $k < n$ our randomized hashing scheme takes any $n$-bit file $F$ and computes a $O(k \log \frac{n}{k})$-bit summary from which one can reconstruct $F$ given a related file $F'$ with edit distance $ED(F,F') \leq k$. The size of our summary is information-theoretically order optimal for all values of $k$, positively answering a question of Orlitsky. It also is the first non-trivial solution when a small constant fraction of symbols have been edited, producing an optimal summary of size $O(H(\delta)n)$ for $k=\delta n$. This concludes a long series of better-and-better protocols which produce larger summaries for sub-linear values of $k$. In particular, the recent break-through of [Belazzougi, Zhang; STOC'16] assumes that $k < n^\epsilon$ and produces a summary of size $O(k\log^2 k + k\log n)$. We also give an efficient derandomization with near optimal summary size $O(k \log^2 \frac{n}{k})$ improving, for every $k$, over a deterministic $O(k^2 + k \log^2 n)$ document exchange scheme by Belazzougi. This directly leads to near optimal systematic error correcting codes which efficiently can recover from any $k$ insertions and deletions while having $\Theta(k \log^2 \frac{n}{k})$ bits of redundancy. For the setting of $k=n\epsilon$ this $O(\epsilon \log^2 \frac{1}{\epsilon} \cdot n)$-bit redundancy is near optimal and a quadratic improvement over the binary codes of Guruswami and Li and Haeupler, Shahrasbi and Vitercik which have redundancy $\Theta\left(\sqrt{\epsilon} \log^{O(1)} \frac{1}{\epsilon} \cdot n\right)$.