Efficient Deterministic Single Round Document Exchange for Edit Distance

TL;DR

This paper introduces the first efficient deterministic protocol for single-round document exchange under edit distance, enabling effective error correction even with many adversarial errors, with optimal time and message size.

Contribution

It presents a novel deterministic protocol for document exchange with optimal efficiency in time and message size, improving over previous randomized approaches.

Findings

Protocol runs in $O(n\,\text{polylog}(n))$ time.

Message size is $O(k^2 + k\log^2 n)$ bits.

Enables efficient error correction for large adversarial edit errors.

Abstract

Suppose that we have two parties that possess each a binary string. Suppose that the length of the first string (document) is $n$ and that the two strings (documents) have edit distance (minimal number of deletes, inserts and substitutions needed to transform one string into the other) at most $k$. The problem we want to solve is to devise an efficient protocol in which the first party sends a single message that allows the second party to guess the first party's string. In this paper we show an efficient deterministic protocol for this problem. The protocol runs in time $O(n\cdot \mathtt{polylog}(n))$ and has message size $O(k^2+k\log^2n)$ bits. To the best of our knowledge, ours is the first efficient deterministic protocol for this problem, if efficiency is measured in both the message size and the running time. As an immediate application of our new protocol, we show a new error correcting code that is efficient even for large numbers of (adversarial) edit errors.