Coding Schemes for Document Exchange under Multiple Substring Edits

TL;DR

This paper introduces efficient document exchange schemes for binary strings under multiple bounded substring edits, achieving near-optimal encoding lengths with low computational complexity and analyzing average case performance.

Contribution

It presents a low-complexity encoding scheme for multiple substring edits and analyzes average encoding lengths, extending prior work from single to multiple edits.

Findings

Encoding length of 4t log n + o(log n) bits with low complexity.

Average encoding length of (4t - 1) log n + o(log n) for uniform strings.

Prior schemes only addressed single substring edits.

Abstract

We study the document exchange problem under multiple substring edits. A substring edit in a string $\mathbf{x}$ occurs when a substring $\mathbf{u}$ of $\mathbf{x}$ is replaced by an arbitrary string $\mathbf{v}$. The lengths of $\mathbf{u}$ and $\mathbf{v}$ are bounded from above by a fixed constant. Let $\mathbf{x}$ and $\mathbf{y}$ be two binary strings that differ by multiple substring edits. The aim of document exchange schemes is to construct an encoding of $\mathbf{x}$ with small length such that $\mathbf{x}$ can be recovered using $\mathbf{y}$ and the encoding. We construct a low-complexity document exchange scheme with encoding length of $4t\log n+o(\log n)$ bits, where $n$ is the length of the string $\mathbf{x}$. The best known scheme achieves an encoding length of $4t \log n+O(\log\log n)$ bits, but at a much higher computational complexity. Then, we investigate the average length of valid encodings for document exchange schemes with uniform strings $\mathbf{x}$ and develop a scheme with an expected encoding length of $(4t-1) \log n+o(\log n)$ bits. In this setting, prior works have only constructed schemes for a single substring edit.