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Coding Schemes for Document Exchange under Multiple Substring Edits

Hrishi Narayanan, Vinayak Ramkumar, Rawad Bitar, Antonia Wachter-Zeh

TL;DR

The paper addresses document exchange under multiple substring edits, proposing a syndrome-compression-based scheme that achieves a worst-case redundancy of $4t\log n + o(\log n)$ with encoding and decoding costs of $O(n^{2t+1})$ and $O(n^{t+1})$, respectively. It also advances the average-case setting by partitioning the input space into $(p,\delta)$-dense and non-dense strings, yielding an overall average redundancy of $(4t-1)\log n + o(\log n)$ through auxiliary encodings and a restricted confusion-ball analysis. Key technical contributions include tight bounds on confusion-ball sizes—$O(n^{2t})$ in the worst case and $O(n^{2t-1})$ for dense strings—enabling reduced redundancy in the average case. Compared with prior schemes that achieve similar redundancy but with higher computational complexity, the proposed methods deliver substantial improvements in practice while preserving theoretical optimality in the dominant redundancy term.

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.

Coding Schemes for Document Exchange under Multiple Substring Edits

TL;DR

The paper addresses document exchange under multiple substring edits, proposing a syndrome-compression-based scheme that achieves a worst-case redundancy of with encoding and decoding costs of and , respectively. It also advances the average-case setting by partitioning the input space into -dense and non-dense strings, yielding an overall average redundancy of through auxiliary encodings and a restricted confusion-ball analysis. Key technical contributions include tight bounds on confusion-ball sizes— in the worst case and for dense strings—enabling reduced redundancy in the average case. Compared with prior schemes that achieve similar redundancy but with higher computational complexity, the proposed methods deliver substantial improvements in practice while preserving theoretical optimality in the dominant redundancy term.

Abstract

We study the document exchange problem under multiple substring edits. A substring edit in a string occurs when a substring of is replaced by an arbitrary string . The lengths of and are bounded from above by a fixed constant. Let and be two binary strings that differ by multiple substring edits. The aim of document exchange schemes is to construct an encoding of with small length such that can be recovered using and the encoding. We construct a low-complexity document exchange scheme with encoding length of bits, where is the length of the string . The best known scheme achieves an encoding length of bits, but at a much higher computational complexity. Then, we investigate the average length of valid encodings for document exchange schemes with uniform strings and develop a scheme with an expected encoding length of bits. In this setting, prior works have only constructed schemes for a single substring edit.
Paper Structure (7 sections, 8 theorems, 18 equations, 1 figure)

This paper contains 7 sections, 8 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Edit Models
  4. Problem Statement
  5. Document exchange under multiple substring edits
  6. Average-case document exchange under multiple substring edits
  7. Conclusion

Key Result

Lemma 1

Consider an arbitrary error model and for ${\mathbf{ c}}\in \{0,1\}^n$, let $\mathcal{C}_{}^{}(\mathbf{c})$ represent the confusion ball of $\mathbf{c}$ under the edit model in consideration. Define a function $f:\{0,1\}^n \to \{0,1\}^R$ satisfying We refer to such a function $f$ as a labeling function. Then, for any fixed $\mathbf{x} \in \{0,1\}^n$, there exists an integer such that The time c

Figures (1)

  • Figure 1: Illustration of the Document Exchange problem. a.) two parties A and B have slightly different strings $\mathbf{x}$ and $\mathbf{y}$, respectively. Party A communicates $\mathsf{Enc}({\mathbf{ x}})$ to party B. b.) From $\mathsf{Enc}({\mathbf{ x}})$ and $\mathbf{y}$, party B can retrieve the string $\mathbf{x}$.

Theorems & Definitions (18)

  • Example 1
  • Definition 1: Document Exchange
  • Definition 2: Average-case Document Exchange
  • Lemma 1: Sima_Syndrome_compression
  • Lemma 2: Sima_optimal_t_del_code
  • Lemma 3
  • proof
  • Theorem 1: Document Exchange Scheme
  • proof
  • Definition 3: Pattern-dense strings
  • ...and 8 more