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Convertible Codes for Data and Device Heterogeneity

Anina Gruica, Benjamin Jany, Stanislav Kruglik

TL;DR

This work addresses converting linear codes in distributed storage to adapt to both data heterogeneity and time-varying device reliability. It introduces convertible codes in the merge regime, establishes general lower bounds on read and write costs for linear conversions, and applies the framework to Reed-Muller codes to achieve efficient, dual-heterogeneity conversions. The RM-based constructions leverage RM structure and the Plotkin operation to merge multiple small codes into a larger one while controlling unchanged, read, and write-symbol counts, yielding partial optimality with respect to the derived bounds. Future work includes tightening RM-specific bounds by exploiting locality/availability properties and exploring bandwidth optimization and non-linear conversion schemes.

Abstract

Distributed storage systems must handle both data heterogeneity, arising from non-uniform access demands, and device heterogeneity, caused by time-varying node reliability. In this paper, we study convertible codes, which enable the transformation of one code into another with minimum cost in the merge regime, addressing the latter. We derive general lower bounds on the read and write costs of linear code conversion, applicable to arbitrary linear codes. We then focus on Reed-Muller codes, which efficiently handle data heterogeneity, addressing the former issue, and construct explicit conversion procedures that, for the first time, combine both forms of heterogeneity for distributed data storage.

Convertible Codes for Data and Device Heterogeneity

TL;DR

This work addresses converting linear codes in distributed storage to adapt to both data heterogeneity and time-varying device reliability. It introduces convertible codes in the merge regime, establishes general lower bounds on read and write costs for linear conversions, and applies the framework to Reed-Muller codes to achieve efficient, dual-heterogeneity conversions. The RM-based constructions leverage RM structure and the Plotkin operation to merge multiple small codes into a larger one while controlling unchanged, read, and write-symbol counts, yielding partial optimality with respect to the derived bounds. Future work includes tightening RM-specific bounds by exploiting locality/availability properties and exploring bandwidth optimization and non-linear conversion schemes.

Abstract

Distributed storage systems must handle both data heterogeneity, arising from non-uniform access demands, and device heterogeneity, caused by time-varying node reliability. In this paper, we study convertible codes, which enable the transformation of one code into another with minimum cost in the merge regime, addressing the latter. We derive general lower bounds on the read and write costs of linear code conversion, applicable to arbitrary linear codes. We then focus on Reed-Muller codes, which efficiently handle data heterogeneity, addressing the former issue, and construct explicit conversion procedures that, for the first time, combine both forms of heterogeneity for distributed data storage.
Paper Structure (11 sections, 12 theorems, 31 equations)

This paper contains 11 sections, 12 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Convertible Codes in the Merge Regime
  3. Preliminaries
  4. General Bounds on Write and Read Costs
  5. Bounds on the Write Cost
  6. Bounds on the Read Cost
  7. Conversion of Reed-Muller Codes
  8. Definitions and Properties
  9. Reed-Muller Convertible Codes
  10. Comparison
  11. Discussion and Future Directions

Key Result

Lemma 1

Let $(\mathcal{C}_\textbf{I},\mathcal{C}_F,\sigma)$ be a convertible code with parameters $(n_\textbf{I},k_\textbf{I},n_F,k_F)$, and for $i \in [\lambda]$ let $G_{I_i} \in \mathbb{F}_q^{k_{I_i} \times n_{I_i}}$ be a generator matrix of $\mathcal{C}_{I_i}$, respectively, and let $G_{F} \in \mathbb{F} If the conversion procedure $\sigma$ is linear, then it can represented as a matrix $\mathcal{Y}$ w

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 5
  • Example 6
  • Lemma 1
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 22 more