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Regenerating codes with minimal disk I/O cost achieving optimal tradeoff between storage and repair bandwidth

Minhan Gao, Kenneth Shum

TL;DR

This work tackles the problem of regenerating codes for distributed storage with minimal disk I/O and optimal repair bandwidth. It proposes uncoded functional repair (repair-by-transfer) at the minimum repair bandwidth points along the trade-off, and frames the design as a dynamical system using signal flow graphs alongside matroid theory, specifically gammoids, to guarantee independence properties. The main contributions are a construction that achieves the optimal storage-bandwidth trade-off points for all $eta=1$ with $d=n-1$, an algorithm for packet selection that preserves the $(n,k)$ recovery property under unlimited single-node failures, and a field-size bound ensuring linear representability of the underlying gammoids. This approach yields practical, low-complexity repair procedures with provable robustness, applicable over finite fields of size independent of the number of failures.

Abstract

There are multiple performance metrics in the design of coding schemes for distributed storage systems. The first metric is called repair bandwidth, which measures the network resources required during the repair process. Another critical metric for repair efficiency is disk I/O cost, defined as the amount of data packets accessed at helper nodes to repair the failed node. In an encoding scheme with optimal I/O cost, the number of packets sent to the newcomer is exactly the same as the number of packets read from memory. This mode of repair is referred to as uncoded repair, as no coding operations are performed at the helper node. In addition to minimizing disk I/O cost, an uncoded repair mechanism has the advantage of incurring minimal computational overhead at the helper node. In this paper, we demonstrate that for single node failures, if all surviving nodes participate in the repair of the failed node, we can achieve all points on the fundamental tradeoff curve between storage and repair bandwidth. The design of the proposed encoding scheme is based on the theory of gammoids, a specialized class of graph-based matroids. We prove that this scheme can tolerate an unlimited number of node repair iterations over a field of fixed size.

Regenerating codes with minimal disk I/O cost achieving optimal tradeoff between storage and repair bandwidth

TL;DR

This work tackles the problem of regenerating codes for distributed storage with minimal disk I/O and optimal repair bandwidth. It proposes uncoded functional repair (repair-by-transfer) at the minimum repair bandwidth points along the trade-off, and frames the design as a dynamical system using signal flow graphs alongside matroid theory, specifically gammoids, to guarantee independence properties. The main contributions are a construction that achieves the optimal storage-bandwidth trade-off points for all with , an algorithm for packet selection that preserves the recovery property under unlimited single-node failures, and a field-size bound ensuring linear representability of the underlying gammoids. This approach yields practical, low-complexity repair procedures with provable robustness, applicable over finite fields of size independent of the number of failures.

Abstract

There are multiple performance metrics in the design of coding schemes for distributed storage systems. The first metric is called repair bandwidth, which measures the network resources required during the repair process. Another critical metric for repair efficiency is disk I/O cost, defined as the amount of data packets accessed at helper nodes to repair the failed node. In an encoding scheme with optimal I/O cost, the number of packets sent to the newcomer is exactly the same as the number of packets read from memory. This mode of repair is referred to as uncoded repair, as no coding operations are performed at the helper node. In addition to minimizing disk I/O cost, an uncoded repair mechanism has the advantage of incurring minimal computational overhead at the helper node. In this paper, we demonstrate that for single node failures, if all surviving nodes participate in the repair of the failed node, we can achieve all points on the fundamental tradeoff curve between storage and repair bandwidth. The design of the proposed encoding scheme is based on the theory of gammoids, a specialized class of graph-based matroids. We prove that this scheme can tolerate an unlimited number of node repair iterations over a field of fixed size.
Paper Structure (11 sections, 13 theorems, 37 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 11 sections, 13 theorems, 37 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Uncoded functional repair
  3. Linear Regenerating Code as a Dynamical System
  4. Matroid and Gammiod
  5. Matroids
  6. Linkings and Gammoids
  7. Signal Flow Graph Representation
  8. Selecting the Packets To Be Sent to the Newcomer
  9. Proof for Theorem \ref{['thm: linear_repre']}
  10. Proof of the Correctness of Algorithm 1
  11. Proof of Lemma \ref{['lem: dependent']}

Key Result

Theorem 1

For any $t$, the repair of the failed node in stage $t$ is I/O-cost-optimal.

Figures (3)

  • Figure 1: Tradeoff curve between storage and repair bandwidth for distributed storage system with parameters $B=1$, $k=4$, and $d=8$.
  • Figure 2: The directed graph that defines the gammoid in Example \ref{['ex:gammoid']}.
  • Figure 3: An example of signal flow graph. At stage 0, node 2 fails. It is regenerated by downloading the first packet from each of the surviving nodes. Nodes 3 fails at stage 1, and is regenerated by downloading the second packets in the other nodes.

Theorems & Definitions (27)

  • Example 1
  • Example 2
  • Remark
  • Remark
  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • ...and 17 more