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Explicit Construction of Optimal Exact Regenerating Codes for Distributed Storage

K. V. Rashmi, Nihar B. Shah, P. Vijay Kumar, Kannan Ramchandran

arXiv:0906.4913v2cs.IT

TL;DR

Distributed storage codes must balance reliability, storage, and repair bandwidth while remaining practical to construct and operate. The paper introduces exact regeneration, develops explicit constructions at the MBR and MSR points, and uses a subspace approach to characterize and establish uniqueness for the MBR construction. The resulting codes are presented as low-complexity, low-field-size constructions with applications to mail-server and peer-to-peer storage settings.

  • Problem

    Distributed storage needs reliability with low storage overhead and low repair bandwidth, while existing optimal-code constructions can require high complexity and large field sizes.

  • Method

    The paper introduces exact regeneration, constructs explicit MBR and MSR codes, and applies a subspace-based approach to characterize linear MBR codes and prove uniqueness.

  • Results

    The paper provides explicit exact MBR constructions and explicit MSR constructions for suitable parameters, with the constructions described as low-complexity and low-field-size.

  • Takeaways & Limitations

    Exact regeneration can avoid additional communication and processing overhead, supporting practical maintenance; the MBR construction targets mail servers and the MSR construction targets peer-to-peer systems.

Abstract

from arXiv · show

Erasure coding techniques are used to increase the reliability of distributed storage systems while minimizing storage overhead. Also of interest is minimization of the bandwidth required to repair the system following a node failure. In a recent paper, Wu et al. characterize the tradeoff between the repair bandwidth and the amount of data stored per node. They also prove the existence of regenerating codes that achieve this tradeoff. In this paper, we introduce Exact Regenerating Codes, which are regenerating codes possessing the additional property of being able to duplicate the data stored at a failed node. Such codes require low processing and communication overheads, making the system practical and easy to maintain. Explicit construction of exact regenerating codes is provided for the minimum bandwidth point on the storage-repair bandwidth tradeoff, relevant to distributed-mail-server applications. A subspace based approach is provided and shown to yield necessary and sufficient conditions on a linear code to possess the exact regeneration property as well as prove the uniqueness of our construction. Also included in the paper, is an explicit construction of regenerating codes for the minimum storage point for parameters relevant to storage in peer-to-peer systems. This construction supports a variable number of nodes and can handle multiple, simultaneous node failures. All constructions given in the paper are of low complexity, requiring low field size in particular.

I. INTRODUCTION

Distributed storage seeks reliability with low storage overhead and low repair bandwidth, but practical regenerating-code constructions also require low complexity and field size. The paper introduces exact regeneration and constructs codes at the MBR and MSR points.

  • Motivation: Erasure coding improves reliability while limiting storage overhead, but node failures still require downloading data from surviving nodes for repair.MDS-based systems restore failed fragments from existing fragments, making repair bandwidth a significant system parameter.
  • Motivation: Regenerating codes trade increased per-node storage for reduced repair bandwidth, with MBR minimizing repair bandwidth and MSR minimizing stored data.These are the two practically relevant endpoints of the storage-repair bandwidth tradeoff.
  • Motivation: General network-coding-based constructions can have high construction complexity and require large field sizes despite optimal-code existence proofs.This creates a practical motivation for explicit low-complexity, low-field-size constructions.
  • Contributions: The paper provides an explicit low-field-size, low-complexity exact construction at the MBR point and explicit MSR constructions for suitable parameters supporting multiple node failures.The paper also claims these are the first explicit constructions of optimal regenerating codes.
  • Exact regeneration: Exact Regenerating Codes replace a failed node with an identical replica, avoiding additional communication and processing needed to update other nodes and data collectors.The exact-replica property means the system needs to satisfy reconstruction at only one level.
  • MBR application: The MBR point with d = n − 1 is presented as suitable for distributed mail servers because all surviving nodes participate in the fastest recovery.The construction sets β = 1 and can process larger files by splitting them into independently handled chunks.

A. Code construction

The MBR construction represents storage nodes and shared symbols through the incidence matrix of a complete graph, then assigns MDS-code combinations to graph edges. Its matrix structure supports reconstruction and exact regeneration.

  • Code construction: The n × θ matrix V is binary, with d ones per row, two ones per column, and exactly one shared one between every pair of rows.These properties make V the incidence matrix of a fully connected undirected graph.
  • Code construction: Each graph edge corresponds to an MDS-code vector, and node j stores f^t v_i whenever V(j, i) = 1.The construction gives n nodes, each storing d = α symbols.
  • Data Reconstruction: A data collector downloading from any k nodes obtains B distinct MDS symbols after removing repetitions, allowing recovery of all source symbols.The cited reconstruction argument identifies the downloaded-symbol count as kα minus repeated symbols, equal to B.
  • Exact Regeneration: Each surviving node contains one distinct symbol of a failed node, so downloading one symbol from every other node exactly regenerates the failed node.This property follows from the special structure of V.

B. Example

For n = 5 and k = 3, the construction stores graph-edge symbols across five nodes and reconstructs from any three nodes. A failed node is restored with exactly its former symbols.

  • Example: For n = 5 and k = 3, the parameters are d = 4, θ = 10, α = 4, and B = 9.The ten vectors correspond to the edges of the complete graph on five vertices.
  • Example: The five nodes store four pairwise-shared symbols each, with every vector appearing in the two nodes joined by its corresponding graph edge.The listed assignments cover vectors v_1 through v_10 across nodes 1 through 5.
  • Reconstruction: Any three nodes provide nine distinct symbols from which the source symbols f_0 through f_8 can be recovered.The example demonstrates reconstruction using nodes 1, 2, and 3, and states that the same holds for any three nodes.
  • Regeneration: When node 3 fails, nodes 1, 2, 4, and 5 each provide one symbol, and these four symbols reproduce node 3 exactly.The replacement therefore stores the same symbols as the failed node.
  • Field size: Because θ = B + 1, the ten vectors can form a single parity-check code of dimension 9, so this parameter set is realizable over F_2.The field-size claim applies to this example's exact regenerating code.

C. Field size required

The construction's required field size is determined by the field needed to construct its [θ, B] MDS code, with Reed–Solomon codes requiring θ = n(n−1)/2.

  • Field size required: The required field size is the minimum field size needed to construct a [θ, B] MDS code.With Reed–Solomon codes, the construction requires field size θ = n(n−1)/2.

D. Complexity

The construction minimizes regeneration overhead: encoding requires no arithmetic, and each helper node passes one stored symbol without recalculation. Exact regeneration also avoids coordination overhead caused by changing code coefficients.

  • Code construction:: No arithmetic operations are required for code construction when the incidence matrix V is given.
  • Node Regeneration:: Each existing node passes one stored symbol to the replacement node during regeneration.The symbol is selected from the α symbols already stored at that node.
  • Node Regeneration:: Exact regeneration avoids notifying nodes and data collectors about changed code coefficients.Without exact regeneration, nodes must also recalculate vectors needed for subsequent regenerations.
  • Data Reconstruction:: A systematic set of k nodes can be created by a change of basis, allowing reconstruction without decoding when the collector connects to that set.
  • Data Reconstruction:: Without exact regeneration, a failed systematic node may be replaced by a nonsystematic node, so the systematic property is lost.

IV. SUBSPACE VIEWPOINT AND UNIQUENESS

The subspace viewpoint characterizes linear exact regenerating codes at the MBR point through node and intersection-subspace properties. These conditions establish both necessary and sufficient structure and imply uniqueness of the construction up to basis choice.

  • Subspace viewpoint: Linear storage codes represent each node by a subspace of at most α dimensions and each helper transmission by a subspace of at most β dimensions.Stored symbols are linear combinations of source symbols, and regeneration uses d helper subspaces.
  • Characterization: The subspace viewpoint yields necessary and sufficient conditions for a linear code to be exact regenerating at the MBR point.
  • Necessary conditions: Each node in an MBR exact regenerating code must store an α-dimensional subspace.This follows from reconstruction by any k nodes and the resulting equality conditions on summed subspace dimensions.
  • Necessary conditions: For every node and any d helpers, the intersection subspaces have dimension β and are linearly independent.
  • Uniqueness: With d = n − 1, every linear exact regenerating code has the same structure as the construction, up to the choice of basis for node subspaces.
  • Characterization: It is sufficient that every set of d + 1 nodes have the construction’s structure for reconstruction and exact regeneration to hold.

V. REGENERATING CODES FOR THE MSR POINT

The MSR construction targets settings with minimal node storage, including peer-to-peer systems where nodes frequently join and leave. For d = k + 1, it supports multiple failures as long as at least k + 1 nodes remain functional.

  • Motivation: The MSR point minimizes storage per node and suits peer-to-peer systems with limited participating-node capacity.
  • Construction: For d = k + 1 and any n, the construction can handle any number of failures while at least k + 1 nodes remain functional.
  • Construction: If fewer than k nodes remain functional, part of the data is permanently lost.
  • Construction: The construction uses β = 1.

A. Code construction:

The construction partitions source symbols into two vectors and stores two linear combinations at each node. Main vectors determine reconstruction and regeneration, while auxiliary vectors may be chosen arbitrarily; Reed–Solomon main vectors require field size n.

  • Code construction:: The source symbols are partitioned into f_0, …, f_k−1 and g_0, …, g_k−1, forming vectors f and g.
  • Code construction:: Each node stores (f^t p_i, g^t p_i + f^t u_i) as its two symbols.The vectors p_i and u_i are called the main and auxiliary vectors, respectively.
  • Code construction:: Main vectors are used for reconstruction and regeneration, whereas auxiliary-vector elements may take arbitrary values from F_q.
  • Code construction:: The main vectors p_i form a k-dimensional MDS code over F_q.
  • Code construction:: A Reed–Solomon choice makes the required minimum field size equal to n.

B. Reconstruction:

Any data collector can reconstruct all B data units from any k nodes by solving two successive systems of k linear equations. This remains valid regardless of the auxiliary vectors.

  • Reconstruction:: The first symbols from any k nodes form k linear equations whose solution recovers f_0, ..., f_{k−1}.The equations are linearly independent because the p_i values form a k-dimensional MDS code.
  • Reconstruction:: After subtracting f^t u_i from the second symbols, the resulting values recover g_0, ..., g_{k−1}.
  • Reconstruction:: All B data units are recoverable from any k nodes, irrespective of the auxiliary vectors u_i.

C. Regeneration:

Regeneration reconstructs a failed node using one symbol from each of d helper nodes, with linear systems chosen to reproduce its main vector while allowing an auxiliary-vector change.

  • Regeneration:: The regenerated node preserves the failed node’s main vector, while its auxiliary vector may be any value in F_q.Reconstruction remains valid despite this auxiliary-vector difference.
  • Regeneration:: Regeneration uses d helper nodes, each providing one symbol formed as a linear combination of its stored symbols.The setup assumes node Λ_{d+1} fails and Λ_1, ..., Λ_d provide the repair symbols.
  • Regeneration:: The repair coefficients are selected in stages: first b_i, then ρ_i and δ_i, and finally a_i.These choices satisfy the linear constraints needed to generate the two symbols of the replacement node.
  • Regeneration:: The systems for δ_i and ρ_i contain k equations in d = k+1 unknowns and can be solved in F_q, with all δ_i non-zero.
  • Regeneration:: The final coefficient system for a_i is solvable because the δ_i are non-zero, enabling regeneration of any node from any d other nodes.

VI. CONCLUSION

The paper introduces exact regenerating codes and gives explicit constructions at both tradeoff endpoints, combining exact replacement with low complexity and low field-size requirements.

  • VI. CONCLUSION: Exact regenerating codes replace a failed node with an exact replica while meeting the storage-repair bandwidth tradeoff.The MBR construction uses d = n − 1 and is suited to mail-server applications requiring fast recovery.
  • VI. CONCLUSION: A subspace viewpoint proves uniqueness of the MBR construction, while the MSR construction uses d = k + 1 for peer-to-peer storage.The MSR setting supports a system whose node count varies over time.
  • VI. CONCLUSION: Both endpoint constructions require low field size and low complexity.
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