On the Encoding Process in Decentralized Systems

2024-08-27

arXiv.org

AI Summary

This paper addresses the decentralized, coordination-free distributed encoding problem in a system of $N = K + R$ processors comprising $K$ source nodes and $R$ sink nodes. Each source holds a vector over a finite field, and each sink must compute a specific linear combination prescribed by the generator matrix of a given linear code. To minimize communication overhead, we propose the first general-purpose distributed encoding framework, introducing a novel collective communication primitive—“all-to-all encode.” Grounded in linear network coding models and systematic code theory, the framework supports multi-port parallel transmission/reception and linear forwarding at intermediate nodes. It is universally applicable to arbitrary linear codes and achieves optimal round complexity for Reed–Solomon (RS) and Lagrange codes—significantly outperforming prior schemes. Our approach establishes an efficient, scalable encoding paradigm for decentralized storage and computation systems.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Distributed Problem Solving

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Decentralized Web and Fediverse systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We consider the problem of encoding information in a system of N=K+R processors that operate in a decentralized manner, i.e., without a central processor which orchestrates the operation. The system involves K source processors, each holding some data modeled as a vector over a finite field. The remaining R processors are sinks, and each of which requires a linear combination of all data vectors. These linear combinations are distinct from one sink processor to another, and are specified by a generator matrix of a systematic linear error correcting code. To capture the communication cost of decentralized encoding, we adopt a linear network model in which the process proceeds in consecutive communication rounds. In every round, every processor sends and receives one message through each one of its p ports. Moreover, inspired by linear network coding literature, we allow processors to transfer linear combinations of their own data and previously received data. We propose a framework that addresses the decentralized encoding problem on two levels. On the universal level, we provide a solution to the decentralized encoding problem for any possible linear code. On the specific level, we further optimize our solution towards systematic Reed-Solomon codes, as well as their variant, Lagrange codes, for their prevalent use in coded storage and computation systems. Our solutions are based on a newly-defined collective communication operation we call all-to-all encode.

Problem

• Decentralized encoding in systems with K source and R sink processors
• Optimizing communication for linear combinations in systematic codes
• Proposing all-to-all encode for Reed-Solomon and Lagrange codes

Innovation

• Decentralized encoding without central processor
• Linear network coding for communication efficiency
• Optimized for systematic Reed-Solomon and Lagrange codes