Constructing Low-Redundancy Codes via Distributed Graph Coloring

2025-12-03

AI Summary

This work addresses the construction of low-redundancy error-correcting codes. We propose a novel coding framework based on distributed graph coloring, leveraging the one-to-one correspondence between independent sets in confusion graphs and uniquely decodable codewords to achieve efficient encoding and decoding within the LOCAL model. To enhance list-decoding capability, we introduce hypergraph labeling and design a progressive synchronization mechanism adaptive to unknown edit distances, enabling correction of long burst errors. Our algorithm improves upon Linial’s coloring by integrating distributed graph coloring with confusion graph modeling, yielding uniquely decodable codes with redundancy only twice the Gilbert–Varshamov bound. For list decoding, our approach significantly outperforms syndrome-based compression across diverse parameter regimes and achieves asymptotic performance within a factor of eight of the optimal lower bound.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We present a general framework for constructing error-correcting codes using distributed graph coloring under the LOCAL model. Building on the correspondence between independent sets in the confusion graph and valid codes, we show that the color of a single vertex - consistent with a global proper coloring - can be computed in polynomial time using a modified version of Linial's coloring algorithm, leading to efficient encoding and decoding. Our results include: i) uniquely decodable code constructions for a constant number of errors of any type with redundancy twice the Gilbert-Varshamov bound; ii) list-decodable codes via a proposed extension of graph coloring, namely, hypergraph labeling; iii) an incremental synchronization scheme with reduced average-case communication when the edit distance is not precisely known; and iv) the first asymptotically optimal codes (up to a factor of 8) for correcting bursts of unbounded-length edits. Compared to syndrome compression, our approach is more flexible and generalizable, does not rely on a good base code, and achieves improved redundancy across a range of parameters.

Problem

• Constructing low-redundancy error-correcting codes via distributed graph coloring
• Developing efficient encoding and decoding using modified Linial's coloring algorithm
• Achieving improved redundancy for various error types and edit scenarios

Innovation

• Distributed graph coloring for error-correcting codes
• Hypergraph labeling for list-decodable code construction
• Incremental synchronization with reduced communication for edits