Overlapped Arithmetic Codes

2025-02-28

AI Summary

This paper addresses longstanding challenges in distributed source coding, joint source-channel coding, and distributed joint coding. We propose Overlapping Arithmetic Coding (OAC), a novel generalization of arithmetic coding that relaxes the classical requirement of disjoint, gapless subintervals. By mapping source symbols to *overlapping*, *forbidden*, and *hybrid* subintervals, OAC establishes the first arithmetic coding framework supporting non-exclusive interval partitions. Leveraging an interval-mapping reconstruction framework—integrated with information-theoretic duality analysis and overlapping modeling—we establish a direct correspondence between coding structure and distributed/joint coding tasks. OAC not only enables efficient distributed source coding but also provides a unified, scalable theoretical framework for multi-terminal collaborative coding. It fundamentally extends arithmetic coding beyond traditional constraints, offering a new paradigm for structured, distributed compression and joint design.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography

Abstract

Arithmetic codes are usually deemed as the most important means to implement lossless source coding, whose principle is mapping every source symbol to a sub-interval in [0, 1). For every source symbol, the length of its mapping sub-interval is exactly equal to its probability. With this symbol-interval mapping rule, the interval [0,1) will be fully covered and there is neither overlapped sub-interval (corresponds to more than one source symbol) nor forbidden sub-interval (does not correspond to any source symbol). It is well-known that there is a duality between source coding and channel coding, so every good source code may also be a good channel code meanwhile, and vice versa. Inspired by this duality, arithmetic codes can be easily generalized to address many coding problems beyond source coding by redefining the source-interval mapping rule. If every source symbol is mapped to an enlarged sub-interval, the mapping sub-intervals of different source symbols will be partially overlapped and we obtain overlapped arithmetic codes, which can realize distributed source coding. On the contrary, if every source symbol is mapped to a narrowed sub-interval, there will be one or more forbidden sub-intervals in [0, 1) that do not correspond to any source symbol and we obtain forbidden arithmetic codes, which can implement joint source-channel coding. Furthermore, by allowing the coexistence of overlapped sub-intervals and forbidden sub-intervals, we will obtain hybrid arithmetic codes, which can cope with distributed joint source-channel coding.

Problem

• Extends arithmetic codes for distributed source coding.
• Introduces forbidden arithmetic codes for joint source-channel coding.
• Proposes hybrid arithmetic codes for distributed joint source-channel coding.

Innovation

• Overlapped arithmetic codes enable distributed source coding.
• Forbidden arithmetic codes implement joint source-channel coding.
• Hybrid arithmetic codes handle distributed joint source-channel coding.