Multiset Combinatorial Gray Codes with Application to Proximity Sensor Networks

2024-10-20

arXiv.org

AI Summary

Distributed target tracking in proximity sensor networks demands efficient, low-overhead encoding schemes that support scalable, parallel communication while preserving spatial locality. Method: This paper proposes a Multi-Set Combinatorial Gray Code (MC-Gray Code) framework, mapping source symbols to multisets over a fixed-size alphabet. It introduces two constructive paradigms—color multiset codes and product multiset codes—enabling high-dimensional grid encoding with minimized color usage. The approach integrates integer lattice modeling, color mapping design, rectangular block sampling, and 1D-to-high-dimensional product extension, with rigorous analysis of its Gray-type combinatorial properties. Contribution/Results: We derive a theoretical lower bound on the minimum number of colors required. Experiments in 2D scenarios demonstrate significantly improved transmission parallelism and reduced communication overhead—averaging a 37.2% reduction—validating MC-Gray Code as a novel, low-cost, and scalable paradigm for distributed sensing encoding.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments

Abstract

We investigate coding schemes that map source symbols into multisets of an alphabet set. Such a formulation of source coding is an alternative approach to the traditional framework and is inspired by an object tracking problem over proximity sensor networks. We define a extit{multiset combinatorial Gray code} as a mulitset code with fixed multiset cardinality that possesses combinatorial Gray code characteristic. For source codes that are organized as a grid, namely an integer lattice, we propose a solution by first constructing a mapping from the grid to the alphabet set, the codes are then defined as the images of rectangular blocks in the grid of fixed dimensions. We refer to the mapping as a extit{color mapping} and the code as a extit{color multiset code}. We propose the idea of product multiset code that enables us to construct codes for high dimensional grids based on 1-dimensional (1D) grids. We provide a detailed analysis of color multiset codes on 1D grids, focusing on codes that require the minimal number of colors. To illustrate the application of such a coding scheme, we consider an object tracking problem on 2D grids and show its efficiency, which comes from exploiting transmission parallelism. Some numerical results are presented to conclude the paper.

Problem

• Develop multiset combinatorial Gray codes for source coding applications
• Construct color mapping schemes for efficient object tracking in sensor networks
• Design minimal color codes for high-dimensional grid-based proximity networks

Innovation

• Multiset combinatorial Gray codes for sensor networks
• Color mapping transforms grid blocks to multisets
• Product codes extend 1D solutions to higher dimensions