Multiset Combinatorial Gray Codes with Application to Proximity Sensor Networks

TL;DR

This paper investigates coding schemes that map source symbols into multisets of an alphabet set, and provides a detailed analysis of color multiset codes on 1D grids, focusing on codes that require the minimal number of colors.

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 \textit{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 \textit{color mapping} and the code as a \textit{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.

Figures (7)

• Figure 1: A 1D color mapping on $\mathcal{G}_{10}$ with $4$ colors.
• Figure 2: A 2D color mapping on $\mathcal{G}_{5,7}$ with $4$ colors.
• Figure 3: (a) An Eulerian circuit of $K_5$. The vertices that it travels is $1,2,3,4,5,1,3,5,2,4$ in order. (b) An Eulerian circuit of $K_6-F$, where $F$ contains the three edges $\{1,2\}, \{3,4\}$ and $\{5,6\}$. The vertices that it travels is $132451625364$ in order.
• Figure 4: $L = 6 \delta$. Sensors are deployed in the grid topology to cover the monitored area $L \times L$, indicated by the magenta square. A sensor at $(i \delta, j \delta)$ would detect the presence of object in the vicinity $x \in ((i-0.5) \delta, (i+0.5) \delta]$ and $y \in ((j-0.5) \delta, (j+0.5) \delta]$, and then transmits the pre-assigned unique ID to signal. In the above example, sensor 4 is triggered and sends its ID, denoted by $\text{C}_4$, to inform a remote observer.
• Figure 5: $L = 6 \delta$ and $m = 2$. Sensors are deployed for monitoring the same area with same size $L \times L$, indicated by the magenta square. A sensor at $(i \delta, j \delta)$ would detect the presence of object in the vicinity $x \in ((i-1) \delta, (i+1) \delta]$ and $y \in ((j-1) \delta, (j+1) \delta]$, and then transmits the pre-assigned ID to signal. Here, sensors 1, 2, 3 and 4 are triggered as $m = 2$. In contrary to the setup in Fig. \ref{['fig:Fig1']}, we do not have to apply distinct ID for each sensor. As shown above, only 5 distinct IDs (indicated as 5 colors) are required for the sensors since the remote observer can distinguish where the object is located by each set of $m^2$-neighboring sensor IDs.

Theorems & Definitions (34)

• Definition 1
• Example 1
• Proposition 1
• proof
• Proposition 2
• Example 2
• Proposition 3
• Proposition 4
• proof
• Remark 1