# DELER: Distributed Clustering with Local Direction Centrality and Density Measure
## 2 Preliminaries
### Z-Value Filling Curve  
The \(z\)-value filling curve maps the coordinates of multidimensional data points into a one-dimensional space, thereby reducing data dimensionality. Figure 5 provides an illustration of the \(z\)-value filling curve in a two-dimensional space. The process of \(z\)-value transformation is as follows:  
First, the coordinate values of the data points in the high-dimensional space are converted into binary numbers with the same bit length. Then, the binary bits at the same positions across all dimensions are interleaved to form a new binary number. This binary number is subsequently converted into a decimal number, which serves as the \(z\)-value coordinate of the data object in the one-dimensional space. Specifically, for a data point \(p\) in a \(d\)-dimensional space, let its original coordinates be denoted as \(p(x_{1},x_{2},\ldots,x_{d})\). Each coordinate value is converted into an mmm-bit binary number. For example, if \(x_{1}=(a_{11}a_{12}\ldots a_{1m})_{2}\), the data point is represented in binary form as \(p((a_{11}a_{12}\ldots a_{1m})_{2},(a_{21}a_{22}\ldots a_{2m})_{2},\ldots(a_{d1 }a_{d2}\ldots a_{dm})_{2})\). The binary sequence is then mapped to a one-dimensional coordinate using Equation (2).  
\[z_{p}=\sum_{i=1}^{m}\sum_{j=1}^{d}2^{d(i-1)+j-1}*a_{ji} \tag{2}\]  
Figure 4: The Illustration of \(DCM\) value calculation.  
Figure 5: Illustration of \(z\)-value filling curve.  
Taking the transformation process of the two-dimensional point \(d\) in Figure 5 as an example, first, the coordinate of the data point \(d\) in the two-dimensional space is \(\left(3,1\right)\). Next, each dimension's decimal coordinate is converted into its binary representation. Since \(3=\left(011\right)_{2},1=\left(001\right)_{2}\), the binary coordinates of point \(d\) become \(\left(\left(011\right)_{2},\left(001\right)_{2}\right)\). Following this, the coordinate transformation process begins by interleaving the binary bits. The first binary bit from each dimension is extracted, yielding \(00\). Then, the second binary bit from each dimension is taken, resulting in \(10\), and so on. Concatenating these bits produces the binary number \(001011\), which is then converted into its decimal equivalent. Thus, the \(z\)-value coordinate of point \(d\) on the \(z\)-value axis is \(z_{d}=11\). This process is illustrated in Figure 6.  
Based on the above coordinate transformation, the six data points in Figure 6 can be mapped into one-dimensional space, as shown in Figure 7.  
From Figures 5 and 7, the \(z\)-value transformation process exhibits the following characteristics. 1. Local Order-Preserving Property: The distribution of data points on the \(z\)-value axis roughly preserves the proximity of points in the original space. That is, points that are close to each other in the original space tend to remain close on the \(z\)-value axis. 2. Disruptive Changes: In certain cases, the relative distribution of some points may change significantly on the \(z\)-value axis. For example, this can be observed in the relationship between points \(c\) and \(m\) in the figures. The local order-preserving property provides a foundation for using \(z\)-values to efficiently identify the range of \(k\)-nearest neighbors (KNNs). However, the occurrence of disruptive changes necessitates the design of filtering criteria to ensure the accurate identification of the KNN range.