## 4 Basic Definitions and Properties  
Fundamental formal definitions and properties are presented in this section. These are necessary to introduce the SMEP method in Sections 5-7, and the decompression and recovery algorithms in Section 8.  
Figure 1: A communication framework for sensor data loss reduction.  
\(I_{m-1}\)) is called as a compression interval covering on \(S\). When the size of every compression interval in the compression interval covering \(C_{S}\) is the same as \(k\), we call \(C_{S}\) the k-size compression interval covering._  
From definitions, we can get the relationship between a sensor data sequence and its 2-size compression interval covering as the below. For the proof refer to Appendix B.  
Theorem 4.1: _Let \(S\) and \(C_{S}\) = (\(I_{0}\), \(I_{1}\), \(I_{2}\),..., \(I_{m-1}\)) be a sensor sequence and a 2-size compression interval covering on \(S\), respectively. Then, the following propositions are true:_  
Proposition 1: \(|S|=2m+1\)_._  
Proposition 2: _Given a compression interval, \(I_{i}=S_{p}\), \(S_{q}\), \(S_{r}\), for some integer \(i\), \(0\leq i\leq m-1\), positions of \(S_{p}\), \(S_{q}\), and \(S_{r}\), is \(2i\), \(2i+1\), and \(2i+2\), respectively, in \(S\)._  
In the _SDSC_ mode already introduced in Section 3, in order for a sensor node not to loss sensor data generated periodically from its sensor, the sensor node has to have its own data queue to save lost sensor data in the _SDSC_ mode. However, whenever the data queue is full in the _SDSC_ mode, the _2 MC_ method executes a compression on a sensor data sequence of its data queue. Now, let us define a _compression round_ as follows:  
Definition 8: _A round compression is an operation defined as follows and a compression round is the number of the compression executions:_  
1. A ground sequence is in the _0-th round compression_.
2. The \(i\)-th _round compression_ is a compression executed when \(S\) is a sensor data sequence corresponding to a full data queue and each sensor data in \(S\) have been derived through the _i-1_ th compression.  
At the time when a compression is executed, some of sensor data sequence are selected and the others are discarded. Here, the selected become the compressed sensor data but the discarded becomes the lost sensor data. In order for each of lost sensor data to be restored, its position should be known or found. The SMEP method uses position information about the selected sensor data to find positions the sensor data lost in compression.  
Besides the above definitions more definitions are introduced in the following sections whenever necessary. All of main notations and terms are listed in Table 1 with their definition numbers. Refer to definitions if necessary.  
\begin{table}
\begin{tabular}{c c c} \hline
**Notations** & **Terms** & **Definitions** \\ \hline \(S\) & _sensor data sequence_ & \\ \(S_{i}\) & _i_-th sensor data of \(S\) & Definition 1 \\ \hline - & _ground sequence (ground data sequence)_ & Definition 2 \\ - & _base sequence_ & Definition 3 \\ \(I=S_{i}S_{i+1}\),..., \(S_{i+m}=[I_{i}i+1\ldots i+m]\) & _compression interval_ & Definition 4 \\  & _consecutive compression intervals_ & Definition 5 \\ \(I\bullet J\) & _merged compression interval of \(I\) and \(J\)_ & Definition 6 \\ \(C_{S}=(I_{0},I_{1},I_{2},\ldots,I_{m-1})\) & _compression interval covering_ on \(S\) & Definition 7 \\ \(I_{i}\) & _i_-th compression interval in \(C_{S}\) & _i_-th compression interval in \(C_{S}\) \\ - & _round compression_ & Definition 8 \\ \(S_{i}\) & _sensor data sequence with the \(c\)-th round compression_ & - \\ \(S_{c,i}\) & _i_-th sensor data in \(S^{c}\) & - \\ \(I_{c,i}\) (= \([p,q]_{c}=[2i,2i+1]_{c}\)) & _i_-th 2-size compression interval in \(C_{S}\)\({}^{c}\) & - \\  & _covers_ & Definition 9, Definition 10 \\ _I-bit position_\((S_{c,i})\) & _I-bit position of \(S_{c,i}\) & Definition 11 \\ \(B_{p}\) & _I-bit position sequence of \(S\)_ & Definition 12 \\ \hline \end{tabular}
\end{table}
Table 1: Notations and terms.