## 7 SMEP Algorithms
### AuxDataQueue Compression Algorithm
** For \(k=0\), \(S_{c,0}\) is always selected as a first sensor data unconditionally from \(S_{0,0}\). Therefore, \(i=0\). For \(k\geq 1\), according to corollary of Theorem 5, the position of \(S_{c,k}\) exists in \([(k-1)2^{c},\,k2^{c}]_{0}\), the k \(-1\) th compression interval of \(S^{0}\). Thus, the absolute position of the first sensor data in the interval \([(k-1)2^{c},\,k2^{c}]_{0}\) is \((k-1)2^{c}\). Since the relative position of \(S_{c,k}\) is \((I\)-_bit position\((S_{c,k}))_{10}+1\), its absolute position \(i\) is \((k-1)2^{c}+(I\)-_bit position\((S_{c,k}))_{10}+1\). \(\square\)

## Appendix L Proof Lemma 5

**Proof. ** Given \(B_{p}\), in the definition, we can get the absolute position \(i^{\prime}\) and \(j^{\prime}\) for \(S_{c,i}\) and \(S_{c,j}\), respectively, using \(B_{p}\) by the theorem. Moreover, we can get the absolute position \(p^{\prime}\) and \(q^{\prime}\) for \(S_{c,p}\) and \(S_{c,q}\) in the same way. In fact, the absolute positions of \(i^{\prime},j^{\prime},p^{\prime}\) and \(q^{\prime}\) are

Figure 12:5 Figure 12:6 Figure 12:7 Figure 12:8Meanwhile, line equations \(\overline{S_{c,p}S_{c,j}}\) and \(\overline{S_{c,j}S_{c,q}}\) by using absolute positions are as follows:

Figure 12:9 Figure 12:0 Figure 12:1

Since in Figure 7a,b are the points \(q^{\prime}\) and \(p^{\prime}\) at on the line \(\overline{S_{0,p^{\prime}}S_{0,i^{\prime}}}\) and \(\overline{S_{0,i^{\prime}}S_{0,q^{\prime}}}\), respectively, now, we can compute \(A\) and \(B\) such that

Figure 12:2

By the way, since \(Sp\) and \(Sq\) is the second and the third senor data in \(I_{c,2k}\), \(p=4k+1\) and \(q=4k+2\), replacing them with \(p\) and \(q\) in (Equation (6)), we can get

Figure 12:3 Figure 12:4

\(\square\)

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