## 7 SMEP Algorithms
### AuxDataQueue Compression Algorithm
Now, divide \(S\) into two same size sensor data sequences, \(P=S_{0}\), \(S_{1}\),..., \(S_{2}\)\({}^{m-1}\) and \(Q=S_{2}\)\({}^{m-1}\), \(S_{2}\)\({}^{m-1}\)\({}_{+1}\),..., \(S_{2}\)\({}^{m-1}\)\({}_{+2}\)\({}^{m-1}\), respectively, where \(S_{2}\)\({}^{m-1}\) is common. Therefore, each size of the two is \(2^{m-1}+1\). Consider two 2-size compression intervals, \(C_{P}=(J_{0},J_{1},\)..., \(J_{P})\) and \(C_{Q}=(H_{0},H_{1},\)..., \(H_{q})\) with respect to \(P\) and \(Q\), respectively. Then, by assumption, \(p=q=2^{m-2}-1\), thus number of compression intervals in both of \(C_{P}\) and \(C_{Q}\) is \(2^{m-2}\). Moreover, \((J_{0},J_{1},\)..., \(J_{P},\)\(H_{0},\)\(H_{1},\)..., \(H_{q})=C_{s}=(I_{0},I_{1},\)..., \(I_{b})\), and since the number of compression intervals in \(C_{s}\) is \(2^{m-2}+2^{m-2}=2^{m-1}\), \(l=2^{m-1}-1\) and the theorem has been proved. \(\square\)

## Appendix D Proof of Theorem 5

Proof. : We prove the theorem by induction. At first, for \(k=1\), \(S_{1,i}\) covers \(I_{0,i-1}=I_{0,(i-1)2}\)\({}^{\circ}=I_{k,(i-1)2}\)\({}^{k-1}\) by Lemma 1. Assume that the theorem is true for integer \(k\leq c\). By Lemma 2, \(S_{c,i}\) covers \(I_{c-1,i-1}\) and \(I_{c-1,j-1}\) covers \(I_{c-2,2(i-1)}\bullet I_{c-2,2i-1}\). By assumption, \(I_{c-2,2(i-1)}\) covers \(I_{0,2\cdot 2(i-1)2}\)\({}^{c-3}\bullet I_{0,2\cdot 2(i-1)2}\)\({}^{c-3}\)\({}_{+1}\)\(\bullet\)\(...\)\(\bullet\)\(I_{0,2\cdot(2i-1)+12}\)\({}^{c-3}\)\({}_{-1}=I_{0,(i-1)2}\)\({}^{c-1}\)\(\bullet\)\(I_{0,(i-1)2}\)\({}^{c-1}\)\({}_{+1}\)\(\bullet\)\(\cdots\)\(\bullet\)\(I_{0,(2i-1)2}\)\({}^{c-2}\)\({}_{-1}\) and \(I_{c-2,2i-1}\) covers \(I_{0,2(2i-1)2}\)\({}^{c-3}\)\({}_{+1}\)\(\bullet\)\(...\)\(\bullet\)\(I_{0,2\cdot 2i}\)\({}^{c-3}\)\({}_{-1}=I_{0,(2i-1)2}\)\({}^{c-2}\)\({}^{c-1}\)\({}_{+1}\)\(\bullet\)\(\cdots\)\(\bullet\)\(I_{0,2i-1)2}\)\({}^{c-1}\)\({}_{+1}\)\(\cdots\)\(\bullet\)\(I_{0,2i}\)\({}^{c-1}\)\({}_{-1}\). Thus, \(S_{c,i}\) covers \(I_{c-2,2i-1}\)\(\bullet\)\(I_{c-2,2i-1}=I_{0,(i-1)2}\)\({}^{c-1}\)\(\bullet\)\(I_{0,(i-1)2}\)\({}^{c-1}\)\({}_{+1}\)\(\bullet\)\(\cdots\)\(\bullet\)\(I_{0,(2i-1)2}\)\({}^{c-2}\)\({}_{-1}\)\(\bullet\)\(I_{0,(2i-1)2}\)\({}^{c-2}\)\({}_{+1}\)\(\bullet\)\(\cdots\)\(\bullet\)\(I_{0,2i}\)\({}^{c-1}\)\({}_{-1}\) and the theorem has been proved. \(\square\)

## Appendix E Proof of Theorem 7

Proof. : Since \(p=4k+1\) and \(q=4k+2\), by definition,

\[E_{p}=E_{4k+1}=\left|S_{c,q}-A\ \right|=\left|S_{c,4k+2}-A\right|=\left|S_{c,4k+2}- \left(\frac{S_{c,i}-S_{c,4k+1}}{f^{\prime}-p^{\prime}}(q^{\prime}-p^{\prime})+ S_{c,4k+1}\right)\ \right|\] \[=\left|-\frac{S_{c,i}-S_{c,4k+1}}{f^{\prime}-p^{\prime}}(q^{ \prime}-p^{\prime})+(S_{c,4k+2}-S_{c,4k+1})\right|\] \[=\left|\frac{S_{c,i}-S_{c,4k+1}}{f^{\prime}-p^{\prime}}(q^{ \prime}-p^{\prime})-(S_{c,4k+2}-S_{c,4k+1})\right|\] \[=\left|\frac{S_{c,i}-S_{c,4k+2}}{f^{\prime}-q^{\prime}}(q-p^{ \prime})-\ (S_{c,4k+2}-S_{c,4k+1})\right|\]

\(\square\)

## Appendix F Proof of Theorem 8

Proof. : In proof of (_i_), above all, for any \(j\) such that \(0\leq j\leq m-1\) the _DataQueue_ size covered by _Zones[j]_ is \(2^{m-j-1}\). If _DataQueue[i]_ exists in the zone \(k\), then \(\sum_{j=0}^{j=k-1}2^{m-j-1}<i<\sum_{j=0}^{j=k+1}2^{m-j-1}\) since \(k-1<k<k+1\). Accordingly, \(i-\sum_{j=0}^{j=k-1}2^{m-j-1}>0\) and \(i-\sum_{j=0}^{j=k+1}2^{m-j-1}<0\). Thus, \(k\) is the maximum number to satisfy \(i-\sum_{j=0}^{j=k-1}2^{m-j-1}>0\), this is, \(k=max\Big{\{}\ t\ \big{|}\ 0<i-\sum_{j=0}^{j=t-1}2^{m-j-1}\Big{\}}\). Since \(\sum_{j=0}^{j=t-1}2^{m-j-1}=2^{m-1}+2^{m-2}+\ldots+2^{m-t}=2^{m}-2^{m-t}=2^{m-t} \left(2^{t}-1\right)\), \(k=max\big{\{}\ t\ \big{|}\ 0<\big{(}i-2^{m-t}\cdot(2^{t}-1)\big{)}\big{\}}\). If \(i=2^{m}\), \(i\) is the last position. Hence, \(i\) exists in the last zone \(m-1\). In the (_ii_) proof, at first, if \(i=0\), \(P_{i}=0\), since \(S_{c,0}=S_{0,0}\) for any number of compressions \(c\). Now, we prove the case for \(i\
eq 0\) and \(k\geq 0\). Let \(k\) be the zone in which \(i\) position is located.