## 7 SMEP Algorithms
### AuxDataQueue Compression Algorithm
As further work, we are going to research not only how to design this method, but also how many consecutive compression intervals are optimal for performance and how much its performance is improved depending on the number of consecutive compression intervals. This research was a part of the project titled Development of the wide-band underwater mobile communication systems funded by the Ministry of Oceans and Fisheries, Korea. [PERSON] proposed the idea of the SMEP method and carried out its formal approach; [PERSON] and [PERSON] performed conceived and designed experiments; [PERSON] performed the experiments; [PERSON] and [PERSON] analyzed the data; [PERSON] wrote the paper. The authors declare no conflict of interest. ## Appendix A Proof of Theorem 1

Proof. : Since \(I=[i,j]\) and \(J=[k,l]\) are interval \(i<j\) and \(k<l\). Moreover, since compression intervals \(I\) and \(J\) are consecutive, either \(j=k\) or \(l=i\) is true. But, if \(l=i\)-then \(k<l=i<j\) and this is a contradiction to the conditions of the theorem, \(i<k\) or \(i<l\) or \(j<l\). Thus, \(j=k\) is true. Therefore, by the definition of consecutive compression interval, \(S_{i}\), \(S_{i+1},\ldots,\)\(S_{j-1}\), \(S_{j}\) and \(S_{k},\ldots,\)\(S_{i-1}\), \(S_{i}\) are consecutive sensor data, respectively, and since \(S_{j}=S_{k}=S_{i+m}\), \([i,l]=S_{i}\), \(S_{i+1},\ldots,\)\(S_{i+m-1}\), \(S_{j}\), \(S_{k+1},\ldots,\)\(S_{l+1}\), \(S_{l}\) are success sensor data and the number of them is \(n+m\). By definition, \(J=[i,l]\) is a compression interval with the size \(m+n\) with respect to \(S\). ## Appendix B Proof of Theorem 3

Proof of Proposition 1.: In initial step, if m = 1, \(C_{S}=(I_{0})\) and \(I_{0}=S\). Thus, \(\mid I_{0}\mid=\mid S\mid=\)3=2\(\cdot\)1+1. So, when m = 0, _Proposition 1_ is true. Assume that the proposition is true for an integer \(k\), \(I\)<\(k\) = \(m\). For \(k=m\), \(\mid S\mid=\mid I_{0}\bullet I_{1}\bullet I_{2}\bullet\ldots\bullet I_{m} \mid=\mid(I_{0}\bullet I_{1}\bullet I_{2}\bullet\ldots\bullet I_{m-1})\bullet I _{m-1}\mid\) (by Theorem 2) = \(\mid I_{0}\bullet I_{1}\bullet I_{2}\bullet\ldots\bullet I_{m-1}\mid=\)\(\bullet\ldots\bullet I_{m-1}\mid+1_{m-1}\mid\) (by Theorem 1) = \((2(m-1)\)+1)+2 (_since \(\mid I_{0}\bullet I_{1}\bullet I_{2}\bullet\ldots\bullet I_{m-1}\mid=\)\(2(m-1)\)+1 by the assumption) = \(2m+1\)._ By induction, _Proposition 1_ is true. Proof of Proposition 2.: For \(p\), \(q\), \(r\) of \(S_{p}\), \(S_{q}\), \(S_{r}\) in \(I_{i}\), \(q=p+1\) and \(r=q+1=p+2\), since they are consecutive in a compression interval. \(p=0\) for \(i=0\) since \(I_{0}\) is the first compression interval and \(S_{p}\) is the first sensor data and so, \(p=0=2\)\(\cdot\)\(0\), \(q=1\) = \(2\)\(\cdot\)\(0\) + \(1\), \(r=2\) = \(2\)\(\cdot\)\(0\) + \(2\). Thus, _Proposition 2_ is true for \(i=0\). Assume that the proposition is true for \(0<k<i\). For \(k=i-1\), the position of the last sensor data in the previous compression interval \(I_{i-1}\) is \(2(I-1)+2\) (by the assumption) = \(2i\). Since \(I_{i-1}\) and \(I_{i}\) are consecutive compression intervals, the last sensor data in \(I_{i}\) is shared as the same one as the first sensor data in \(I_{i}\). Thus, \(p=2i\). Consequently, \(q=2i+1\) and \(r=2i+2\). Thus, by induction, _Proposition 2_ is true for an integer, i, \(0\leq i\leq m-1\). ## Appendix C Proof of Theorem 4

Proof. : Let the size of \(S\) be \(n\). If \(m=1\), then \(n=3\), \(I_{0}=S=S_{0}\), \(S_{1}\), \(S_{2}\), \(C_{S}=(I_{0})\), and \(k=0=2^{1-1}-1\). Thus, the theorem is true for \(m=1\). Assume that the theorem is true for any integer \(k\), \(k<m\). If \(k=m\)the size of \(S\) is \(2^{m}+1\). So, let \(S=S_{0}\), \(S_{1}\),..., \(S_{2}\)\({}^{m-1}\)\({}_{-1}\), \(S_{2}\)\({}^{m-1}\), \(S_{2}\)\({}^{m-1}\)\({}_{+1}\),..., \(S_{2}\)\({}^{m}\). Since \(2m=2\)\({}^{m-1}\)\(=2\)\({}^{m-1}\)\(=2\)\({}^{m-1}\), \(S=S_{0}\), \(S_{1}\),..., \(S_{2}\)\({}^{m-1}\)\({}_{+1}\),..., \(S_{2}\)\({}^{m-1}\)\({}_{+2}\)\({}^{m-1}\).