## 7 SMEP Algorithms
### AuxDataQueue Compression Algorithm
The where \(P_{B}\) is the number of absolute positions covered by zones from the zone 0 to the zone \(k-1\), \(P_{k,\ i-1}\) is the number of absolute positions within a zone \(k\) which covered by sensor data from 1 st sensor data to the \(i-1\) th positioned sensor data in the zone, \(P_{k,I_{i}}\) is an _I-bit position_ address of the \(i\)-th position within the compression interval in which the \(i\)-th position is located. Each zone \(j\) has \(2^{m-j-1}\) sensor data in _DataQueue_ and each of them has been selected among \(2^{Zones[j]}\). Therefore, each zone \(i\) covers \(2^{m-j-1}.2^{Zones[j]}\) absolute positions. Since zones from the zone \(0\) to the zone \(k-1\) are located before the zone \(k\),

\[P_{B}=\sum\limits_{j=0}^{k-1}2^{m-j-1}.2^{Zones[j]} \tag{10}\]

Meanwhile, the number of relative positions in the zone \(k\) for \(I-1\) position is \(j-1-\sum\
olimits_{j=0}^{k-1}2^{m-j-1}.2^{Zones[j]}\) and each of \(j-1-\sum\
olimits_{j=0}^{k-1}2^{m-j-1}.2^{Zones[j]}\) sensor data covers \(2^{Zones[k]}\) in the zone \(k\) since it has passed through _Zones[k]_ compressions. Thus,

\[P_{k,i-1}=(j-1-\sum\limits_{j=0}^{k-1}2^{m-j-1}).2^{Zones[k]} \tag{11}\]

The relative position of the position \(i\) in the zone \(k\) is \(j-\sum\
olimits_{j=0}^{k-1}2^{m-j-1}.2^{Zones[j]}\). Using this position, the I-bit position sequence _ZoneBPseq[k]_ in the zone \(k\), and the _BitSubstring_ function,

\[P_{k,I_{i}}\ =\ (BitSubstring(ZoneBPseq[k],\ Zones[k],\ i-\sum\limits_{j=0}^{j =k-1}2^{m-j-1}))\ \ \ +1 \tag{12}\]

Replacing Equation (1) with Equation (2)-(4) finally,

Figure 12:0 Figure 12:1 Figure 12:2

We have completed the proof of the theorem. \(\square\)

## Appendix G Proof of Theorem 9

**Proof. ** Let \(L\) be the size of a sensor data sequence from the StartZone zone to the last zone. Then,

Figure 12:3

Because \(L=2^{m}\) - the number of _DataQueue_ positions in zones before the _StartZone_ zone

Figure 12:4

Given a sensor data sequence with even number size, the sensor data sequence after compression becomes to be reduced to the half size. Thus, the compressed sensor data sequence with the size \(L\) is reduced to the size \(\frac{1}{2}L=\frac{1}{2}.2^{m-StatZone}=2^{m-StatZone-1}\), which is the same as the size of _StartZone_ zone. Thus, the sensor data sequence in consecutive equivalent zones from the _StartZone_ zone to the last \(m-1\) zone is compressed to the _Startzone_ zone. \(\square\)

## Appendix H Proof of Lemma 1

**Proof. ** By induction, for \(k=0\), the theorem is true because a sensor data selected in \(I_{c,0}\) is the one selected among sensor data selected in \(I_{c-1,0}\)=\(I_{c,2\cdot\ 0}\) and \(I_{c-1,1}=I_{c,2\cdot\ 0+1}\) so that \(I_{c,0}\) covers \(I_{c-1,2\cdot\ 0}\bullet I_{c-1,2\cdot\ 0+1}\). Assume that the theorem is true for any integer \(i\leq k-1\). Consider \(I_{c-1,2k-2}\), \(I_{c-1,2k-1}\), \(I_{c-1,2k}\), and \(I_{c-1,2k+1}\). They are consecutive depending on the order of compression intervals in \(C_{s}^{c}\). Here, \(I_{c-1,2k-2}=I_{c-1,2(k-1)}\) and \(I_{c-1,2k-1}=I_{c-1,2(k-1)+1}\), \(I_{c-1,2k-2}\) and therefore, \(I_{c,k-1}\) covers \(I_{c-1,2k-2}\bullet I_{c-1,2k-1}\). Thus, the \(I_{c,k}\) must cover \(I_{c-1,2k}\bullet I_{c-1,2k+1}\). Consequently, for \(i\leq k-1\), the theorem is true. \(\square\)

## Appendix I Proof of Lemma 2

**Proof. **\(S_{c,i}\) is selected in \(I_{c-1,j-1}\). Thus, \(I_{c-1,i-1}=I_{c-1,j}\) and \(j=i-1\). Moreover, since \(S_{c,i}\) covers \(I_{c-1,i-1}\), by lemma 1, \(I_{c-1,j}\) covers \(I_{c-2,2(i-1)}\bullet I_{c-2,2i-1}=I_{c-2,2(i-1)}\bullet I_{c-2,2(i-1)+1}\). \(\square\)

## Appendix J Proof of Lemma 3

**Proof. ** The lemma 3 is trivial. Because the I-bit position of \(S_{c,k}\) is positioned at the \(k\)-th I-bit position in the I-bit position sequence, the length of I-bit position is c, and I-bit position of \(S_{c,k}\) is located at the c(\(k-1\)) th position in the I-bit position sequence (note that the first position in I-bit position sequence is 0). \(\square\)

## Appendix K Proof of Lemma 4

**Proof.