# Cluster-Based Data Aggregation in Flying Sensor Networks Enabled Internet of Things
## 4 Data Aggregation and Communication
### Upper Bound Proof \(|A(Y)-A(X)|1=O(lognlog^{*}n)d(X,Y)\)  
**Proof. ** Express this bound on \(L1\) space, assume the impact of the edit procedures, and show that all adds a role to the \(L1\) distance limited by \(O(lognlog^{*}n)\). Editing processes are permitted to overlay on blocks. We present a Lemma applicable to all data, except with no contiguous repetitive alphabets. \(\square\)  
**Lemma 5**. : _The nearby landmark to any character of \(X_{loc}\) is computed by at most five successive characters of \(X_{loc}\) to the right and at most \(log^{*}|\Sigma_{loc}|+5\) repeated characters of \(X_{i}\) to the left._  
Figure 4: Representation of node structure hierarchically in parse tree based on string S. **Proof. ** Assume that a character of \(X_{loc}\), i.e., \(X_{loc}[loc_{2}]\), indicates the mechanism to identify the nearby landmark. Type 1 Iterating mb: A lengthy duplication of a character, \(c\), is considered a distinct and larger landmark. \(X_{loc}[loc_{2}]\) is contained within such an mb if \(X_{loc}[loc_{2}]=X_{loc}[loc_{2}+1]\) or if \(X_{loc}[loc_{2}]=X_{loc}[loc_{2}-1]\). We also assume \(X_{loc}[loc_{2}]\) to be part of a reiterating partial data if \(X_{loc}[loc_{2}-1]=X_{loc}[loc_{2}-2];X_{loc}[loc_{2}+1]=X_{loc}[loc_{2}+2]\); and \(X_{loc}[loc_{2}]6=X_{loc}[loc_{2}+1]\) and \(X_{loc}[loc_{2}]6=X_{loc}[loc_{2}-1]\). Type 2 and type 3 Nonrepeating mbs: When \(X_{loc}[loc_{2}]\) is not a character in re-counting mb, formerly choose it either in a long or short mb. We investigate the partial data \(X_{loc}[loc_{2}-log*|\Sigma_{loc}|-3\ldots loc_{2}-1]\). When there is a \(k\) such that \(X_{loc}[loc_{3}]=X_{loc}[loc_{3}-1]\) and \(loc_{3}\) is more significant than all, a repeating mb ending at location \(loc_{3}\) exists. In the landmark, thus, we analyze \(X_{loc}[loc_{2}]\) as a short mb part, beginning at \(X[loc_{3}+1]\). Inspecting the partial data \(X[loc_{2}+1\ldots loc_{2}+5]\) permits us to find if there is an additional repeating mb nearby location, \(loc_{2}\), and we can decide the formation of a node comprising \(X_{loc}[loc_{2}]\). When no repeating mb is marked in \(X_{loc}[loc_{2}-log^{*}|\Sigma_{loc}|-3\ldots loc_{2}-1]\), it is possible to use the alphabet decrease procedure to find a landmark. The ability to find a nearby landmark to a character through observing only a limited number of successive nearby codes demonstrates that if a modification occurs outside this area, a similar landmark will be found; hence, a similar node will be shaped comprising that symbol. This permits the verification of the subsequent lemma. \(\square\)  
Lemma 6: _Inserting \(loc_{3}\leq log^{*}n+10\) contiguous symbols into \(X\) to obtain \(X^{\prime}\) means \(|A_{i}(X)-A_{i}(X^{\prime})|1\leq 2(log^{*}n+10)\in\mathcal{C}\) levels._  
**Proof. ** We have a contribution after the insertion itself into the \(L1\) distance and its effect on the nearby vicinity. Assume that the aggregate of characters at the level \(i\) break down to distinct nodes next to addition equated to earlier nodes. Consider the total characters at a level \(i\) break down in a different way as a result of the insertion \(P_{i}\). Lemma 4.5 proves that in a non-overlapping mb, any character \(log^{*}|\Sigma_{loc}|+5\) indexes to the right or above five indexes to the left of each character can be altered and will identify the identical nearby landmark; hence, it will be shaped according to similar nodes. Hence, it will not pay for \(P_{i}\). In the same way, for an overlapping mb, any character within the block will be parsed into a similar node (i.e., into a tripartite of that character), excluding the last four characters and depending on the block size. Thus, for an overriding mb, \(P_{i}\leq 4\). The total characters that are lower-level parsed in a different way into nodes as a result of the insertion are max \(P_{i}-\frac{1}{2}\), and there is an area of 5 characters at maximum to the left, plus \(log^{*}|\Sigma_{i}|+5\) characters on the right side are parsed differently at a level \(i\).