# Cluster-Based Data Aggregation in Flying Sensor Networks Enabled Internet of Things
## 4 Data Aggregation and Communication
### Upper Bound Proof \(|A(Y)-A(X)|1=O(lognlog^{*}n)d(X,Y)\)
As noted earlier, \(|\Sigma_{i}|\leq|X|\leq n\), and the recurrence \(P_{i}\leq P_{i}-\frac{1}{2}+log^{*}n+10\) can consequently form. If \(P_{i}-1\leq 2(log^{*}n+10)\), then \(P_{i}\leq 2(log^{*}n+10)\). Insertion point, \(P_{0}\leq log^{*}n+10\). To conclude, \(|A_{i}(X)-A_{i}(X^{\prime})|1\leq 2\left(P_{i}-\frac{1}{2}\right)\), so then we might miss \(P_{i}-\frac{1}{2}\) ancient nodes and obtain these various new nodes. \(\square\)  
Lemma 7: _Removing \(k<log^{*}n+10\) contiguous characters from \(X\) to obtain \(X^{\prime}\) means \(|A_{i}(X)-A_{i}(X^{\prime})|1\leq 2(log^{*}n+10)\)._  
**Proof. ** Notice that the removal of a series of symbols is exactly the double to addition of that series at the similar index. Assume that a series of symbols pop in and then are removed; the resulting data are the same as the original data. Hence, the modified nodes number must be essentially limited by its equal quantity to the insert, as proved in Lemma 6. We merge both lemmas to confirm that modifying processes limited the parse-tree influences. \(\square\)  
Lemma 8: _When a particular allowed operation of edit distance alters data \(X\) to \(X^{\prime}\), then \(|A(X)-A(X^{\prime})|1\leq 8 logn(log^{*}n+10)\)._  
**Proof. ** Each permissible procedure is assumed. Character edit processes  
The scenario for inserting follows straightaway from Lemma 6, as the character insertion operation will influence the breakdown of \(\max 2(log^{*}n+10)\) characters at all levels and the maximum levels \(log2n\). Overall, \(|A(X)-A(X^{\prime})|\,1\leq 2 logn(log^{*}n+10)\). Likewise, the scenario for deletion follows instantaneously from Lemma 7. To conclude, the case for substitution is presented by noticing that a symbol substitution assumed that can be used for removing instantly contiguous to an addition. Partial data Moves  
If the size of partial data being moved is \(log^{*}n+10\) at maximum, then a move can be said to be a deletion of the partial data following its reinsertion somewhere else. From Lemma 6 and Lemma 7, we see that \(|A(X)-A(X^{\prime})|\,1\leq 4 logn(log^{*}n+10)\). Otherwise, we assume parsing of partial data by ESP. Assume a symbol in a non-iterating mb greater than \(log^{*}+5\) symbols from the beginning of partial data and greater than 5 symbols from the tail. As per Lemma 5, only symbols inside the partial data being moved decide how that symbol is parsed. Therefore, the parsing of these symbols and, hence, the contribution to \(A(X)\) are free of the position of partial data in original data. Only the \(log^{*}n+5\) symbols at the start and 5 symbols of the partial data at the end will influence the data-parsing procedure. We can consider these to be the deletion of two partial data of the size \(k\leq log^{*}n+10\) and their reinsertion somewhere else. Lemma 8 demonstrates that each permissible action influences the \(L1\) conversion distance by \(\underline{8}\underline{8}\underline{8}\underline{8}\underline  
**Proof. ** Note that for all \(j\) values in lemma, \(d(t,s[loc\ldots loc_{2}])\geq|(Y-loc+1)-m|\), as these multiple symbols must be deleted or inserted. By triangle inequality of match with moves, for all, \(loc_{2}d(t,s[loc\ldots loc+m-1]\). \[\leq d(t,s[loc\ldots loc_{2}])+d(s[loc\ldots loc_{2}],s[loc\ldots loc +m-1]\] \[=d(t,s[loc\ldots loc_{2}])+|(Y-loc+1)-m|\] \[\leq 2d(t,s[loc\ldots loc_{2}])\]  
The lengthiest common prefix followed to assume \(s[loc\ldots loc_{2}]\) and \(s[loc\ldots loc+m-1]\). The above Lemma importance is that it serves to estimate only \(O(n)\) distances, such as \(d(t,sloc\ldots loc+m-1)\)  for all \(i\) values, to solve the data-match problem DMM, equal to a factor 2 approximation. Therefore, it trims away candidates from the \"quadratic\" number of distance calculations that an honest process would require. \(\square\)