# Cluster-Based Data Aggregation in Flying Sensor Networks Enabled Internet of Things
## 4 Data Aggregation and Communication
### Upper Bound Proof \(|A(Y)-A(X)|1=O(lognlog^{*}n)d(X,Y)\)
#### 4.3.2 ESP Sub-Trees  
We estimate distance among partial data s and the pattern t by matching the parsing ESP of two patterns. On the other hand, it would be costly and useless to parse the partial data s. In this subsection, we indicate that an ESP tree is assigned to a datum, and being inspired by a partial datum means that the subtree will have similar editing sensitivity features as the complete one. **Definition 2**. : _Let \(ET_{loc}(X)_{loc_{2}}\) at \(loc_{2}\), such that \(loc\) represents the level of UAV in the breakdown of \(X\)._  
The set of values \(X[a_{1}\ldots a_{n}]\) to the tags on last level of the sub-tree, where (root) \(ET_{loc}(X)_{loc_{2}}\) is defined as range, \(\big{(}ET_{loc}(X)_{loc_{2}}\big{)}\), and relates to the partial data, \(X[a_{1}\ldots a_{n}]\). The ESP subtree of data \(X\) can be defined \(EST(X,l,r)\) as the sub-tree \(ET(X)\) that comprises all \(X[loc]\), such that \(l\leq\ loc\leq r\), plus all parent UAVs. Formally, we search and discover UAVs of \(ET_{loc}(X)_{loc_{2}}\), where \([l\ldots r]\cap range(ET_{loc}(X)loc_{2})6=\emptyset\). A name node is derived from \(ET_{loc}(X)_{loc_{2}}\), which is \(loc,hXrange(ET_{loc}(X)loc_{2})\cap[a\ldots b))\). This results in an appropriate sub-tree of \(ET(X)\), as a node is part of the sub-tree if one of its children is counted in as a minimum. As stated earlier, we can define an array that represents this \(ET\). **Definition 3**. : _Define \(AX(X,\,l,\,r)\) as the characteristic array of \(EST(X)\) by similarity with \(A(X)\); namely \(AX(X,\,l,\,r)[x]\) represents the the total times the partial data \(x\) are denoted as a node in \(EST(X,\,l,\,r)\). In this regard, \(EST(X,\,1,\,|X|)=ET(X)\); however, if not, generally \(EST(X,\,l,\,r)=ET(X[l\ldots r])\). Through \(EST(X,\,l,\,r)\), it send the features of the edit-sensitive parsing. Similar to Theorem 2, Theorem 4 is stated as follows._  
**Theorem 4**. : _Let \(d\) be \(d(Y[l_{p}\ldots r_{p}],\,X\,[l_{q}\ldots r_{q}])\). Then, \(d\leq 2||A.XY,\,l_{p},\,r_{p})-A \(X(X,\,l_{q},\,r_{q}])||1=O(logn\,log^{*}n)d\)._  
Proof. : Definitely, as Lemma 9 does not assume anything regarding the tree structure, the lower bound exists, meaning that the edit distance is simply double the length of the difference amongst the ESP subtrees. For the higher limit, assume implementing the required editing processes to the partial data of \(X\) and \(Y\). The impact on the ordinary ESP trees is observed, \(ET(Y)\) and \(ET(X)\). Theorem 2 verified that all editing processes could create a divergence of \(O(lognlog^{*}n)\) maximum among \(A(X)\) and \(A(X^{\prime})\). Following this, the variance in \(AS(X,l,r)\) should be limited as a result of similar volume: it seems impossible to remove additional UAVs, as the UAVs of \(EST(X,l,r)\) is the subset of UAVs of \(ET(X)\). Hence, as demonstrated in Theorem 2, the aggregate divergence \(\big{|}|A.X(Y,l_{p},r_{p})-A.X(X,l_{q},r_{q})\big{|}|1=d\cdot O(lognlog^{*}n)\). \(\square\)  
**Lemma 10**. : \(A\,X(X,\,l+1,\,r+1)\) _can be calculated from \(A\,X(X,\,l,\,r)\) in time \(O(\,log\,|X|)\)._  
**Proof. ** Remember that a UAV is contained within \(EST(X,l,r)\) once any dependent is last-level analogous to \(X[i]\) contained when \(i\in[l\ldots r]\). It results in an easy method for discovering \(EST(X,l+1,r+1)\) from \(EST(X,l,r)\) and \(AX(X,l+1,r+1)\). We need to eliminate \(X\) and some parents that do not cover \(X[l]\) from \(EST(X,l,r)\) to generate \(AX(X,l+1,r+1)\). Each parent \(X\) is adjusted to guarantee that its tag is accurate. The UAV located at i(level) matching to the partial data \(X[loc_{2}\ldots loc_{3}]\), which is a parent of \(X\), was earlier denoted in the sub-tree via the \((loc,h(X[l\ldots loc_{3}]))\); it is required to be substituted with \((loc,h(X[l+1\ldots loc_{3}]))\). Suppose \(y\) is a _UAV_ conforming to \(X[r+1]\) in \(ET(X)\) at the right.