# Cluster-Based Data Aggregation in Flying Sensor Networks Enabled Internet of Things
## 4 Data Aggregation and Communication
### Data Embedding
#### 4.1.2 Type 2: Long Data without Duplications  
We assume a dataset where two consecutive symbols are duplicates and represent an mb of type 2. Assuming a structure, \(X\), without duplications (i.e., \(X[loc]6=X[loc+1]\) for \(loc=1\ldots|X|-1\)), we select as a maximum \(|X|/2\) and minimum \(|X|/3\) partial data of \(X\) as a node. We obtain \(X\) upon concatenating these nodes. The first phase comprises repeating the reduction process of an alphabet. Reduction of alphabet (\(\sum\)): For every character \(C[loc]\), calculate a new tag. \(C[loc-1]\) is the symbol located to the left of \(C[loc]\), and assume \(C[loc]\) and \(C[loc-1]\) are denoted as binary numbers. The least-significant bit (LSB) key where \(C[loc]\) differs from \(C[loc-1]\) is represented by \(Le\), and assume bit (\(Le,C[i]\)) is the value of \(C[i]\) at the \(Le^{th}\) bit position. For example, the location of bit \(Le\) is next to the character at the former index, i.e., form label (\(C[loc]\)) as \(Le+bit(Le,C[loc])\). For some loc, if \(C[loc]\
eq C[loc+1]\), then \(C[loc]\
eq label(C[loc+1])\). Suppose the LSB location, where \(C[loc]\) differs from \(C[loc+1]\), is similar in a way where \(C[loc]\) also differs from \(C[loc-1]\) (else, \(labelC[loc]\
eq label(C[loc+1])\)). However, the bit character at this position in every symbol must be different; therefore, \(labelC[loc]\
eq label(C[loc+1])\). Adopting this method, we create an innovative series. If the existing alphabets have length, \(\
u\), then the extracted alphabets have size, \(2 log[\eta]\). We currently repeat (repetition is orthogonal to the duplication that produces an EST tree of \(X\); repeating on \(C\) that is a subseries with no matching contiguous characters) and make the character decrease until the length of the alphabet is unable to shrink. This will take \(log^{*}|\eta|\) repetitions. Note that the labels for the first \(log^{*}|\eta|\) symbols will not exist. The lemma states that if a symbol at position \(C[loc]\) is not equal to the symbol at position \(C[loc+1]\), then the label assigned to \(C[loc]\) is not equal to the label assigned to \(C[loc+1]\). Lemma 1's goal is to establish a connection between symbols and their corresponding labels based on how they differ from one another. Lemma 1's proof demonstrates that if there is a difference between two symbols, (\(C[loc]\) and \(C[loc+1]\)), their labels will also be different. The subsequent actions and procedures in the text are theoretically justified by this lemma. \(\square\)  
After the last reiteration of the \(\sum\)reduction, the size of \(\sum\)is \(6.\
u\). Upon every repetition of cataloguing, the size of the alphabet is reduced from \(|\sum|\) to \(2[log|\sum|]\). If \(|\sum|>6\), then \(2[log|\sum|]\) is firmly smaller than \(6\). \(A\) has no duplicate symbols contiguously, nor do the final order of tags on \(A\) by Lemma 1 iteratively. Lemma 2 plays a crucial role in establishing the size of the alphabet (\(\sum\)) after the last iteration of cataloguing in the given text. It states that the size of \(\sum\) is \(6\) times the original alphabet size (\(\
u\)) after the last repetition of cataloguing. The proof of Lemma 2 demonstrates that if the initial alphabet size is greater than \(6\), then the reduction process reduces the alphabet size to a value that is strictly smaller than \(6\). Lastly, three passes over the order of labels were accomplished to decrease the alphabet from {0, 1, 2}: initially, we substitute every 3 with the minimum item from {0, 1, 2} that is not in the neighborhood of 3, and then we perform the same operation for every 4 and 5. This produces a series of symbols extracted from the \(\sum\{0,1,2\}\), where no contiguous labels are equal. We designate this series \(A^{\prime}\). We currently choose distinct positions, known as landmarks, from the structures that are closely related to each other. We initially chose some location, I, and a local maximum as a landmark, such as \(A^{\prime}[loc-1]<A^{\prime}[loc]>A^{\prime}[loc+1]\).