# An Efficient Model-Based Clustering via Joint Multiple Sink Placement for WSNs
## 3 Preliminaries
### Ant Clustering Algorithm  
This sub-section provides a general summary of the ant clustering algorithm; later we will propose an improved version for the WSN context. The ant clustering algorithm was introduced by [62], and it is a swarm-intelligent approach for clustering issues inspired by the ant-like behaviour exhibited [63]. The algorithm is based on the foraging behaviour of real ants hunting for food. The method uses the ants to deposit an amount of pheromone that reflects the problem to be grouped during their movements. Hence, an ant is placed at each data point site, and the ants are free to move about the search space in order to identify locations with high pheromone density. The quantity of pheromone deposited at various locations across the search space controls the movements of the ant. The introduction of a pheromone function to the grid enhances the reward for dropping objects in more dense areas of the grid. Therefore, the more pheromone is deposited, the more ants congregate. This results in the formation of homogeneous clusters of data.  
The grid size is proportional to the number of objects. The agent ants are free to travel around the grid, picking up or discarding objects based on their resemblance to other objects,  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & \multicolumn{1}{c}{**Description**} \\ \hline N & Number of sensor nodes \\ M & Number of sinks \\ S\({}_{\text{i}}\) & The sensor i \\ d(S\({}_{\text{i}}\), S\({}_{\text{j}}\)) & Distance between the sensor S\({}_{\text{i}}\) and the sensor S\({}_{\text{j}}\). \\ min\_d(S_{\text{i}}\), S\({}_{\text{j}}\) & Minimum Distance between the sensor S\({}_{\text{i}}\) and the sensor S\({}_{\text{j}}\). \\ CHs & All cluster head nodes in the network. \\ BS & Base station/Sink. \\ CH\({}_{\text{i}}\) & Cluster head of the j\({}^{\text{th}}\) cluster. \\ C\({}_{\text{P}_{\text{i}}}\) & Chance of picking a node \(i\) factor. \\ C\({}_{\text{range}}\)(S\({}_{\text{i}}\), CH\({}_{\text{j}}\)) & Communication range between a node S\({}_{\text{i}}\) and the CH\({}_{\text{j}}\) node. \\ clust\({}_{\text{i}}\) & The i\({}^{\text{th}}\) cluster. \\ \(\text{l clust}_{\text{i}}\) & Number of nodes in the cluster i. \\ S(i) & A set of alive neighboors of node i. \\ MNC & The most neighbors clusters. \\ \(\text{|MNC|}\) & Number of nodes in the cluster MNC. \\ Rj & The j\({}^{\text{th}}\) multi path route. \\ BR & The route between CH and a node with minimum cost value (Best Route). \\ R\({}_{\text{energy}}\) & The residual energy. \\ min\_R\({}_{\text{energy}}\) & The minimum residual energy in the considering network. \\ max\_R\({}_{\text{energy}}\) & The maximum residual energy in the considering network. \\ E\({}_{\text{R}}\) & The energy required for reception data. \\ E\({}_{\text{T}}\) & The energy required for transmission data. \\ Emp & The energy required in multi path model. \\ E\({}_{\text{O}}\) & The energy required in Direct path model. \\ N\({}_{\text{ant}}\) & Number of ants \\ N\({}_{\text{Iteration}}\) & Number of iterations \\ \hline \hline \end{tabular}
\end{table}
Table 3: Symbol denotation.  
Figure 2: Data transmission using cluster head.  
and the density of the objects in their current local area, as well as the status of the agent ant (whether it is loaded or not). Low-density neighbourhoods (low amounts of pheromones) increase the likelihood of an ant picking up an object, but a remarkable resemblance between things in the neighbourhood area (high amount of pheromone) decreases it. On the other hand, high-density communities increase the chances of dropping an object. The grid's agent ants and objects can be in one of two states: (a) the agent ant is holding an object i and calculating the dropping probability at its current location; (b) another agent ant is unloading and calculating the probability of picking up an object. Finally, the agent ants aggregate the objects on the grid into heaps containing objects similar