## 4 Maintaining Maximum Coverage and Connectivity Scheme
### Design of the MCC-ACO Algorithm
If \(\vec{X}_{u}\) represents the location of an unfamiliar monitoring point, then \(EffiSet\) can be expressed as:_  
\[EffiSet=\left\{\vec{X}^{\prime}_{q}\left|\left\|\vec{X}^{\prime}_{q}-\vec{ X}_{u}\right.\right\|\leq r^{s},\vec{X}^{\prime}_{q}\in CandiSet\right.\right\} \tag{8}\]  
_where \(\vec{X}^{\prime}_{q}\) is the coordinate of the candidate grid point in \(EffiSet\)._  
In the actual design of the algorithm, a matrix variable \(Allowed\) is used to represent the set of grid points that the ant can select next, and then, ant colony search with the greedy strategy is expressed as:  
\[Allowed=\left\{ \begin{array}{ll}EffiSet&EffiSet\
eq\left\{\phi\right\}\\ CandiSet&EffiSet=\left\{\phi\right\}\end{array} \right. \tag{9}\]  
The core idea of the greedy strategy is to select the grid points that can cover the \"unfamiliar monitoring points\" from the ant's next set of optional grid points as the candidate computing grid points. If for some reason, such candidate grid points do not exist, the algorithm discards the greedy strategy and uses the original set of candidate grid points for calculation. Path-selection probabilities  
The initial task of the ant colony optimization algorithm is the solution of NP problems. Therefore, a specific application model needs to be established in accordance with the specific problem when introducing the problem into the sensor node deployment design. The ant colony optimization model for the MCC-ACO algorithm is shown as follows:  
In the MCC-ACO, the movement probability of the \(k\) node from the grid point \(i\) to the grid point \(j\) is defined as shown in Equation (10) [20; 21; 25]:  
\[P^{k}_{ij}\left(t\right)=\left\{ \begin{array}{ll}\frac{\left[\gamma_{ij} \left(t\right)\right]^{a}\cdot\left[\gamma_{ij}\right]^{\beta}}{\sum\limits _{i\leq allowed_{k}}\left[\gamma_{ii}\left(t\right)\right]^{a}\cdot\left[\gamma _{ii}\right]^{b}}&j\in allowed_{k}\\ 0&others\end{array} \right. \tag{10}\]  
where \(\gamma_{ij}\left(t\right)\) represents the pheromone density in the \(\left(i,j\right)\) path at time \(t\). Heuristic information \(\eta_{ij}\) is used to guide the node to move towards the grid point using a shorter path. According to the theory of the ant colony optimization algorithm, \(\alpha\) and \(\beta\) show the relative importance between the previous empirical data and this independent search. Here, the heuristic information factor \(\alpha\) reflects the relative importance of the cumulative amount of information (residual information content) of the node during the movement in the node search process. The expected heuristic factor \(\beta\) affects the relative importance of the heuristic information (expectation)of the node during the movement in the guiding node search. From Equation (10), we can see that if \(\alpha=0\), then Equation (10) degrades into a completely greedy algorithm, i.e., every ant only chooses the best current grid point to which to move. If \(\beta=0\), then every ant selects the next grid point to hop to, based on existing experience. This limits the innovative search capabilities of the ants. In MCC-ACO, if the value of the influence factor is constant, it will affect the convergence of the algorithm. In order to speed up the convergence of the initial stage and avoid rapid local convergence, we modify the value of the impact factor \(\alpha\) as follows. \[\alpha\left(n\right)=\lambda\left(1+e^{-\gamma n}\right),0\leq n\leq N_{\max} \tag{11}\]  
where \(\lambda\) and \(\gamma\) are constants and \(\lambda\in[0,1]\),\(\gamma\in[0,1]\). \(n\) refers to current search times. \(N_{max}\) represents the maximum number of search. The value of \(\alpha\left(n\right)\) will decrease as the number of searches \(n\) increases. In this way, a relatively fast search speed is employed in the early stages of the search. As the algorithm progresses, the search speed will gradually slow down.