Algorithm 3 A fast algorithm for constructing combinations.

Input: A set of POIs $P=\{1,\cdots ,m\}$ on the line segment with positions $p_1⪯p_2⪯\cdots ⪯p_m$,        where $(p_j,0)$ is the position for $j\in P$, and a real number r that is the radii of the sensors;

Output: The set of combinations $\mathsf{\Phi }:=\{c_1,\cdots ,c_t\}$.

1: Set $\mathsf{\Phi }:=\varnothing$, $p_0:=-2r$ and $p_{m+1}=p_m+2r$;

2: For POI $j=1$ to $\left|P\right|$ do

3:    For $i:=j$ to $\left|P\right|$ do

4:        If $p_i-p_j\le 2r$ and $p_{i+1}-p_{j-1}\ge 2r$ then

5:            $\mathsf{\Phi }:=\mathsf{\Phi }\cup \left\{\right\{j,\cdots ,i\left\}\right\}$;

6:       EndIf

7:    EndFor

8: EndFor

9: Return $\mathsf{\Phi }$ and terminate.