Algorithm 2 The whole algorithm for optimal LBTC.

Input: A movement bound $D\in {\mathbb{Z}}^{+}$, a set of sensors $\mathsf{\Gamma }=\{1,\cdots ,n\}$ with positions $\left\{(x_i,y_i)\right|i\in {\left[n\right]}^{+}\}$,        the sensing radii r, a set of POIs $P=\{1,\cdots ,m\}$ with positions $p_1⪯p_2⪯\cdots ⪯p_m$, where $p_j$ is        the position for $j\in P$;

Output: The minimized maximum movement of the sensors together with new positions $\left\{x_i^{\prime }\right|i\in {\left[n\right]}^{+}\}$

1: Set $ub:=d_{max}$ and $lb:=1$, where $d_{max}$ is the maximum distance between the sensors and the POIs;

2: If there exists no coverage returned from calling Algorithm 1 wrt $D=d_{max}$ then

3:     Return “infeasible”;

4: EndIf

5: Set $tmp:=⌈\frac{lb+ub}{2}⌉$;

6: While $ub-lb>1$ do

7:     If there exists no coverage returned from calling Algorithm 1 wrt $D=tmp$ then

8:        Set $lb:=tmp$ and $tmp:=⌈\frac{lb+ub}{2}⌉$;

9:     Else

10:        Set $ub:=tmp$ and then $tmp:=⌈\frac{lb+ub}{2}⌉$;

11:     EndIf

12: End While

13: Return the result of call Algorithm 1 wrt $D=tmp$.