Algorithm 4 A fast algorithm for LBTC.

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

Output: A minimum movement D under which the sensors can be relocated to covered all POIs in P.

1: Set $\mathsf{\Psi }:=\varnothing$ and compute $\mathsf{\Phi }:=\{c_1,\cdots ,c_t\}$ the collection of combinations by Algorithm 3;

2: For each sensor i do

3:    For each combination $c_j\in \mathsf{\Phi }$ do

4:       Compute $d_{ij}$, the minimum movement needed when using sensor i to cover $c_j$;

5:       Set $\mathsf{\Psi }:=\mathsf{\Psi }\cup \left\{d_{ij}\right\}$;

6:    EndFor

7: EndFor

8: Sort $\mathsf{\Psi }$ in a non-decreasing order and set $lb:=1$ and $ub:=\left|\mathsf{\Psi }\right|$, to prepare for binary search.

9: Use $\mathsf{\Psi }\left[1\right]$ as the movement bound (i.e., D) and call Algorithm 1;

/*Note that $\mathsf{\Psi }\left[1\right]$ is the smallest element in $\mathsf{\Psi }$. */

10: If there exists a feasible coverage under movement bound $\mathsf{\Psi }\left[1\right]$ then

11:    Return $\mathsf{\Psi }\left[1\right]$ as the optimum movement bound;

12: End if

13: While $ub-lb>1$ do

14:    Set $idx:=⌈\frac{lb+ub}{2}⌉$;

15:    Use $\mathsf{\Psi }\left[idx\right]$ as the movement bound (i.e., D) and call Algorithm 1;        /*$\mathsf{\Psi }\left[idx\right]$ is the $idx$th smallest element in $\mathsf{\Psi }$. */

16:    If there exists a feasible coverage under movement bound $\mathsf{\Psi }\left[idx\right]$ then

17:       Set $ub:=idx$;

18:    Else

19:       Set $lb:=idx$;

20:    End if

21: Endwhile

22: Return $\mathsf{\Psi }\left[idx\right]$ as the optimum movement bound.