Algorithm 1 A greedy algorithm for decision LBTC

Input: A bound $D\in {\mathbb{Z}}^{+}$ on maximum movement, a set of sensors $\mathsf{\Gamma }=\{1,\cdots ,n\}$ with original       positions $\left\{(x_i,y_i)\right|i\in {\left[n\right]}^{+}\}$ and r being the sensing radii, 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: New positions $\left\{x_i^{\prime }\right|i\in {\left[n\right]}^{+}\}$ for the sensors or return “infeasible”.

1: Set $\mathcal{I}:=\mathsf{\Gamma }$ and $s:=p_1$, where s is the leftmost point of the uncovered part of the barrier.

2: For each sensor i do

3:    Compute the leftmost position $l_i$ and the rightmost position $g_i$ that sensor i can monitor;

4: EndFor

5: While $\mathcal{I}\ne \varnothing$ do

6:    If there exists $i^{\prime }\in \mathcal{I}$, such that $l_{i^{\prime }}\le s\le g_{i^{\prime }}$ then

7:       Select the sensor with minimum $g_i$ among all the sensors $\left\{i^{\prime }\right|l_{i^{\prime }}\le s\le g_{i^{\prime }}\}$, which is to find             sensor $i\in \mathcal{I}$ that $g_i={min}_{i^{\prime }:l_{i^{\prime }}\le s\le g_{i^{\prime }}}\left\{g_{i^{\prime }}\right\}$;

8:       Set $t:=min\{s+2r,g_i\}$ and $x_i^{\prime }:=t-r$;

9:       Set $s:=min\left\{p_j\right|p_j>t\}$ and $\mathcal{I}:=\mathcal{I}\setminus \left\{i\right\}$;

10:    Else

11:       Return “infeasible”;

12:    End if

13:    If $t\ge p_m$ then /*All POIs have been covered. */

14:       Return “feasible” together with the new positions $\left\{x_i^{\prime }\right|i\in \mathsf{\Gamma }\}$;

15:    Endif

16: Endwhile