Algorithm 2 Optimal Lattice Monitor Path Algorithm

1:Input: $s_j\in \mathcal{S}\in {\mathcal{S}}_c$ 2:Compute all the neighbor links ${\mathcal{E}}_{ij}$ and sequence it by length in an increasing way 3:Initialize the ${\mathcal{M}}_i$, let ${\mathcal{M}}_i={\mathcal{M}}_i^{\Delta }\left(1\right)$ 4: $j=1:\left|{\mathcal{E}}_{ij}\right|$ 5: if $(rank({\mathcal{M}}_i)\le 2(\left|{\mathcal{I}}_i^{〞}\right|+1)-3)$ then 6: ${\mathcal{M}}_i=\left[\begin{array}{c}{\mathcal{M}}_i\\ {\mathcal{M}}_i^{\Delta }(j+1)\end{array}\right]$ 7: if $({\mathcal{M}}_i$ is full rank) then 8:  According to ${\mathcal{M}}_i$, record the edge ${\mathcal{E}}_{ij}^{\prime }$ 9: end if 10: end if 11:for $l=\{{\mathcal{I}}_i^{〞}$, $i\}$do 12: for $k=\{{\mathcal{I}}_i^{〞}$, i, $k\ne l\}$ do 13:  if $((k$,$l)\notin {\mathcal{E}}_{ij}^{\prime })$ then 14:   Delete $(k$,$l)$ 15:   Draw the edge of ${\mathcal{E}}_{ij}^{\prime }$ 16:  end if 17: end for 18: end for