Algorithm 3. Our proposed algorithm.

Input: $G(V,E)$

Output: $Tree\{1,2,\dots M\}$, $Optr$

1 randomly select $M$ sensor nodes $g_1,g_2,\dots ,g_M$ in the network probability $\frac{M}{N}$, generate $\mathsf{\Phi }$

2 for $i=1:M$

3   query candidate nodes $Ca{n}_i$(${\varphi }_i{}_j\ne 0$) of projection nodes $i$

4   initialization $Tre{e}_i\leftarrow i$, $tem{p}_i\leftarrow i$

5   while !empty(temp) do

6     $CanNode\leftarrow Del(temp)$

7     if $CanNode$ is $i$’s candidate node

8      $T\mathrm{ree}\leftarrow CanNode$

9      $temp\leftarrow CanNode$

10     $Del(In{t}_i,CanNode)$

11    end if

12   end while

13  while !empty($Ca{n}_i$) do

14    for all residual candidate nodes $r\in Ca{n}_i$

15    $Path(r)\leftarrow$ find a shortest path to $Tre{e}_i$ using the $Dijkstra$ algorithm

16      if $shortestpath$ > $Path(r)$

17         $shortestpath\leftarrow$$Path(r)$

18      end if

19    end for

20   $Tree\leftarrow shortestpath$

21   $temp\leftarrow shortestpath$

22   $Del(Ca{n}_i,shortestpath)$

23   while !empty(temp) do

24   go back to steps 7–11

25   end while

26  end while

27 $Optr\leftarrow$ Optimal routing from $i$ to the sink node using Algorithm 2

28  return $Tree\{1,2,\dots M\}$

29 end for