Algorithm 1. Find minimum-cost path of $b_{\lambda }$ for collaborative task execution.

1.    Input:

2.               $S, D, T_d$

3.    $p_b\leftarrow PathAlgorithm\left(S,D,b\right)$

4.    If     $d\left(p_b\right)\le T_d$ then

5.              return $p_b$

6.    end if

7.    $p_d\leftarrow PathAlgorithm\left(S,D,d\right)$

8.    If $d\left(p_d\right)>T_d$ then

9.              return “There is no solution”

10.    end if

11.    while true do

12.              $\mathsf{\lambda }\leftarrow \frac{b\left(p_b\right)-b\left(p_d\right)}{d\left(p_d\right)-d\left(p_b\right)}$

13.              $p_{\lambda }\leftarrow PathAlgorithm\left(S,D,b_{\lambda }\right)$

14.              If $b_{\lambda }\left(p_{\lambda }\right)=b_{\lambda }\left(p_b\right)$ then

15.                    return $p_d$

16.              else

17.                    if $d\left(p_{\lambda }\right)\le T_d$ then

18.                       $p_d\leftarrow p_{\lambda }$

19.                    else

20.                       $p_b\leftarrow p_{\lambda }$

21.                    end If

22.              end If

23.    end while

24.    Output: $p_{\lambda }^{*}$