Algorithm 3LDP

Input: Initialized shape assembly xp
Output: Resulting shape assembly xo and the associated oriented boundaries.
begin
1. For each object Oi(1≤i≤m), do steps 2 to 4.
2. Determine $P_1^{O_i},P_2^{O_i},\dots .,P_{n_i}^{O_i}$ sets of points for Oi in I.
3. Determine costs κ(u,v) via first level DP for all directed arcs in Fig. 1.
4. For each point u in $P_1^{O_i}$, determine the best directed path from u back to u via the second level DP, the corresponding shape xOi, and its total cost Ki(xOi).
5. Compute arc costs $IC\left(B_{\ell }^{O_i},B_k^{O_i}\right)$ in Fig. 2. Find the best path xo which has minimal total cost in the graph of Fig. 2 via third level DP.
6. Output the optimum shape assembly xo and the corresponding oriented boundaries.
end