Figure 6

The incorporation of cavity matching subroutines in the DP algorithm. In (a), the DP matrix H is illustrated, where the solid red entries correspond to entries in the sub-matrix$H_{u,u^{\prime }}$ for some u ∈ T and u′∈S. In this example, N(u) = v ₁ ,v ₂ ,v ₃ , and$N\left(u^{\prime }\right)=v_1^{\prime },v_2^{\prime },v_3^{\prime },v_4^{\prime }$, where$\left|T_u^{v_1}\right|\le \left|T_u^{v_2}\right|\le \left|T_u^{v_3}\right|$, and$\left|S_{u^{\prime }}^{v_1^{\prime }}\right|\le \left|S_{u^{\prime }}^{v_2^{\prime }}\right|\le \left|S_{u^{\prime }}^{v_3^{\prime }}\right|\le \left|S_{u^{\prime }}^{v_4^{\prime }}\right|$. The solid blue entries correspond to computed values of the form${\mathit{\text{Rooted-HSA}}}^{-r}(T_{v_i}^u,S_{v_j^{\prime }}^{u^{\prime }})$, which are required for the computation of term III in Equation 7 for entries in$H_{u,u^{\prime }}$. The waived entries correspond to computed values of the form${\mathit{\text{Rooted-HSA}}}^{-r}(T_{v_i}^u,S_{u^{\prime }}^{v_j^{\prime }})$ and${\mathit{\text{Rooted-HSA}}}^{-r}(T_u^{v_i},S_{v_j^{\prime }}^{u^{\prime }})$, which are required for the computation of terms I and II in Equation 7 for entries in$H_{u,u^{\prime }}$. (a) The computation of${\mathit{\text{Rooted-HSA}}}^{-r}(T_u^{v_1},S_{u^{\prime }}^{v_1^{\prime }})$ according to Equation 7. The entry corresponding to this sub-instance is marked with a solid black circle. In order to compute term I of the equation, there is a need to examine values of the form${\mathit{\text{Rooted-HSA}}}^{-r}(T_{v_j}^u,S_{u^{\prime }}^{v_1^{\prime }})$ for v _j  ∈ N(u) ∖ {v ₁ }. As each$T_{v_j}^u$ is a subtree of$T_u^{v_1}$, the rows corresponding to these subtrees have smaller indices than the row of$T_u^{v_1}$, and so the required solutions are already computed and stored in H (waved entries marked with a dashed green circle, in the same column above the computed entry). Similarly, for computing term II, there is a need to examine solutions${\mathit{\text{Rooted-HSA}}}^{-r}(T_u^{v_1},S_{v_j^{\prime }}^{u^{\prime }})$ for$v_j^{\prime }\in N\left(u^{\prime }\right)\setminus \left\{v_1^{\prime }\right\}$, appearing at the same row and to the left of the computed entry. For computing term III, there is a need to construct the matching cost function$w_{u,u^{\prime }}^{v_1,v_1^{\prime }}$, which assigns for each v _j  ∈ N(u) ∖ {v ₁ } and$v_{j^{\prime }}^{\prime }\in N\left(u^{\prime }\right)\setminus \left\{v_1^{\prime }\right\}$ the matching cost$w_{u,u^{\prime }}^{v_1,v_1^{\prime }}(v_j,v_{j^{\prime }}^{\prime })={\mathit{\text{Rooted-HSA}}}^{-r}(T_{v_j}^u,S_{v_{j^{\prime }}^{\prime }}^{u^{\prime }})$ (the corresponding matching instance is shown in (b)). These required values appear in blue entries marked by a dashed black circle. Due to the order in which entries are being traversed, all required values were previously computed by the algorithm and stored in H, thus it is possible to compute${\mathit{\text{Rooted-HSA}}}^{-r}(T_u^{v_1},S_{u^{\prime }}^{v_1^{\prime }})$ at this stage.