Figure 3

(A) Sparsification of secondary structure folding. Suppose the optimal solution L_i,j is obtained from the optimal solutions L_i,k, L_{k + 1,q} and L_{q + 1,j}. Based on the recursions of the secondary structures, L_i,kand L_{k + 1,q} produce an optimal solution of L_i,q. Similarly, L_{k + 1,q} and L_{q + 1,j} produce an optimal solution of L_{k + 1,j}. Now, in order to obtain an optimal solution of L_i,j it is sufficient to consider either the grouping L_i,q and L_{q + 1,j} or L_i,k and L_{k + 1,j}. (B) General idea of sparsification: L _v is alternatively realized via$L_{v_1}$ and$L_{v_2^{\prime }}$, or$L_{v_1^{\prime }}$ and$L_{v_3}$. Thus it is sufficient to only consider one of the computation paths.