Fig. 3.

(A) Example alignment. Due to stacking effects, the probability of a base pair a₃ in the loop closed by a₂ (loop of single structure) is much higher than its probability in the loop closed by a₁ [loop of the consensus structure $\{(a_1,b_1),(a_3,b_2)\}$]. (B) Computing represented entries in the sparsified algorithm. We show the matrix M^ab; the rounded bars left and on top of the matrix symbolize the represented positions for a and b; the gray areas contain the represented entries for M^ab. In our example, the entry $M^{ab}(i,k)$ recurses (solid arrows) to unrepresented entries $M^{ab}(i-1,k-1),M^{ab}(i,k-1)$, $M^{ab}(i-1,k)$ and $M^{ab}(a_1^{\text{L}}-1,b_1^{\text{L}}-1)$ (white boxes); the latter via matching base pairs; their left ends correspond to the dashed box at $(a_1^{\text{L}},b_1^{\text{L}})$. The numbers 1-4 at the arrow heads refer to the respective recursion case. The unrepresented entries are computed from represented entries (dashed arrows to black boxes), each in constant time