Improved spliced alignment algorithm

Define $P(i,j) = \max \limits _{B_l \in \mathcal{B}[i]} S({\rm Last}(B_l),j,l)$. The recursive relation for S(i,j,k) can be presented as:

$S(i,j,k) = \max \cases{ S(i,j-1,k) + (-,t_j) & if $i \geq {\rm First}(B_k)$\s... ..._k)$\space \cr P(i,j) + (g_i, -) & if $i = {\rm First}(B_k)$\space \cr }$


The value of P(i,j) can be achieved by an alignment of $\Gamma$ ending before the (i-1)'th position (=P(i-1,j)) or exactly at the (i-1)'th position (= $\max \limits _k S(i,j,k)$). Therefore, the recursive relation for P(i,j) is:

$P(i,j)=\max \{P(i-1,j), \max \limits _k S(i,j,k) \}$

The space complexity remains O(nmL), however the time complexity is now reduced to O(nmL) (instead of O(nmL²)).