Local Structural Alignment algorithm complexity

From the problem definition in the section2 which asks to find correspondence $C_{1,2}$ between $P_1$ and $P_2$ (assume each of length of $n$ ), a naive algorithm to find a correspondence of length of $l$ would test each of $O(l!*n^2)$ correspondences to see if it can find a rigid transformation $R$ (Rotational), $T$ (Transformational) which can transform one set in the correspondence to the other set, and to find the correspondence with maximum $l$ we to try for all the possible $k{\leq}l{\leq}n$ and hence the naive algorithm needs to generate all $\sum_{l=k}^{n}O(l!*n^2)$ correspondences and for each of such correspondence check if there is a rigid transformation $R$ and $T$ which can transform one set to the other, note the $l!$ comes from ordering of the atoms in each of the residue, as we described in the section3.1 the atoms in each residue can be ordered in any order so we need to consider all the $l!$ orderings. Now we have addressed the complexity of the problem and lets see how we can solve this in the next section.