Table 4 A polynomial time algorithm for the MAD problem

Input:

sequences P = (p₁,…,p _n ), Q = (q₁,…,q _m ) and $\ell \in \mathbb{I}$.

Without loss of generality assume m ≥ n.

Output:

(i) subsets P^′ ⊆ P, Q^′ ⊆ Q, |P^′| = |Q^′|, and

(ii) mapping f:P^′↦Q^′, fulfilling the following conditions:

(A) |P^′| = ℓ,

(B) d = RMSD(P^′,f(p^′)) is minimized.

1.

For each translation t ∈ {I/ℓ| - ℓ c _max  ≤ I ≤ ℓ c _max }³,

For each 3 × 3 matrix M, where ∀e ∈ M, e ∈ {I/ℓ²|-,

$\{4{\ell }^3{c}_{\mathit{\text{max}}}^2\le I\le 4{\ell }^3{c}_{\mathit{\text{max}}}^2\}$

Compute rotation matrix R from M.

$Q\leftarrow \mathit{RQ}-t$.

Apply an algorithm for the case where the superposition

is known to P and $Q$ (as discussed in the ‘Complexity Of

The LCP And MAD When The Optimal Superposition Is

Known’ section), and denote the solution MAD(P, $Q$).

2.

Output the MAD(P, $Q$) of the smallest RMSD as the solution.