Problem Definition of Local structural alignment

INPUT: Given two protein structures $P_1=(a_{1,1},a_{1,2},a_{1,3}{\ldots})$ and $P_2=(b_{1,1},b_{1,2},b_{1,3}\ldots)$ where $a_{i,j}$ represents $j^{th}$ atom of $i^{th}$ residue $P_1$ and $b_{p,q}$ is $q^{th}$ atom in $p^{th}$ residue of $P_2$ . The value in $a_{i,j}$ and $b_{p,q}$ corresponds to 3D-coordinate of that atom, $a_{i,j}=(X_{j},Y_{j},Z_{j})$ and $b_{p,q}=(X_{q},Y_{q},Z_{q})$ .
OUTPUT: Define correspondence between $P_1$ and $P_2$ as $C_{1,2} = ((a_{p,q},b_{r,s}),(a_{m,n},b_{k,l}),{\ldots})$ , the Local structural alignment problem asks to find a correspondence ($C_{1,2}$ ) between $P_1$ ,$P_2$ along with a rotational matrix $R$ and translation matrix $T$ such that when you apply $R$ and $T$ to one set of coordinates ( $a_{p,q},a_{m,n},\ldots$ ) we would be able to produce the other set in the correspondence ( $b_{r,s},b_{k,l},\ldots$ ), the optimization version asks for $\vert C_{1,2}\vert$ to be maximal.