On Complexity of Protein Structure Alignment Problem under Distance Constraint

We study the well-known Largest Common Point-set (LCP) under Bottleneck Distance Problem. Given two proteins o and 6 (as sequences of points in three-dimensional space) and a distance cutoff σ, the goal is to find a spatial superposition and an alignment that maximizes the number of pairs of points from a and b that can be fit under the distance σ from each other. The best to date algorithms for approximate and exact solution to this problem run in time O(n⁸) and O(n³²), respectively, where n represents protein length. This work improves runtime of the approximation algorithm and the expected runtime of the algorithm for absolute optimum for both order-dependent and order-independent alignments. More specifically, our algorithms for near-optimal and optimal sequential alignments run in time O(n⁷log n) and O(n¹⁴ log n), respectively. For nonsequential alignments, corresponding running times are O(n^7.5) and O(n^14.5).