Preprocessing Chains for Fast Dihedral Rotations Is Hard or Even Impossible
<|reference_start|>Preprocessing Chains for Fast Dihedral Rotations Is Hard or Even Impossible: We examine a computational geometric problem concerning the structure of polymers. We model a polymer as a polygonal chain in three dimensions. Each edge splits the polymer into two subchains, and a dihedral rotation rotates one of these chains rigidly about this edge. The problem is to determine, given a chain, an edge, and an angle of rotation, if the motion can be performed without causing the chain to self-intersect. An Omega(n log n) lower bound on the time complexity of this problem is known.   We prove that preprocessing a chain of n edges and answering n dihedral rotation queries is 3SUM-hard, giving strong evidence that solving n queries requires Omega(n^2) time in the worst case. For dynamic queries, which also modify the chain if the requested dihedral rotation is feasible, we show that answering n queries is by itself 3SUM-hard, suggesting that sublinear query time is impossible after any amount of preprocessing.<|reference_end|>
@article{soss2002preprocessing,
    title={Preprocessing Chains for Fast Dihedral Rotations Is Hard or Even
  Impossible},
    author={Michael Soss, Jeff Erickson, and Mark Overmars},
    journal={arXiv preprint arXiv:cs/0204042},
    year={2002},
    archivePrefix={arXiv},
    eprint={cs/0204042},
    primaryClass={cs.CG}
}