Geometric Algebra to Describe the Exact Discretizable Molecular Distance Geometry Problem for an Arbitrary Dimension

Abstract

The K-Discretizable Molecular Distance Geometry Problem (\n $$^{\\textit{K}}\\hbox {DMDGP}$$\n ) is a subclass of the Distance Geometry Problem\xa0(DGP), whose complexity is NP-hard, such that the search space is finite. In this work, the authors describe it completely using Conformal Geometric Algebra (CGA), exploring a Minkowski space that provides a natural interpretation of hyperspheres, hyperplanes, points and pair of points as computational primitives, which are largely relevant to the $$^{\\textit{K}}\\hbox {DMDGP}$$\n . It also presents a theoretical approach to solve the $$^{\\textit{K}}\\hbox {DMDGP}$$\n using ideas from classic Branch-and-Prune\xa0(BP) algorithm in this new fashion. Time complexity analysis and practical computational results showed that the naive implementation of the CGA is not as efficient as classical formulation. In order to illustrate this, preliminary results are displayed at the end and, also, directions to future developments.