Abstract
The K-Discretizable Molecular Distance Geometry Problem (\(^{\textit{K}}\hbox {DMDGP}\)) is a subclass of the Distance Geometry Problem (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 \(^{\textit{K}}\hbox {DMDGP}\). It also presents a theoretical approach to solve the \(^{\textit{K}}\hbox {DMDGP}\) using ideas from classic Branch-and-Prune (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.
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References
Alves, R., Lavor, C.: Geometric algebra to model uncertainties in the discretizable molecular distance geometry problem. Adv. Appl. Clifford Algebra 27, 439–452 (2017)
Alves, R., Lavor, C., Souza, C., Souza, M.: Clifford algebra and discretizable distance geometry. Math. Methods Appl. Sci. 41, 4063–4073 (2018)
Andreas, W.M.D., Havel, T.F.: Distance geometry and geometric algebra. Found. Phys. 23(10), 1357–1374 (1993)
Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.: Julia: a fresh approach to numerical computing. SIAM Rev. 59(1), 65–98 (2017)
Charrier, P., Klimek, M., Steinmetz, C., Hildenbrand, D.: Geometric algebra enhanced precompiler for C++, OpenCL and Mathematica’s OpenCLLink. Adv. Appl. Clifford Algebras 24(2), 613–630 (2014)
Coope, I.: Reliable computation of the points of intersection of n spheres in n-space. ANZIAM J. 42, 461–477 (2000)
Crippen, G., Havel, T.: Distance Geometry and Molecular Conformation. Research Studies Press Ltda, New York (1988)
Dorst, L.: Boolean combination of circular arcs using orthogonal spheres. Adv. Appl. Clifford Algebra 29 (2019) (online)
Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry (The Morgan Kaufmann Series in Computer Graphics). Morgan Kaufmann Publishers Inc., (2007)
Dubbs, C., Siegel, D.: Computing determinants. Coll. Math. J. 18(1), 48–50 (1987)
Faria, V.R., Castelani, E.V., da Silva, J., Camargo, V.S., Shirabayashi, W.V.I. evcastelani/liga.jl dev version. GitHub Repository: https://github.com/evcastelani/Liga.jl (2018)
Fernandes, L.A.F.: Geometric Algebra Template Library. GitHub Repository. https://github.com/laffernandes/gatl (2017)
Fidalgo, F., Gonçalves, D., Lavor, C., Liberti, L., Mucherino, A.: A symmetry-based splitting strategy for discretizable distance geometry problems. J. Glob. Optim. 71, 717–733 (2018)
Gonçalves, D.S.: A least-squares approach for discretizable distance geometry problems with inexact distances. Optim. Lett. (2017) (to appear)
Grassmann, H.: Die Lineale Ausdehnungslehre. Verlag von Otto Wigand, Leipzig (1878)
Lavor, C., Alves, R.: Oriented conformal geometric algebra and the molecular distance geometry problem. Adv. Appl. Clifford Algebra 29, 439–452 (2019)
Lavor, C., Alves, R., Figueiredo, W., Petraglia, A., Maculan, N.: Clifford algebra and the discretizable molecular distance geometry problem. Adv. Appl. Clifford Algebra 25, 925–942 (2015)
Lavor, C., Liberti, L., Maculan, N.: A note on a branch-and-prune algorithm for the molecular distance geometry problem. Int. Trans. Oper. Res. 18(6), 751–752 (2011)
Lavor, C., Liberti, L., Maculan, N., Mucherino, A.: The discretizable molecular distance geometry problem. Comput. Optim. Appl. 52, 115–146 (2012)
Li, H., Hestenes, D., Rockwood, A.: Generalized homogeneous coordinates for computational geometry. In: Sommer, G. (ed.) Geometric Computing with Clifford Algebras, pp. 27–59. Springer, Berlin (2001)
Liberti, L.: Private communication, may (2019)
Liberti, L., Lavor, C.: Euclidean Distance Geometry: An Introduction. Springer, Cham, Switzerland (2017)
Liberti, L., Lavor, C., Maculan, N.: A branch-and-prune algorithm for the molecular distance geometry problem. Int. Trans. Oper. Res. 15(1), 1–17. Janeiro (2008)
Liberti, L., Lavor, C., Maculan, N., Mucherino, A.: Euclidean distance geometry and applications. SIAM Rev. 56(1), 3–69 (2014)
Liberti, L., Masson, B., Lee, J., Lavor, C., Mucherino, A.: On the number of realizations of certain henneberg graphs arising in protein conformation. Disc. Appl. Math. (2013) (Article in Press)
Maioli, D.S., Lavor, C., Gonçalves, D.S.: A note on computing the intersection of spheres in \({\mathbb{R}}^{n}\). ANZIAM J. 59(2), 271–279 (2017)
Moré, J. J., W. Z.: Global continuation for distance geometry problems. SIAM J. Comput. 7(3), 814–836 (1997)
Moré, J. J., W.Z.: Distance geometry optimization for protein structures. J. Glob. Optim. 15(3), 219–234 (1999)
Mucherino, A., Lavor, C., Liberti, L.: Exploiting symmetry properties of the discretizable molecular distance geometry problem. J. Bioinf. Comput. Biol. 10, 3 (2012)
Perwass, C.B.U.: Geometric Algebra with Applications in Engineering. Springer, Berlin (2009)
Rojas, N., Thomas, F.: Distance-based position analysis of the three seven-link assur kinematic chains. Mech. Mach. Theory 46(2), 112–126 (2011)
Saxe, J.B.: Embeddability of weighted graphs in k-space is strongly np-hard. In Proc. 17th Allerton Conference in Communications, Control and Computing, pp. 480–489 (1979)
Watt, D.A., Findlay, W.: Programming language design concepts. Wiley, Hoboken (2004)
Worley, B., Delhommel, F., Cordier, F., Malliavin, T.E., Bardiaux, B., Wolff, N., Nilges, M., Lavor, C., Liberti, L.: Tuning interval branch-and-prune for protein structure determination. J. Glob. Optim. 72, 109–127 (2018)
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The authors would like to thank CNPq, FAPERJ, UFSC, UFF, and UEM for financial support and Leo Liberti for precious research information.
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This article is part of the Topical Collection on Proceedings of AGACSE 2018, IMECC-UNICAMP, Campinas, Brazil, edited by Sebastià Xambó-Descamps and Carlile Lavor.
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Camargo, V.S., Castelani, E.V., Fernandes, L.A.F. et al. Geometric Algebra to Describe the Exact Discretizable Molecular Distance Geometry Problem for an Arbitrary Dimension. Adv. Appl. Clifford Algebras 29, 75 (2019). https://doi.org/10.1007/s00006-019-0995-7
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DOI: https://doi.org/10.1007/s00006-019-0995-7