Solving Distance Geometry Problem with Inexact Distances in Integer Plane

  • Piyush K. BhunreEmail author
  • Partha Bhowmick
  • Jayanta Mukhopadhyay
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9667)


Given the pairwise distances for a set of unknown points in a known metric space, the distance geometry problem (DGP) is to compute the point coordinates in conformation with the distance constraints. It is a well-known problem in the Euclidean space, has several variations, finds many applications, and so has been attempted by different researchers from time to time. However, to the best of our knowledge, it is not yet fully addressed to its merit, especially in the discrete space. Hence, in this paper we introduce a novel variant of DGP where the pairwise distance between every two unknown points is given a tolerance zone with the objective of finding the solution as a collection of integer points. The solution is based on characterization of different types of annulus intersection, their equivalence, and cardinality bounds of integer points. Necessary implementation details and useful heuristics make it attractive for practical applications in the discrete space.


  1. 1. Accessed 27 Sept 2015
  2. 2.
    Blumenthal, L.M.: Theory and Application of Distance Geometry. Oxford University Press, Oxford (1953)zbMATHGoogle Scholar
  3. 3.
    Brandes, U., Pich, C.: An experimental study on distance-based graph drawing. In: GD 2008, Revised Papers, pp. 218–229 (2008)Google Scholar
  4. 4.
    Bulusu, N., Estrin, D., Heidemann, J.: Scalable coordination for wireless sensor networks self-configuring localization systems. In: Proceedings of the ISCTA (2001)Google Scholar
  5. 5.
    Cheng, L., Wu, C., Zhang, Y., Wu, H., Li, M., Maple, C.: A survey of localization in wireless sensor network. Int. J. Distrib. Sens. Netw., 324–357 (2012)Google Scholar
  6. 6.
    Civril, A., Magdon-Ismail, M., Bocek-Rivele, E.: SDE: graph drawing using spectral distance embedding. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 512–513. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Dong, Q., Wu, Z.: A linear-time algorithm for solving the molecular distance geometry problem with exact inter-atomic distance. J. Global Optim. 22, 365–375 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dong, Q., Wu, Z.: A geometric build-up algorithm for solving the molecular distance geometry problem with sparse distance data. J. Global Optim. 26, 321–333 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gibson, H., Faith, J., Vickers, P.: A survey of two-dimensional graph layout techniques for information visualization. Inf. Vis. 12(3–4), 324–357 (2012)Google Scholar
  10. 10.
    Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)zbMATHGoogle Scholar
  11. 11.
    Lavor, C., Liberti, L., Maculan, N., Mucherino, A.: Recent advances on the discretizable molecular distance geometry problem. Computational Optimization and Applications (2012)Google Scholar
  12. 12.
    Lavor, C., Liberti, L., Mucherino, A.: The interval branch-and-prune algorithm for the discretizable molecular distance geometry problem with inexact distances. J. Global Optim. 56, 855–871 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liberti, L., Lavor, C., Maculan, N.: A branch-and-prune algorithm for the molecular distance geometry problem. Intl. Trans. Oper. Res. 15, 1–17 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Liberti, L., Lavor, C., Mucherino, A.: Euclidean distance geometry and applications. SIAM Rev. 56, 3–69 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Liberti, L., Lavor, C., Mucherino, A., Maculan, N.: Molecular distance geometry methods:from continuous to discrete. Intl. Trans. Oper. Res. 18, 33–51 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mor\(\acute{e}\), J.J., Wu, Z.: Distance geometry optimization for protein structures. J. Global Optim. 15, 219–234 (1999)Google Scholar
  17. 17.
    Mucherino, A., Lavor, C., Liberti, L., Maculan, N.: Distance Geometry: Theory, Methods, and Applications, 1st edn. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  18. 18.
    Savvides, A., Han, C.C., Strivastava, M.B.: Dynamic fine-grained localization in ad-hoc networks of sensors. In: Proceedings of the MobiCom 2001, pp. 166–179 (2001)Google Scholar
  19. 19.
    Saxe, J.B.: Embeddability of weighted graphs in k-space is strongly NP-hard. In: Proceedings of the 17th Allerton Conference in Communications, Control & Computing, pp. 480–489 (1979)Google Scholar
  20. 20.
    Sit, A.: Solving distance geometry problems for protein structure determination. P.h.D thesis, Iowa State University (2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Piyush K. Bhunre
    • 1
    Email author
  • Partha Bhowmick
    • 1
  • Jayanta Mukhopadhyay
    • 1
  1. 1.Department of Computer Science and EngineeringIndian Institute of TechnologyKharagpurIndia

Personalised recommendations