An Optimal Algorithm for Reconstructing Point Set Order Types from Radial Orderings

  • Oswin Aichholzer
  • Vincent Kusters
  • Wolfgang Mulzer
  • Alexander Pilz
  • Manuel Wettstein
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9472)


Given a set P of n labeled points in the plane, the radial system of P describes, for each \(p\in P\), the radial ordering of the other points around p. This notion is related to the order type of P, which describes the orientation (clockwise or counterclockwise) of every ordered triple of P. Given only the order type of P, it is easy to reconstruct the radial system of P, but the converse is not true. Aichholzer et al. (Reconstructing Point Set Order Types from Radial Orderings, in Proc. ISAAC 2014) defined T(R) to be the set of order types with radial system R and showed that sometimes \(|T(R)|=n-1\). They give polynomial-time algorithms to compute T(R) when only given R.

We describe an optimal \(O(n^2)\) time algorithm for computing T(R). The algorithm constructs the convex hulls of all possible point sets with the given radial system, after which sidedness queries on point triples can be answered in constant time. This set of convex hulls can be found in O(n) time. Our results generalize to abstract order types.


Convex Hull Rotation System Radial System Order Type Generalize Configuration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was initiated during the ComPoSe Workshop on Order Types and Rotation Systems held in February 2015 in Strobl, Austria. We thank the participants for valuable discussions.

O.A. and A.P. are partially supported by the ESF EUROCORES programme EuroGIGA - ComPoSe, Austrian Science Fund (FWF): I 648-N18. W.M. is supported in part by DFG grants MU-3501/1 and MU-3501/2.


  1. 1.
    Aichholzer, O., Cardinal, J., Kusters, V., Langerman, S., Valtr, P.: Reconstructing point set order types from radial orderings. In: Ahn, H.-K., Shin, C.-S. (eds.) ISAAC 2014. LNCS, vol. 8889, pp. 15–26. Springer, Heidelberg (2014) Google Scholar
  2. 2.
    Balko, M., Fulek, R., Kynčl, J.: Crossing numbers and combinatorial characterization of monotone drawings of \(K_n\). Discrete Comput. Geom. 53(1), 107–143 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chazelle, B., Guibas, L.J., Lee, D.T.: The power of geometric duality. BIT 25(1), 76–90 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Edelsbrunner, H., O’Rourke, J., Seidel, R.: Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput. 15(2), 341–363 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Goodman, J.E.: Proof of a conjecture of Burr, Grünbaum, and Sloane. Discrete Math. 32(1), 27–35 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Goodman, J.E., Pollack, R.: Semispaces of configurations, cell complexes of arrangements. J. Combin. Theor. Ser. A 37(3), 257–293 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Knuth, D.E.: Axioms and Hulls. LNCS, vol. 606. Springer, Heidelberg (1992) zbMATHGoogle Scholar
  8. 8.
    Kynčl, J.: Simple realizability of complete abstract topological graphs in P. Discrete Comput. Geom. 45(3), 383–399 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Pilz, A., Welzl, E.: Order on order types. In: Proceedings 31st International Symposium on Computational Geometry (SOCG 2015), pp. 285–299. LIPICS (2015)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Oswin Aichholzer
    • 1
  • Vincent Kusters
    • 2
  • Wolfgang Mulzer
    • 3
  • Alexander Pilz
    • 1
  • Manuel Wettstein
    • 2
  1. 1.Institute for Software TechnologyGraz University of TechnologyGrazAustria
  2. 2.Department of Computer ScienceETH ZürichZurichSwitzerland
  3. 3.Institut für InformatikFreie Universität BerlinBerlinGermany

Personalised recommendations