Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Optimal Deterministic Shallow Cuttings for 3-d Dominance Ranges


Shallow cuttings are one of the most fundamental tools in range searching as many problems in the field admit efficient static data structures due to their application. We present the first efficient deterministic algorithms that, given n three-dimensional points, construct optimal-size (single and multiple) shallow cuttings for orthogonal dominance ranges. Specifically, we show how to construct a single shallow cutting in \(O\left( n\log n\right) \) worst-case time, using \(O\left( n\right) \) space. We also show that the same complexity suffices to construct simultaneously a logarithmic number of shallow cuttings on the input pointset. Our algorithms are optimal in the comparison and algebraic-comparison models, and constitute an important step forwards as the first improvement over previous deterministic polynomial-time guarantees by Matoušek (Comput Geom 2(3):169–186, 1992) and Agarwal et al. (SIAM J Comput 29(3):912–953, 2000) matching the complexity of the optimal deteministic algorithm for the more general 3-d halfspace ranges by Chan and Tsakalidis (Discrete Comput Geom 56(4):866–881, 2016). Our methods yield worst-case efficient preprocessing algorithms for a series of important orthogonal range searching problems in the pointer machine and the word-RAM models, where such shallow cuttings are utilized to support queries efficiently.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4


  1. 1.

    Afshani, P.: On dominance reporting in 3D. In: Proceedings of 16th European Symposium on Algorithms, pp. 41–51 (2008)

  2. 2.

    Afshani, P., Arge, L., Larsen, K.D.: Orthogonal range reporting in three and higher dimensions. In: Proceedings of 50th IEEE Symposium on Foundations of Computer Science, pp. 149–158 (2009)

  3. 3.

    Afshani, P., Arge, L., Larsen, K.D.: Orthogonal range reporting: query lower bounds, optimal structures in 3-d, and higher-dimensional improvements. In: Proceedings of 26th ACM Symposium on Computational Geometry, pp. 240–246 (2010)

  4. 4.

    Afshani, P., Arge, L., Larsen, K.G.: Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model. In: Proceedings of 21st ACM Symposium on Computational Geometry, pp. 323–332 (2012)

  5. 5.

    Afshani, P., Brodal, G.S., Zeh, N.: Ordered and unordered top-K range reporting in large data sets. In: Proceedings of 22nd ACM/SIAM Symposium on Discrete Algorithms, pp. 390–400 (2011)

  6. 6.

    Afshani, P., Chan, T.M.: Optimal halfspace range reporting in three dimensions. In: Proceedings of 20th ACM/SIAM Symposium on Discrete Algorithms, pp. 180–186 (2009)

  7. 7.

    Afshani, P., Chan, T.M., Tsakalidis, K.: Deterministic rectangle enclosure and offline dominance reporting on the RAM. In: Proceedings of 41st International Colloquium on Automata, Languages, and Programming, LNCS, vol. 8572, pp. 77–88. Springer (2014)

  8. 8.

    Afshani, P., Hamilton, C., Zeh, N.: A general approach for cache-oblivious range reporting and approximate range counting. Comput. Geom. 43(8), 700–712 (2010). Special Issue on the 25th Annual Symposium on Computational Geometry (SoCG’09)

  9. 9.

    Afshani, P., Tsakalidis, K.: Optimal deterministic shallow cuttings for 3D dominance ranges. In: Proceedings of 25th ACM/SIAM Symposium on Discrete Algorithms, pp. 1389–1398 (2014)

  10. 10.

    Agarwal, P.K., Efrat, A., Sharir, M.: Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM J. Comput. 29(3), 912–953 (2000)

  11. 11.

    Chan, T.M.: Random sampling, halfspace range reporting, and construction of \((< k)\)-levels in three dimensions. SIAM J. Comput. 30(2), 561–575 (2000)

  12. 12.

    Chan, T.M.: Persistent predecessor search and orthogonal point location on the word ram. ACM Trans. Algorithms 9(3), 22:1–22:22 (2013)

  13. 13.

    Chan, T.M., Larsen, K.G., Pătraşcu, M.: Orthogonal range searching on the RAM, revisited. In: Proceedings of 27th ACM Symposium on Computational Geometry, pp. 1–10 (2011)

  14. 14.

    Chan, T.M., Tsakalidis, K.: Optimal deterministic algorithms for 2-d and 3-d shallow cuttings. Discrete Comput. Geom. 56(4), 866–881 (2016)

  15. 15.

    Chan, T.M., Wilkinson, B.: Adaptive and approximate orthogonal range counting. In: Proceedings of 24th ACM/SIAM Symposium on Discrete Algorithms, pp. 241–251 (2013)

  16. 16.

    Chazelle, B.: Filtering search: a new approach to query answering. SIAM J. Comput. 15(3), 703–724 (1986)

  17. 17.

    Clarkson, K.L., Shor, P.W.: Applications of random sampling in computational geometry, II. Discrete Comput. Geom. 4, 387–421 (1989)

  18. 18.

    Frederickson, G.N.: An optimal algorithm for selection in a min-heap. Inf. Comput. 104(2), 197–214 (1993)

  19. 19.

    Makris, C., Tsakalidis, K.: An improved algorithm for static 3D dominance reporting in the pointer machine. In: Proceedings of 23rd International Symposium on Algorithms and Computation, pp. 568–577 (2012)

  20. 20.

    Matoušek, J.: Cutting hyperplane arrangements. Discrete Comput. Geom. 6, 385–406 (1991)

  21. 21.

    Matoušek, J.: Reporting points in halfspaces. Comput. Geom. 2(3), 169–186 (1992)

  22. 22.

    McCreight, E.M.: Priority search trees. SIAM J. Comput. 14(2), 257–276 (1985)

  23. 23.

    Nekrich, Y.: A data structure for multi-dimensional range reporting. In: Proceedings of 23rd ACM Symposium on Computational Geometry, pp. 344–353 (2007)

  24. 24.

    Nekrich, Y.: Data structures for approximate orthogonal range counting. In: Proceedings of 20th Latin American Theoretical Informatics Symposium, vol. 5878, pp. 183–192 (2009)

  25. 25.

    Ramos, E.A.: On range reporting, ray shooting and k-level construction. In: Proceedings of 15th ACM Symposium on Computational Geometry, pp. 390–399 (1999)

  26. 26.

    Vengroff, D.E., Vitter, J.S.: Efficient 3-D range searching in external memory. In: Proceedings of 28th ACM Symposium on Theory of Computation, pp. 192–201 (1996)

Download references

Author information

Correspondence to Konstantinos Tsakalidis.

Additional information

A preliminary version of this work appeared in the proceedings of the 25th annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2014) [9].

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Afshani, P., Tsakalidis, K. Optimal Deterministic Shallow Cuttings for 3-d Dominance Ranges. Algorithmica 80, 3192–3206 (2018).

Download citation


  • Shallow cuttings
  • Orthogonal range searching
  • Computational geometry