Faster Algorithms for Largest Empty Rectangles and Boxes

Author Timothy M. Chan

Thumbnail PDF


  • Filesize: 0.75 MB
  • 15 pages

Document Identifiers

Author Details

Timothy M. Chan
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA


I thank David Zheng for discussions on the 2D problem.

Cite AsGet BibTex

Timothy M. Chan. Faster Algorithms for Largest Empty Rectangles and Boxes. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We revisit a classical problem in computational geometry: finding the largest-volume axis-aligned empty box (inside a given bounding box) amidst n given points in d dimensions. Previously, the best algorithms known have running time O(nlog²n) for d = 2 (by Aggarwal and Suri [SoCG'87]) and near n^d for d ≥ 3. We describe faster algorithms with running time - O(n2^{O(log^*n)}log n) for d = 2, - O(n^{2.5+o(1)}) time for d = 3, and - Õ(n^{(5d+2)/6}) time for any constant d ≥ 4. To obtain the higher-dimensional result, we adapt and extend previous techniques for Klee’s measure problem to optimize certain objective functions over the complement of a union of orthants.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Largest empty rectangle
  • largest empty box
  • Klee’s measure problem


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Pankaj K. Agarwal and Jeff Erickson. Geometric range searching and its relatives. In B. Chazelle, J. E. Goodman, and R. Pollack, editors, Advances in Discrete and Computational Geometry, pages 1-56. AMS Press, 1999. URL:
  2. Alok Aggarwal and Maria M. Klawe. Applications of generalized matrix searching to geometric algorithms. Discret. Appl. Math., 27(1-2):3-23, 1990. URL:
  3. Alok Aggarwal and Subhash Suri. Fast algorithms for computing the largest empty rectangle. In Proc. 3rd Symposium on Computational Geometry (SoCG), pages 278-290, 1987. URL:
  4. Jonathan Backer and J. Mark Keil. The mono- and bichromatic empty rectangle and square problems in all dimensions. In Proc. 9th Latin American Theoretical Informatics Symposium (LATIN), volume 6034 of Lecture Notes in Computer Science, pages 14-25. Springer, 2010. URL:
  5. Jérémy Barbay, Timothy M. Chan, Gonzalo Navarro, and Pablo Pérez-Lantero. Maximum-weight planar boxes in O(n²) time (and better). Inf. Process. Lett., 114(8):437-445, 2014. URL:
  6. Michael A. Bender and Martin Farach-Colton. The LCA problem revisited. In Gaston H. Gonnet, Daniel Panario, and Alfredo Viola, editors, Proc. 4th Latin American Symposium on Theoretical Informatics (LATIN), volume 1776, pages 88-94, 2000. URL:
  7. Jean-Daniel Boissonnat, Micha Sharir, Boaz Tagansky, and Mariette Yvinec. Voronoi diagrams in higher dimensions under certain polyhedral distance functions. Discret. Comput. Geom., 19(4):485-519, 1998. URL:
  8. Karl Bringmann. An improved algorithm for Klee’s measure problem on fat boxes. Comput. Geom., 45(5-6):225-233, 2012. Preliminary version in SoCG'10. URL:
  9. Timothy M. Chan. Geometric applications of a randomized optimization technique. Discret. Comput. Geom., 22(4):547-567, 1999. URL:
  10. Timothy M. Chan. A (slightly) faster algorithm for Klee’s measure problem. Comput. Geom., 43(3):243-250, 2010. Preliminary version in SoCG'08. URL:
  11. Timothy M. Chan. Klee’s measure problem made easy. In Proc. 54th IEEE Symposium on Foundations of Computer Science (FOCS), pages 410-419, 2013. URL:
  12. Timothy M. Chan. (Near-)linear-time randomized algorithms for row minima in Monge partial matrices and related problems. In Proc. 32nd ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1465-1482, 2021. URL:
  13. Timothy M. Chan, John Hershberger, and Simon Pratt. Two approaches to building time-windowed geometric data structures. Algorithmica, 81(9):3519-3533, 2019. URL:
  14. Timothy M. Chan and Patrick Lee. On constant factors in comparison-based geometric algorithms and data structures. Discret. Comput. Geom., 53(3):489-513, 2015. URL:
  15. Bernard Chazelle, Robert L. (Scot) Drysdale III, and D. T. Lee. Computing the largest empty rectangle. SIAM J. Comput., 15(1):300-315, 1986. URL:
  16. Karen L. Daniels, Victor J. Milenkovic, and Dan Roth. Finding the largest area axis-parallel rectangle in a polygon. Comput. Geom., 7:125-148, 1997. URL:
  17. Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational Geometry: Algorithms and Applications. Springer, 3rd edition, 2008. URL:
  18. Adrian Dumitrescu and Minghui Jiang. On the largest empty axis-parallel box amidst n points. Algorithmica, 66(2):225-248, 2013. URL:
  19. Adrian Dumitrescu and Minghui Jiang. Perfect vector sets, properly overlapping partitions, and largest empty box. CoRR, abs/1608.06874, 2016. URL:
  20. Adrian Dumitrescu and Minghui Jiang. On the number of maximum empty boxes amidst n points. Discret. Comput. Geom., 59(3):742-756, 2018. Preliminary version in SoCG'16. URL:
  21. Adrian Dumitrescu, Joseph S. B. Mitchell, and Micha Sharir. Binary space partitions for axis-parallel segments, rectangles, and hyperrectangles. Discret. Comput. Geom., 31(2):207-227, 2004. URL:
  22. Harold N. Gabow, Jon Louis Bentley, and Robert Endre Tarjan. Scaling and related techniques for geometry problems. In Proc. 16th ACM Symposium on Theory of Computing (STOC), pages 135-143, 1984. URL:
  23. Panos Giannopoulos, Christian Knauer, Magnus Wahlström, and Daniel Werner. Hardness of discrepancy computation and epsilon-net verification in high dimension. J. Complex., 28(2):162-176, 2012. URL:
  24. Maria M. Klawe and Daniel J. Kleitman. An almost linear time algorithm for generalized matrix searching. SIAM J. Discret. Math., 3(1):81-97, 1990. URL:
  25. Lawrence L. Larmore. An optimal algorithm with unknown time complexity for convex matrix searching. Inf. Process. Lett., 36(3):147-151, 1990. URL:
  26. Amnon Naamad, D. T. Lee, and Wen-Lian Hsu. On the maximum empty rectangle problem. Discret. Appl. Math., 8(3):267-277, 1984. URL:
  27. Mark H. Overmars and Chee-Keng Yap. New upper bounds in Klee’s measure problem. SIAM J. Comput., 20(6):1034-1045, 1991. URL:
  28. Seth Pettie and Vijaya Ramachandran. An optimal minimum spanning tree algorithm. J. ACM, 49(1):16-34, 2002. URL:
  29. Micha Sharir and Pankaj K. Agarwal. Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, 1995. Google Scholar
  30. Mario Ullrich and Jan Vybíral. An upper bound on the minimal dispersion. J. Complex., 45:120-126, 2018. URL:
  31. Jean Vuillemin. A unifying look at data structures. Commun. ACM, 23(4):229-239, 1980. URL: