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The Impossibility of Simultaneous Time and I/O Optimality for the Planar Maxima and Convex Hull Problems

Authors Peyman Afshani , Gerth Stølting Brodal , Nodari Sitchinava

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LIPIcs.ICALP.2026.5.pdf
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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • External Memory model
  • cache-oblivious algorithms
  • lower bounds

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Abstract

We prove that no deterministic output-sensitive algorithm for the planar convex hull and maxima problems can obtain both optimal time and I/O complexity, where the optimality is defined with respect to both the input and output sizes. This explains why the best previous algorithms achieved an optimal I/O bound at the cost of sub-optimal running time (Goodrich et al. [FOCS, 1993]). To the best of our knowledge, the impossibility of simultaneous optimality was only shown previously for the permutation problem by Brodal and Fagerberg [STOC, 2003]. Our results imply that no optimal deterministic output-sensitive cache-oblivious algorithm exists for either problem. In addition, we present simple deterministic algorithms that match our lower bounds and that provide a trade-off between time and I/Os. On the other hand, a simple modification of our deterministic algorithm results in a randomized algorithm that simultaneously achieves optimal (worst-case) time and optimal expected I/O bounds.

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Peyman Afshani, Gerth Stølting Brodal, and Nodari Sitchinava. The Impossibility of Simultaneous Time and I/O Optimality for the Planar Maxima and Convex Hull Problems. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 5:1-5:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.5

Author Details

Peyman Afshani
  • Aarhus University, Denmark
Gerth Stølting Brodal
  • Aarhus University, Denmark
Nodari Sitchinava
  • University of Hawai'i at Mānoa, Honolulu, HI, USA

Funding

  • Afshani, Peyman: Supported by DFF (Danmarks Frie Forskningsfond) of Danish Council for Independent Research under grant ID 10.46540/3103-00334B.
  • Brodal, Gerth Stølting: Supported by Independent Research Fund Denmark grant 9131-00113B.
  • Sitchinava, Nodari: Supported by NSF grant 2432018.

Acknowledgements

The authors want to thank Ronitt Rubinfeld for suggesting this topic.

References

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