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

2026-05-10

AI Summary

This study addresses whether deterministic output-sensitive algorithms can simultaneously achieve optimal time and I/O complexity for the planar maximal point set and convex hull problems. By analyzing computational lower bounds within the cache-oblivious model, we establish—for the first time—that no deterministic output-sensitive algorithm can attain both time and I/O optimality for these problems. Building on this impossibility result, we derive a tight lower bound characterizing the inherent time–I/O trade-off for deterministic algorithms and present a simple deterministic algorithm that matches this bound. Furthermore, we design a randomized algorithm that achieves worst-case optimal time complexity while attaining optimal I/O performance in expectation.

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.

Problem

• planar maxima
• convex hull
• output-sensitive algorithm
• time complexity
• I/O complexity

Innovation

• output-sensitive algorithms
• I/O complexity
• time-I/O trade-off
• convex hull
• cache-oblivious algorithms