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

TL;DR

This paper proves the impossibility of achieving both optimal time and I/O complexity simultaneously for planar convex hull and maxima problems, and presents algorithms that approach these bounds.

Contribution

It establishes a theoretical lower bound for deterministic algorithms and introduces algorithms that balance or achieve optimality in time and I/O complexity.

Findings

No deterministic output-sensitive algorithm can be both time and I/O optimal for these problems.

Simple deterministic algorithms match the lower bounds and offer a trade-off between time and I/O.

A randomized algorithm can achieve both optimal worst-case time and expected I/O bounds.

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.