Computing Planar Convex Hulls with a Promise
Sepideh Aghamolaei, Kevin Buchin, Timothy M. Chan, Jacobus Conradi, Ivor Van der Hoog, Vahideh Keikha, Jeff M. Phillips, Benjamin Raichel

TL;DR
This paper presents a sub-$O(n \,\log n)$ time algorithm for computing convex hulls under a specific promise that the hull points are sorted, and proves this promise is tight for faster algorithms.
Contribution
It introduces a deterministic $O(n \,\sqrt{\log n})$-time algorithm and an expected $O(n \log^{\varepsilon} n)$ randomized algorithm for convex hulls under a new promise, and shows this promise is tight.
Findings
Deterministic $O(n \sqrt{\log n})$-time convex hull algorithm under the promise.
Expected $O(n \log^{\varepsilon} n)$ randomized algorithm with the same promise.
Breaking the promise even slightly leads to an $\\Omega(n \log n)$ lower bound.
Abstract
Computing the convex hull of a planar -point set is one of the most fundamental problems in computational geometry. It has an lower bound in the algebraic computation tree model, and many convex hull algorithms match this bound. Classical results show that, under special input assumptions, sub- algorithms are possible. For instance, when the points are given in lexicographic or angular order, the convex hull can be computed in linear time. Even under the weaker assumption that the sequence of points corresponds to the ordered vertices of a simple polygonal chain, linear-time algorithms exist. This naturally raises the question: can the convex hull of a point set be computed in sub- time under weaker input assumptions? We answer this positively. Under the promise that the input sequence contains the convex hull as a subsequence, we…
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