Tight Universal Bounds for Partially Presorted Pareto Front and Convex Hull

TL;DR

This paper establishes tight lower bounds for algorithms computing Pareto fronts and convex hulls, demonstrating that existing algorithms are universally optimal and revealing their connection to QuickSort rather than TimSort.

Contribution

The paper provides matching lower bounds for these algorithms and introduces a new perspective linking them to QuickSort, establishing their universal optimality.

Findings

Algorithms are universally optimal under the new lower bounds.

Existing algorithms are based on QuickSort, not TimSort.

Matching lower bounds confirm the optimality of current methods.

Abstract

TimSort is a well-established sorting algorithm whose running time depends on how sorted the input already is. Recently, Eppstein, Goodrich, Illickan, and To designed algorithms inspired by TimSort for Pareto front, planar convex hull, and two other problems. For each of these problems, they define a Range Partition Entropy; a function $H$ mapping lists $I$ that store $n$ points to a number between $0$ and $\log n$. Their algorithms have, for each list of points $I$, a running time of $O(n(1 + H(I)))$. In this paper, we provide matching lower bounds for the Pareto front and convex hull algorithms by Eppstein, Goodrich, Illickan, and To. In particular, we show that their algorithm does not correspond to TimSort (or related stack-based MergeSort variants) but rather to a variant of QuickSort. From this, we derive an intuitive notion of universal optimality. We show comparison-based lower bounds that prove that the algorithms by Eppstein, Goodrich, Illickan and To are universally optimal under this notion of universal optimality.