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Tight Universal Bounds for Partially Presorted Pareto Front and Convex Hull

Ivor van der Hoog, Eva Rotenberg, Daniel Rutschmann

TL;DR

<3-5 sentence high-level summary> This paper advances the understanding of adaptive, input-sensitive sorting-related problems by formalizing universal-optimality via range-partition entropies for Pareto front and planar convex hull computations. It shows that the Eppstein–Goodrich–Illickan–To algorithms operate as a deterministic QuickSort variant with early termination, and it provides matching universal lower bounds under refined universes built from region-based partitions and downdraft orderings. The work develops a detailed counting framework that injects witness-structures into front- and hull-defining outputs, proving universal optimality for these problems under the proposed models. It also generalizes the notion of sorting regions through generalized compass functions and discusses implications for future work on higher-dimensional analogs and other geometric problems.

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.

Tight Universal Bounds for Partially Presorted Pareto Front and Convex Hull

TL;DR

<3-5 sentence high-level summary> This paper advances the understanding of adaptive, input-sensitive sorting-related problems by formalizing universal-optimality via range-partition entropies for Pareto front and planar convex hull computations. It shows that the Eppstein–Goodrich–Illickan–To algorithms operate as a deterministic QuickSort variant with early termination, and it provides matching universal lower bounds under refined universes built from region-based partitions and downdraft orderings. The work develops a detailed counting framework that injects witness-structures into front- and hull-defining outputs, proving universal optimality for these problems under the proposed models. It also generalizes the notion of sorting regions through generalized compass functions and discusses implications for future work on higher-dimensional analogs and other geometric problems.

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 mapping lists that store points to a number between and . Their algorithms have, for each list of points , a running time of . 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.

Paper Structure

This paper contains 34 sections, 19 theorems, 20 equations, 5 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Entropy.
  3. Universal optimality.
  4. Contribution.
  5. Entropy and Universal Optimality
  6. Universal lower bounds.
  7. An entropy, based only on the input values.
  8. TimSort and QuickSort.
  9. Defining our problems, entropies and universes
  10. QuickSort.
  11. Pareto front.
  12. Convex hull.
  13. Sorting
  14. Pareto Front
  15. The Output.
  16. ...and 19 more sections

Key Result

Theorem 1

The sorting algorithm $\mathcal{A}$ induced by eppstein_entropy-bounded_2025 runs in $O(n(1+H(I)))$ time and $\exists c > 0$ such that $\forall (s_1,\ldots,s_k)$, $\texttt{Universal}(\mathcal{A},(s_1,\ldots,s_k)) \;\le\; c\cdot \texttt{Universal-LB}(s_1,\ldots,s_k)$.

Figures (5)

  • Figure 1: The final step in the proof of \ref{['lemm:pareto_linear_order']}.
  • Figure 2: (a) A point set $P$ with its quadrant tree. (b) We color each $p \in P$ by $u(p)$. (c) For $R=\emptyset$, the normalized downdraft $\psi$ maps every $p \in P - \mathop{\mathrm{pf}}\nolimits(P)$ to a point that dominates it.
  • Figure 3: Cutting a region $r$ into three pieces.
  • Figure 4: Top: the commuting diagram. Bottom: The diagram chasing, starting from $S$.
  • Figure 5: Top: the commuting diagram. Bottom: The diagram chasing, starting from $S$.

Theorems & Definitions (26)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Definition 4
  • Theorem 5
  • Definition 6
  • Lemma 7
  • Definition 8
  • Lemma 9
  • Lemma 10
  • ...and 16 more