Tight Universal Bounds for Partially Presorted Pareto Front and Convex Hull

2025-12-06

AI Summary

Eppstein et al.’s algorithms for computing the Pareto frontier and planar convex hull exhibit empirically adaptive behavior, yet their theoretical optimality—particularly with respect to input structure—remained uncharacterized. Method: This paper establishes the first matching lower bound on comparison complexity for these algorithms and identifies their core mechanism as a variant of quicksort—not a stack-based merge strategy like Timsort. It introduces the formal notion of *universal optimality*, defining optimal comparison complexity as $O(n(1 + H(I)))$, where $H(I)$ is the range-partitioning entropy quantifying input structure. Using information-theoretic and decision-tree analysis, the paper proves that Eppstein et al.’s algorithms achieve this tight bound. Contribution/Results: This work provides the first structural, input-adaptive optimality theory for both the Pareto frontier and planar convex hull problems. Moreover, it establishes a new paradigm for designing universally optimal algorithms—those asymptotically optimal across all input structures—especially for partially ordered or structured inputs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Establishes lower bounds for Pareto front and convex hull algorithms
• Demonstrates algorithms correspond to QuickSort variant, not TimSort
• Proves universal optimality of existing algorithms under new notion

Innovation

• Adapts TimSort-inspired algorithm to Pareto front and convex hull
• Introduces Range Partition Entropy for adaptive runtime analysis
• Proves universal optimality via QuickSort variant lower bounds