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Online Sorting and Online TSP: Randomized, Stochastic, and High-Dimensional

Authors Mikkel Abrahamsen , Ioana O. Bercea , Lorenzo Beretta , Jonas Klausen , László Kozma

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Author Details

Mikkel Abrahamsen
  • University of Copenhagen, Denmark
Ioana O. Bercea
  • KTH Royal Institute of Technology, Stockholm, Sweden
Lorenzo Beretta
  • University of California, Santa Cruz, CA, USA
Jonas Klausen
  • University of Copenhagen, Denmark
László Kozma
  • Institut für Informatik, Freie Universität Berlin, Germany

Funding

Ioana O. Bercea, Lorenzo Beretta and Jonas Klausen: Supported by grant 16582, Basic Algorithms Research Copenhagen (BARC), from the VILLUM Foundation.
  • Abrahamsen, Mikkel: Supported by Starting Grant 1054-00032B from the Independent Research Fund Denmark under the Sapere Aude research career programme and part of Basic Algorithms Research Copenhagen (BARC), supported by the VILLUM Foundation grant 16582.
  • Kozma, László: Supported by DFG grant KO 6140/1-2.

Cite AsGet BibTex

Mikkel Abrahamsen, Ioana O. Bercea, Lorenzo Beretta, Jonas Klausen, and László Kozma. Online Sorting and Online TSP: Randomized, Stochastic, and High-Dimensional. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.5

Abstract

In the online sorting problem, n items are revealed one by one and have to be placed (immediately and irrevocably) into empty cells of a size-n array. The goal is to minimize the sum of absolute differences between items in consecutive cells. This natural problem was recently introduced by Aamand, Abrahamsen, Beretta, and Kleist (SODA 2023) as a tool in their study of online geometric packing problems. They showed that when the items are reals from the interval [0,1] a competitive ratio of O(√n) is achievable, and no deterministic algorithm can improve this ratio asymptotically. In this paper, we extend and generalize the study of online sorting in three directions: - randomized: we settle the open question of Aamand et al. by showing that the O(√n) competitive ratio for the online sorting of reals cannot be improved even with the use of randomness; - stochastic: we consider inputs consisting of n samples drawn uniformly at random from an interval, and give an algorithm with an improved competitive ratio of Õ(n^{1/4}). The result reveals connections between online sorting and the design of efficient hash tables; - high-dimensional: we show that Õ(√n)-competitive online sorting is possible even for items from ℝ^d, for arbitrary fixed d, in an adversarial model. This can be viewed as an online variant of the classical TSP problem where tasks (cities to visit) are revealed one by one and the salesperson assigns each task (immediately and irrevocably) to its timeslot. Along the way, we also show a tight O(log n)-competitiveness result for uniform metrics, i.e., where items are of different types and the goal is to order them so as to minimize the number of switches between consecutive items of different types.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • sorting
  • online algorithm
  • TSP

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