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DOI: 10.4230/LIPIcs.ESA.2024.5

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

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

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

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.

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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)

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@InProceedings{abrahamsen_et_al:LIPIcs.ESA.2024.5,
  author =	{Abrahamsen, Mikkel and Bercea, Ioana O. and Beretta, Lorenzo and Klausen, Jonas and Kozma, L\'{a}szl\'{o}},
  title =	{{Online Sorting and Online TSP: Randomized, Stochastic, and High-Dimensional}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{5:1--5:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.5},
  URN =		{urn:nbn:de:0030-drops-210766},
  doi =		{10.4230/LIPIcs.ESA.2024.5},
  annote =	{Keywords: sorting, online algorithm, TSP}
}

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DOI: 10.4230/LIPIcs.SoCG.2021.6

Online Packing to Minimize Area or Perimeter

Authors: Mikkel Abrahamsen and Lorenzo Beretta

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract

We consider online packing problems where we get a stream of axis-parallel rectangles. The rectangles have to be placed in the plane without overlapping, and each rectangle must be placed without knowing the subsequent rectangles. The goal is to minimize the perimeter or the area of the axis-parallel bounding box of the rectangles. We either allow rotations by 90^∘ or translations only. For the perimeter version we give algorithms with an absolute competitive ratio slightly less than 4 when only translations are allowed and when rotations are also allowed. We then turn our attention to minimizing the area and show that the competitive ratio of any algorithm is at least Ω(√n), where n is the number of rectangles in the stream, and this holds with and without rotations. We then present algorithms that match this bound in both cases and the competitive ratio is thus optimal to within a constant factor. We also show that the competitive ratio cannot be bounded as a function of Opt. We then consider two special cases. The first is when all the given rectangles have aspect ratios bounded by some constant. The particular variant where all the rectangles are squares and we want to minimize the area of the bounding square has been studied before and an algorithm with a competitive ratio of 8 has been given [Fekete and Hoffmann, Algorithmica, 2017]. We improve the analysis of the algorithm and show that the ratio is at most 6, which is tight. The second special case is when all edges have length at least 1. Here, the Ω(√n) lower bound still holds, and we turn our attention to lower bounds depending on Opt. We show that any algorithm for the translational case has a competitive ratio of at least Ω(√{Opt}). If rotations are allowed, we show a lower bound of Ω(∜{Opt}). For both versions, we give algorithms that match the respective lower bounds: With translations only, this is just the algorithm from the general case with competitive ratio O(√n) = O(√{Opt}). If rotations are allowed, we give an algorithm with competitive ratio O(min{√n,∜{Opt}}), thus matching both lower bounds simultaneously.

Cite as

Mikkel Abrahamsen and Lorenzo Beretta. Online Packing to Minimize Area or Perimeter. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 6:1-6:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2021.6,
  author =	{Abrahamsen, Mikkel and Beretta, Lorenzo},
  title =	{{Online Packing to Minimize Area or Perimeter}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{6:1--6:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.6},
  URN =		{urn:nbn:de:0030-drops-138054},
  doi =		{10.4230/LIPIcs.SoCG.2021.6},
  annote =	{Keywords: Packing, online algorithms}
}

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Documents authored by Beretta, Lorenzo

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

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Online Packing to Minimize Area or Perimeter

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