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DOI: 10.4230/LIPIcs.ISAAC.2025.32

Time-Optimal k-Server

Authors: Fabian Frei, Dennis Komm, Moritz Stocker, and Philip Whittington

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

The time-optimal k-server problem minimizes the time spent instead of the distance traveled when serving n requests, appearing one after the other, with k servers in a metric space. The classical distance model was motivated by a hard disk with k heads. Instead of minimal head movements, the time model aims for optimal reading speeds. This paper provides a lower bound of 2k-1 on the competitive ratio of any deterministic online algorithm for the time-optimal k-server problem on a specifically designed metric space. This lower bound coincides with the best known upper bound on the competitive ratio for the classical k-server problem, achieved by the famous work function algorithm. We provide further lower bounds of k+1 for all Euclidean spaces and k for uniform metric spaces. Our most technical result, proven by applying Yao’s principle to a suitable instance distribution, is a lower bound of k+H_k-1 that holds even for randomized algorithms, which contrasts with the best known lower bound for the classical problem, which is polylogarithmic in k. We hope to initiate further intensive study of this natural problem.

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Fabian Frei, Dennis Komm, Moritz Stocker, and Philip Whittington. Time-Optimal k-Server. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{frei_et_al:LIPIcs.ISAAC.2025.32,
  author =	{Frei, Fabian and Komm, Dennis and Stocker, Moritz and Whittington, Philip},
  title =	{{Time-Optimal k-Server}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{32:1--32:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.32},
  URN =		{urn:nbn:de:0030-drops-249407},
  doi =		{10.4230/LIPIcs.ISAAC.2025.32},
  annote =	{Keywords: k-server problem, optimizing time instead of distance, deterministic and randomized algorithms, Yao’s principle}
}

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DOI: 10.4230/LIPIcs.STACS.2024.35

The 2-Attractor Problem Is NP-Complete

Authors: Janosch Fuchs and Philip Whittington

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

Abstract

A k-attractor is a combinatorial object unifying dictionary-based compression. It allows to compare the repetitiveness measures of different dictionary compressors such as Lempel-Ziv 77, the Burrows-Wheeler transform, straight line programs and macro schemes. For a string T ∈ Σⁿ, the k-attractor is defined as a set of positions Γ ⊆ [1,n], such that every distinct substring of length at most k is covered by at least one of the selected positions. Thus, if a substring occurs multiple times in T, one position suffices to cover it. A 1-attractor is easily computed in linear time, while Kempa and Prezza [STOC 2018] have shown that for k ≥ 3, it is NP-complete to compute the smallest k-attractor by a reduction from k-set cover. The main result of this paper answers the open question for the complexity of the 2-attractor problem, showing that the problem remains NP-complete. Kempa and Prezza’s proof for k ≥ 3 also reduces the 2-attractor problem to the 2-set cover problem, which is equivalent to edge cover, but that does not fully capture the complexity of the 2-attractor problem. For this reason, we extend edge cover by a color function on the edges, yielding the colorful edge cover problem. Any edge cover must then satisfy the additional constraint that each color is represented. This extension raises the complexity such that colorful edge cover becomes NP-complete while also more precisely modeling the 2-attractor problem. We obtain a reduction showing k-attractor to be NP-complete and APX-hard for any k ≥ 2.

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Janosch Fuchs and Philip Whittington. The 2-Attractor Problem Is NP-Complete. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 35:1-35:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{fuchs_et_al:LIPIcs.STACS.2024.35,
  author =	{Fuchs, Janosch and Whittington, Philip},
  title =	{{The 2-Attractor Problem Is NP-Complete}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{35:1--35:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.35},
  URN =		{urn:nbn:de:0030-drops-197457},
  doi =		{10.4230/LIPIcs.STACS.2024.35},
  annote =	{Keywords: String attractors, dictionary compression, computational complexity}
}

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Time-Optimal k-Server

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The 2-Attractor Problem Is NP-Complete

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