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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2025.86

Optimal Distance Labeling for Permutation Graphs

Authors: Paweł Gawrychowski and Wojciech Janczewski

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract

A permutation graph is the intersection graph of a set of segments between two parallel lines. In other words, they are defined by a permutation π on n elements, such that u and v are adjacent if an only if u < v but π(u) > π(v). We consider the problem of computing the distances in such a graph in the setting of informative labeling schemes. The goal of such a scheme is to assign a short bitstring 𝓁(u) to every vertex u, such that the distance between u and v can be computed using only 𝓁(u) and 𝓁(v), and no further knowledge about the whole graph (other than that it is a permutation graph). This elegantly captures the intuition that we would like our data structure to be distributed, and often leads to interesting combinatorial challenges while trying to obtain lower and upper bounds that match up to the lower-order terms. For distance labeling of permutation graphs on n vertices, Katz, Katz, and Peleg [STACS 2000] showed how to construct labels consisting of 𝒪(log² n) bits. Later, Bazzaro and Gavoille [Discret. Math. 309(11)] obtained an asymptotically optimal bound by showing how to construct labels consisting of 9log{n}+𝒪(1) bits, and proving that 3log{n}-𝒪(log{log{n}}) bits are necessary. This however leaves a quite large gap between the known lower and upper bounds. We close this gap by showing how to construct labels consisting of 3log{n}+𝒪(1) bits.

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Paweł Gawrychowski and Wojciech Janczewski. Optimal Distance Labeling for Permutation Graphs. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 86:1-86:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{gawrychowski_et_al:LIPIcs.ICALP.2025.86,
  author =	{Gawrychowski, Pawe{\l} and Janczewski, Wojciech},
  title =	{{Optimal Distance Labeling for Permutation Graphs}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{86:1--86:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.86},
  URN =		{urn:nbn:de:0030-drops-234632},
  doi =		{10.4230/LIPIcs.ICALP.2025.86},
  annote =	{Keywords: informative labeling, permutation graph, distance labeling}
}

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DOI: 10.4230/LIPIcs.ISAAC.2020.27

Efficient Labeling for Reachability in Directed Acyclic Graphs

Authors: Maciej Dulęba, Paweł Gawrychowski, and Wojciech Janczewski

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Abstract

We consider labeling nodes of a directed graph for reachability queries. A reachability labeling scheme for such a graph assigns a binary string, called a label, to each node. Then, given the labels of nodes u and v and no other information about the underlying graph, it should be possible to determine whether there exists a directed path from u to v. By a simple information theoretical argument and invoking the bound on the number of partial orders, in any scheme some labels need to consist of at least n/4 bits, where n is the number of nodes. On the other hand, it is not hard to design a scheme with labels consisting of n/2+𝒪(log n) bits. In the classical centralised setting, where a single data structure is stored as a whole, Munro and Nicholson designed a structure for reachability queries consisting of n²/4+o(n²) bits (which is optimal, up to the lower order term). We extend their approach to obtain a scheme with labels consisting of n/3+o(n) bits.

Cite as

Maciej Dulęba, Paweł Gawrychowski, and Wojciech Janczewski. Efficient Labeling for Reachability in Directed Acyclic Graphs. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{duleba_et_al:LIPIcs.ISAAC.2020.27,
  author =	{Dul\k{e}ba, Maciej and Gawrychowski, Pawe{\l} and Janczewski, Wojciech},
  title =	{{Efficient Labeling for Reachability in Directed Acyclic Graphs}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{27:1--27:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.27},
  URN =		{urn:nbn:de:0030-drops-133710},
  doi =		{10.4230/LIPIcs.ISAAC.2020.27},
  annote =	{Keywords: informative labeling scheme, reachability, DAG}
}

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Documents authored by Janczewski, Wojciech

Optimal Distance Labeling for Permutation Graphs

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Efficient Labeling for Reachability in Directed Acyclic Graphs

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