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DOI: 10.4230/LIPIcs.ITCS.2024.35

Collective Tree Exploration via Potential Function Method

Authors: Romain Cosson and Laurent Massoulié

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract

We study the problem of collective tree exploration (CTE) in which a team of k agents is tasked to traverse all the edges of an unknown tree as fast as possible, assuming complete communication between the agents [FGKP06]. In this paper, we present an algorithm performing collective tree exploration in 2n/k+𝒪(kD) rounds, where n is the number of nodes in the tree, and D is the tree depth. This leads to a competitive ratio of 𝒪(√k), the first polynomial improvement over the 𝒪(k) ratio of depth-first search. Our analysis holds for an asynchronous generalization of collective tree exploration. It relies on a game with robots at the leaves of a continuously growing tree extending the "tree-mining game" of [C23] and resembling the "evolving tree game" of [BCR22]. Another surprising consequence of our results is the existence of algorithms {𝒜_k}_{k ∈ ℕ} for layered tree traversal (LTT) with cost at most 2L/k+𝒪(kD), where L is the sum of all edge lengths. For the case of layered trees of width w and unit edge lengths, our guarantee is thus in 𝒪(√wD).

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Romain Cosson and Laurent Massoulié. Collective Tree Exploration via Potential Function Method. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 35:1-35:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cosson_et_al:LIPIcs.ITCS.2024.35,
  author =	{Cosson, Romain and Massouli\'{e}, Laurent},
  title =	{{Collective Tree Exploration via Potential Function Method}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{35:1--35:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.35},
  URN =		{urn:nbn:de:0030-drops-195638},
  doi =		{10.4230/LIPIcs.ITCS.2024.35},
  annote =	{Keywords: collective exploration, online algorithms, evolving tree, competitive analysis}
}

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DOI: 10.4230/LIPIcs.DISC.2023.14

Efficient Collaborative Tree Exploration with Breadth-First Depth-Next

Authors: Romain Cosson, Laurent Massoulié, and Laurent Viennot

Published in: LIPIcs, Volume 281, 37th International Symposium on Distributed Computing (DISC 2023)

Abstract

We study the problem of collaborative tree exploration introduced by Fraigniaud, Gasieniec, Kowalski, and Pelc [Pierre Fraigniaud et al., 2006] where a team of k agents is tasked to collectively go through all the edges of an unknown tree as fast as possible and return to the root. Denoting by n the total number of nodes and by D the tree depth, the 𝒪(n/log(k)+D) algorithm of [Pierre Fraigniaud et al., 2006] achieves a 𝒪(k/log(k)) competitive ratio with respect to the cost of offline exploration which is at least max{{2n/k,2D}}. Brass, Cabrera-Mora, Gasparri, and Xiao [Peter Brass et al., 2011] study an alternative performance criterion, the competitive overhead with respect to the cost of offline exploration, with their 2n/k+𝒪((D+k)^k) guarantee. In this paper, we introduce "Breadth-First Depth-Next" (BFDN), a novel and simple algorithm that performs collaborative tree exploration in 2n/k+𝒪(D²log(k)) rounds, thus outperforming [Peter Brass et al., 2011] for all values of (n,D,k) and being order-optimal for trees of depth D = o(√n). Our analysis relies on a two-player game reflecting a problem of online resource allocation that could be of independent interest. We extend the guarantees of BFDN to: scenarios with limited memory and communication, adversarial setups where robots can be blocked, and exploration of classes of non-tree graphs. Finally, we provide a recursive version of BFDN with a runtime of 𝒪_𝓁(n/k^{1/𝓁}+log(k) D^{1+1/𝓁}) for parameter 𝓁 ≥ 1, thereby improving performance for trees with large depth.

Cite as

Romain Cosson, Laurent Massoulié, and Laurent Viennot. Efficient Collaborative Tree Exploration with Breadth-First Depth-Next. In 37th International Symposium on Distributed Computing (DISC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 281, pp. 14:1-14:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cosson_et_al:LIPIcs.DISC.2023.14,
  author =	{Cosson, Romain and Massouli\'{e}, Laurent and Viennot, Laurent},
  title =	{{Efficient Collaborative Tree Exploration with Breadth-First Depth-Next}},
  booktitle =	{37th International Symposium on Distributed Computing (DISC 2023)},
  pages =	{14:1--14:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-301-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{281},
  editor =	{Oshman, Rotem},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2023.14},
  URN =		{urn:nbn:de:0030-drops-191409},
  doi =		{10.4230/LIPIcs.DISC.2023.14},
  annote =	{Keywords: collaborative exploration, online algorithms, trees, adversarial game, competitive analysis, robot swarms}
}

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Collective Tree Exploration via Potential Function Method

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Efficient Collaborative Tree Exploration with Breadth-First Depth-Next

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