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Collective Tree Exploration via Potential Function Method

Authors Romain Cosson , Laurent Massoulié



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Author Details

Romain Cosson
  • Inria, Paris, France
Laurent Massoulié
  • Inria, Paris, France

Acknowledgements

The authors thank Laurent Viennot for stimulating discussions, the Argo team at Inria, as well as the anonymous reviewers for their suggestions.

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.ITCS.2024.35

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).

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
  • Mathematics of computing → Graph algorithms
Keywords
  • collective exploration
  • online algorithms
  • evolving tree
  • competitive analysis

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References

  1. Allan Borodin, Nathan Linial, and Michael E Saks. An optimal on-line algorithm for metrical task system. Journal of the ACM (JACM), 39(4):745-763, 1992. Google Scholar
  2. Peter Brass, Flavio Cabrera-Mora, Andrea Gasparri, and Jizhong Xiao. Multirobot tree and graph exploration. IEEE Trans. Robotics, 27(4):707-717, 2011. URL: https://doi.org/10.1109/TRO.2011.2121170.
  3. Sébastien Bubeck, Christian Coester, and Yuval Rabani. Shortest paths without a map, but with an entropic regularizer. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 1102-1113. IEEE, 2022. Google Scholar
  4. William R Burley. Traversing layered graphs using the work function algorithm. Journal of Algorithms, 20(3):479-511, 1996. Google Scholar
  5. Marek Chrobak and Lawrence L Larmore. The server problem and on-line games. On-line algorithms, 7:11-64, 1991. Google Scholar
  6. Romain Cosson. Breaking the k/log k barrier in collective tree exploration with tree-mining. In arXiv preprint arXiv:2309.07011, 2023. URL: https://arxiv.org/abs/2309.07011.
  7. Romain Cosson, Laurent Massoulié, and Laurent Viennot. Breadth-first depth-next: Optimal collaborative exploration of trees with low diameter. In 37th International Symposium on Distributed Computing, 2023. Google Scholar
  8. Claude Dellacherie, Servet Martinez, Jaime San Martin, et al. Inverse M-matrices and ultrametric matrices, volume 2118. Springer, 2014. Google Scholar
  9. Dariusz Dereniowski, Yann Disser, Adrian Kosowski, Dominik Pajak, and Przemyslaw Uznanski. Fast collaborative graph exploration. In Fedor V. Fomin, Rusins Freivalds, Marta Z. Kwiatkowska, and David Peleg, editors, Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part II, volume 7966 of Lecture Notes in Computer Science, pages 520-532. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-39212-2_46.
  10. Miroslaw Dynia, Miroslaw Korzeniowski, and Christian Schindelhauer. Power-aware collective tree exploration. In International Conference on Architecture of Computing Systems, pages 341-351. Springer, 2006. Google Scholar
  11. Miroslaw Dynia, Jakub Lopuszanski, and Christian Schindelhauer. Why robots need maps. In Giuseppe Prencipe and Shmuel Zaks, editors, Structural Information and Communication Complexity, 14th International Colloquium, SIROCCO 2007, Castiglioncello, Italy, June 5-8, 2007, Proceedings, volume 4474 of Lecture Notes in Computer Science, pages 41-50. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-72951-8_5.
  12. Amos Fiat, Dean P Foster, Howard Karloff, Yuval Rabani, Yiftach Ravid, and Sundar Vishwanathan. Competitive algorithms for layered graph traversal. SIAM Journal on Computing, 28(2):447-462, 1998. Google Scholar
  13. Pierre Fraigniaud, Leszek Gasieniec, Dariusz R Kowalski, and Andrzej Pelc. Collective tree exploration. In LATIN 2004: Theoretical Informatics: 6th Latin American Symposium, Buenos Aires, Argentina, April 5-8, 2004. Proceedings 6, pages 141-151. Springer, 2004. Google Scholar
  14. Pierre Fraigniaud, Leszek Gasieniec, Dariusz R. Kowalski, and Andrzej Pelc. Collective tree exploration. Networks, 48(3):166-177, 2006. URL: https://doi.org/10.1002/net.20127.
  15. Christian Ortolf and Christian Schindelhauer. A recursive approach to multi-robot exploration of trees. In International Colloquium on Structural Information and Communication Complexity, pages 343-354. Springer, 2014. Google Scholar
  16. Christos H Papadimitriou and Mihalis Yannakakis. Shortest paths without a map. Theoretical Computer Science, 84(1):127-150, 1991. Google Scholar
  17. H Ramesh. On traversing layered graphs on-line. In SODA, pages 412-421, 1993. Google Scholar
  18. Bernd Sturmfels, Caroline Uhler, and Piotr Zwiernik. Brownian motion tree models are toric. arXiv preprint, 2019. URL: https://arxiv.org/abs/1902.09905.
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