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Quasi-Linear Distance Query Reconstruction for Graphs of Bounded Treelength

Authors Paul Bastide , Carla Groenland

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Distance Reconstruction
  • Randomized Algorithm
  • Treelength

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Abstract

In distance query reconstruction, we wish to reconstruct the edge set of a hidden graph by asking as few distance queries as possible to an oracle. Given two vertices u and v, the oracle returns the shortest path distance between u and v in the graph.
The length of a tree decomposition is the maximum distance between two vertices contained in the same bag. The treelength of a graph is defined as the minimum length of a tree decomposition of this graph. We present an algorithm to reconstruct an n-vertex connected graph G parameterized by maximum degree Δ and treelength k in O_{k,Δ}(n log² n) queries (in expectation). This is the first algorithm to achieve quasi-linear complexity for this class of graphs. The proof goes through a new lemma that could give independent insight on graphs of bounded treelength.

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Paul Bastide and Carla Groenland. Quasi-Linear Distance Query Reconstruction for Graphs of Bounded Treelength. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 20:1-20:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.IPEC.2024.20

Author Details

Paul Bastide
  • LaBRI - Université de Bordeaux, France
  • TU Delft, The Netherlands
Carla Groenland
  • TU Delft, The Netherlands

References

  1. Paul Bastide and Carla Groenland. Optimal distance query reconstruction for graphs without long induced cycles. arXiv preprint, 2023. URL: https://doi.org/10.48550/arXiv.2306.05979.
  2. Zuzana Beerliova, Felix Eberhard, Thomas Erlebach, Alexander Hall, Michael Hoffmann, Mat Mihal'ak, and L Shankar Ram. Network discovery and verification. IEEE Journal on selected areas in communications, 24(12):2168-2181, 2006. URL: https://doi.org/10.1109/JSAC.2006.884015.
  3. Rémy Belmonte, Fedor V Fomin, Petr A Golovach, and MS Ramanujan. Metric dimension of bounded tree-length graphs. SIAM Journal on Discrete Mathematics, 31(2):1217-1243, 2017. URL: https://doi.org/10.1137/16M1057383.
  4. Eli Berger and Paul Seymour. Bounded-diameter tree-decompositions. arXiv:2306.13282, 2023. Google Scholar
  5. Reinhard Diestel and Malte Müller. Connected tree-width. Combinatorica, 38(2):381-398, 2018. URL: https://doi.org/10.1007/S00493-016-3516-5.
  6. Thomas Dissaux, Guillaume Ducoffe, Nicolas Nisse, and Simon Nivelle. Treelength of series-parallel graphs. Procedia Computer Science, 195:30-38, 2021. URL: https://doi.org/10.1016/J.PROCS.2021.11.008.
  7. Yon Dourisboure and Cyril Gavoille. Tree-decompositions with bags of small diameter. Discrete Mathematics, 307(16):2008-2029, 2007. URL: https://doi.org/10.1016/J.DISC.2005.12.060.
  8. Wassily Hoeffding. Probability inequalities for sums of bounded random variables. In The collected works of Wassily Hoeffding, pages 409-426. Springer, 1994. Google Scholar
  9. Sampath Kannan, Claire Mathieu, and Hang Zhou. Near-linear query complexity for graph inference. In International Colloquium on Automata, Languages, and Programming (ICALP), pages 773-784, 2015. URL: https://doi.org/10.1007/978-3-662-47672-7_63.
  10. Adrian Kosowski, Bi Li, Nicolas Nisse, and Karol Suchan. k-chordal graphs: From cops and robber to compact routing via treewidth. Algorithmica, 72(3):758-777, 2015. URL: https://doi.org/10.1007/S00453-014-9871-Y.
  11. Claire Mathieu and Hang Zhou. Graph reconstruction via distance oracles. In International Colloquium on Automata, Languages, and Programming (ICALP), pages 733-744, 2013. URL: https://doi.org/10.1007/978-3-642-39206-1_62.
  12. Claire Mathieu and Hang Zhou. A simple algorithm for graph reconstruction. Random Structures & Algorithms, pages 1-21, 2023. URL: https://doi.org/10.1002/rsa.21143.
  13. Lev Reyzin and Nikhil Srivastava. Learning and verifying graphs using queries with a focus on edge counting. In Algorithmic Learning Theory: 18th International Conference, ALT 2007, Sendai, Japan, October 1-4, 2007. Proceedings 18, pages 285-297. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-75225-7_24.
  14. Neil Robertson and Paul Seymour. Graph minors. II. Algorithmic aspects of tree-width. Journal of algorithms, 7(3):309-322, 1986. URL: https://doi.org/10.1016/0196-6774(86)90023-4.
  15. Guozhen Rong, Wenjun Li, Yongjie Yang, and Jianxin Wang. Reconstruction and verification of chordal graphs with a distance oracle. Theoretical Computer Science, 859:48-56, 2021. URL: https://doi.org/10.1016/J.TCS.2021.01.006.
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