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Lower Bounds on Tree Covers

Authors Yu Chen , Zihan Tan , Hangyu Xu

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LIPIcs.ITCS.2026.38.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • Mathematics of computing → Combinatorics
Keywords
  • Tree Covers
  • Combinatorial Fixed-Point Theorems

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Abstract

Given an n-point metric space (X,d_X), a tree cover 𝒯 is a set of |𝒯| = k trees on X such that every pair of vertices in X has a low-distortion path in one of the trees in 𝒯. Tree covers have been playing a crucial role in graph algorithms for decades, and the research focus is the construction of tree covers with small size k and distortion.
When k = 1, the best distortion is known to be Θ(n). For a constant k ≥ 2, the best distortion upper bound is Õ(n^{1/k}) and the strongest lower bound is Ω(log_k n), leaving a gap to be closed. In this paper, we improve the lower bound to Ω(n^{1/(2^{k-1)}}).
Our proof is a novel analysis on a structurally simple grid-like graph, which utilizes some combinatorial fixed-point theorems. We believe that they will prove useful for analyzing other tree-like data structures as well.

Cite As Get BibTex

Yu Chen, Zihan Tan, and Hangyu Xu. Lower Bounds on Tree Covers. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.38

Author Details

Yu Chen
  • National University of Singapore, Queenstown, Singapore
Zihan Tan
  • University of Minnesota Twin Cities, MN, USA
Hangyu Xu
  • University of Science and Technology of China, Hefei, China

Funding

  • Xu, Hangyu: Supported in part by NSFC Grant 62272431 and Quantum Science and Technology-National Science and Technology Major Project (Grant No. 2021ZD0302901).

References

  1. Sunil Arya, Gautam Das, David M Mount, Jeffrey S Salowe, and Michiel Smid. Euclidean spanners: short, thin, and lanky. In Proceedings of the twenty-seventh annual ACM symposium on Theory of computing, pages 489-498, 1995. URL: https://doi.org/10.1145/225058.225191.
  2. Baruch Awerbuch, Shay Kutten, and David Peleg. On buffer-economical store-and-forward deadlock prevention. IEEE transactions on communications, 42(11):2934-2937, 2002. Google Scholar
  3. Baruch Awerbuch and David Peleg. Routing with polynomial communication-space trade-off. SIAM Journal on Discrete Mathematics, 5(2):151-162, 1992. URL: https://doi.org/10.1137/0405013.
  4. Yair Bartal, Ora Nova Fandina, and Ofer Neiman. Covering metric spaces by few trees. Journal of Computer and System Sciences, 130:26-42, 2022. URL: https://doi.org/10.1016/J.JCSS.2022.06.001.
  5. Yair Bartal, Nathan Linial, Manor Mendel, and Assaf Naor. On metric ramsey-type phenomena. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 463-472, 2003. URL: https://doi.org/10.1145/780542.780610.
  6. Artur Bikeev, Andrey Kupavskii, and Maxim Turevskii. Tree covers of size 2 for the euclidean plane, 2025. URL: https://doi.org/10.48550/arXiv.2508.16875.
  7. Béla Bollobás. Extremal graph theory. Courier Corporation, 2004. Google Scholar
  8. T-H Hubert Chan, Anupam Gupta, Bruce M Maggs, and Shuheng Zhou. On hierarchical routing in doubling metrics. ACM Transactions on Algorithms (TALG), 12(4):1-22, 2016. URL: https://doi.org/10.1145/2915183.
  9. Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenkovic, Shay Solomon, and Cuong Than. Covering planar metrics (and beyond): O (1) trees suffice. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 2231-2261. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00139.
  10. Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenkovic, Shay Solomon, and Cuong Than. Resolving the steiner point removal problem in planar graphs via shortcut partitions. arXiv preprint arXiv:2306.06235, 2023. URL: https://doi.org/10.48550/arXiv.2306.06235.
  11. Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenković, Shay Solomon, and Cuong Than. Optimal euclidean tree covers. In 40th International Symposium on Computational Geometry (SoCG 2024), pages 37:1-37:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.SoCG.2024.37.
  12. Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenković, Shay Solomon, and Cuong Than. Shortcut partitions in minor-free graphs: Steiner point removal, distance oracles, tree covers, and more. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 5300-5331. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.191.
  13. Ky Fan. A generalization of tucker’s combinatorial lemma with topological applications. Annals of Mathematics, 56(3):431-437, 1952. Google Scholar
  14. Elyot Grant and Will Ma. A geometric approach to combinatorial fixed-point theorems. In The Seventh European Conference on Combinatorics, Graph Theory and Applications: EuroComb 2013, pages 463-468. Springer, 2013. Google Scholar
  15. Anupam Gupta. Steiner points in tree metrics don't (really) help. In Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms, pages 220-227, 2001. URL: http://dl.acm.org/citation.cfm?id=365411.365448.
  16. Anupam Gupta, Mohammad T Hajiaghayi, and Harald Räcke. Oblivious network design. In Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, pages 970-979, 2006. Google Scholar
  17. Anupam Gupta, Amit Kumar, and Rajeev Rastogi. Traveling with a pez dispenser. In 42nd Annual Symposium on Foundations of Computer Science: Proceedings: October 14-17, 2001, Las Vegas, Nevada, USA, page 148. IEEE Computer Society, 2001. Google Scholar
  18. Hung Le, Lazar Milenković, Shay Solomon, and Tianyi Zhang. Covering the euclidean plane by a pair of trees, 2025. URL: https://doi.org/10.48550/arXiv.2508.11507.
  19. Jiří Matoušek, Anders Björner, and Günter M Ziegler. Using the Borsuk-Ulam theorem: lectures on topological methods in combinatorics and geometry. Springer, 2003. Google Scholar
  20. Manor Mendel and Assaf Naor. Ramsey partitions and proximity data structures. Journal of the European Mathematical Society, 9(2):253-275, 2007. Google Scholar
  21. Frédéric Meunier. Pleins étiquetages et configurations équilibrées: aspects topologiques de l'optimisation combinatoire. PhD thesis, Université Joseph-Fourier-Grenoble I, 2006. Google Scholar
  22. Oleg R Musin. Sperner type lemma for quadrangulations. arXiv preprint, 2014. URL: https://arxiv.org/abs/1406.5082.
  23. Yuri Rabinovich and Ran Raz. Lower bounds on the distortion of embedding finite metric spaces in graphs. Discrete & Computational Geometry, 19(1):79-94, 1998. URL: https://doi.org/10.1007/PL00009336.
  24. A. W. Tucker. Some topological properties of disk and sphere. In Proceedings of the First Canadian Math. Congress (Montreal, 1945), pages 285-309, 1945. Google Scholar
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