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arXiv · 2508.10376

Lower Bounds on Tree Covers

Yu Chen·Zihan Tan·Hangyu Xu
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Abstract

Given an $n$-point metric space $(X,d_X)$, a tree cover $\mathcal{T}$ is a set of $|\mathcal{T}|=k$ trees on $X$ such that every pair of vertices in $X$ has a low-distortion path in one of the trees in $\mathcal{T}$. 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 $\Theta(n)$. For a constant $k\ge 2$, the best distortion upper bound is $\tilde O(n^{\frac 1 k})$ and the strongest lower bound is $\Omega(\log_k n)$, leaving a gap to be closed. In this paper, we improve the lower bound to $\Omega(n^{\frac{1}{2^{k-1}}})$. 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.

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Yu Chen, Zihan Tan, Hangyu Xu. 2025-08-14. Lower Bounds on Tree Covers. https://arxiv.org/abs/2508.10376

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