Lower Bounds on Tree Covers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911272782528512 |
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| author | Chen, Yu Tan, Zihan Xu, Hangyu |
| author_facet | Chen, Yu Tan, Zihan Xu, Hangyu |
| contents | 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 $Θ(n)$. For a constant $k\ge 2$, the best distortion upper bound is $\tilde O(n^{\frac 1 k})$ and the strongest lower bound is $Ω(\log_k n)$, leaving a gap to be closed. In this paper, we improve the lower bound to $Ω(n^{\frac{1}{2^{k-1}}})$.
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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_10376 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lower Bounds on Tree Covers Chen, Yu Tan, Zihan Xu, Hangyu Data Structures and Algorithms Combinatorics 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 $Θ(n)$. For a constant $k\ge 2$, the best distortion upper bound is $\tilde O(n^{\frac 1 k})$ and the strongest lower bound is $Ω(\log_k n)$, leaving a gap to be closed. In this paper, we improve the lower bound to $Ω(n^{\frac{1}{2^{k-1}}})$. 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. |
| title | Lower Bounds on Tree Covers |
| topic | Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2508.10376 |