Lower Bounds on Tree Covers

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 $\hat{I}˜(n)$. For a constant $k\ge 2$, the best distortion upper bound is $\tilde{O}(n^{\frac{1}{k}})$ and the strongest lower bound is $\hat{I}\text{c◯}({\log }_{k}n)$, leaving a gap to be closed. In this paper, we improve the lower bound to $\hat{I}\text{c◯}(n^{\frac{1}{2^{k-1}}})$. 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.

ITCS 2026