Lower Bounds on Tree Covers

TL;DR

This paper establishes a new lower bound on the size of tree covers in metric spaces, improving the understanding of the limitations of such structures for low-distortion path representations.

Contribution

It introduces a stronger lower bound of (n^{1/2^{k-1}}) for the size of tree covers, using novel combinatorial analysis on grid-like graphs.

Findings

Lower bound improved to (n^{1/2^{k-1}})

Utilizes combinatorial fixed-point theorems in analysis

Provides insights for analyzing tree-like data structures

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.