Lower Bounds on Tree Covers

TL;DR

This work tightens the fundamental trade-off for tree covers by proving a universal lower bound $\Omega(n^{1/2^{k-1}})$ on distortion for any size-$k$ tree cover of an $n$-point metric, for all constants $k\ge 1$. The authors develop a grid-like hard instance and connect tree-cover distortion to combinatorial topology via Tucker's lemma, with a dimension-doubling step to $d=2^{k-1}$ and an octahedron-based triangulation to enforce antipodality. The result implies $\Omega(\sqrt{n})$ distortion for $k=2$, nearly matching the known $\tilde{O}(\sqrt{n})$ upper bound, and rules out polylogarithmic lower bounds for constant $k$. An auxiliary contribution provides an alternative near-optimal proof for Ramsey tree covers, using a Tucker-cube framework to obtain $\Omega(n^{1/k})$ lower bounds. Overall, the paper uncovers a deep link between tree covers and fixed-point/topological methods, suggesting new tools 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 $Θ(n)$. 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.

Figures (7)

• Figure 1: Interpolated tree, sequences $\Pi_i$'s, and a bad $\Pi_5$.
• Figure 2: The torus grid, horizontal and vertical cycles, and $T_1$ and $T_2$-bad edges.
• Figure 3: The $\ell_1$ norm ball grid and its decomposition, and the triangulations of $\ell_1$-norm unit balls in $2$ and $3$ dimensions.
• Figure 4: Construction of $\tilde{P}$ by alternately flipping $P$. The two paths have the same length.
• Figure 5: The $\ell_{\infty}$ norm ball grid and its decomposition, and the triangulations of $\ell_{\infty}$-norm unit balls in $2$ and $3$ dimensions.

Theorems & Definitions (19)

• Theorem 1
• Definition 2: Triangulation
• Definition 3: Antipodal Symmetry
• Definition 4: Labeling of Triangulation
• Theorem 5: Tucker's Lemma
• Theorem 6: grant2013geometric
• Lemma 7
• proof
• Definition 8: Tree Index
• Lemma 9