[Paper Review] Covering Metric Spaces by Few Trees

Yair Bartal, Nova Fandina+1 more|arXiv (Cornell University)|Jan 1, 2019

Advanced Topology and Set Theory9 citations

TL;DR

This paper presents deterministic algorithms for constructing tree covers and Ramsey tree covers in general, planar, and doubling metric spaces, achieving nearly optimal distortion for a small number of trees. It introduces a novel hierarchical clustering method for doubling metrics, yielding a (1+ϵ)-distortion tree cover with (1/ϵ)^O(log λ) trees—improving prior work by avoiding Steiner points and matching asymptotic optimality in edge count.

ABSTRACT

A tree cover of a metric space (X,d) is a collection of trees, so that every pair x,y in X has a low distortion path in one of the trees. If it has the stronger property that every point x in X has a single tree with low distortion paths to all other points, we call this a Ramsey tree cover. Tree covers and Ramsey tree covers have been studied by [Yair Bartal et al., 2005; Anupam Gupta et al., 2004; T-H. Hubert Chan et al., 2005; Gupta et al., 2006; Mendel and Naor, 2007], and have found several important algorithmic applications, e.g. routing and distance oracles. The union of trees in a tree cover also serves as a special type of spanner, that can be decomposed into a few trees with low distortion paths contained in a single tree; Such spanners for Euclidean pointsets were presented by [S. Arya et al., 1995]. In this paper we devise efficient algorithms to construct tree covers and Ramsey tree covers for general, planar and doubling metrics. We pay particular attention to the desirable case of distortion close to 1, and study what can be achieved when the number of trees is small. In particular, our work shows a large separation between what can be achieved by tree covers vs. Ramsey tree covers.

Motivation & Objective

• To develop efficient deterministic algorithms for constructing tree covers and Ramsey tree covers in general, planar, and doubling metric spaces.
• To achieve nearly optimal distortion (1+ϵ) with a small number of trees, particularly avoiding Steiner points in Euclidean and doubling spaces.
• To establish tight lower bounds on distortion for Ramsey tree covers, showing a significant separation between tree covers and Ramsey tree covers.
• To improve spanner constructions by decomposing them into few trees with low-distortion paths, achieving asymptotically optimal edge counts.

Proposed method

• Proposes a novel bottom-up hierarchical clustering for doubling metrics, enabling construction of low-distortion tree covers.
• Uses a refined hierarchical partitioning technique to achieve distortion O(α) with O(λ^{1/α} · log λ · log α) trees for any α.
• Introduces a recursive construction of N-cycle graphs (recursive N-cycle graph) to establish lower bounds for Ramsey tree covers.
• Demonstrates that [C_{2N}]^k_β embeds with constant distortion into a series-parallel graph with O(1) doubling dimension.
• Applies probabilistic and inductive arguments to prove lower bounds on Ramsey tree cover distortion, using cycle metrics and coloring-based partitioning.
• Leverages known results on probabilistic tree embeddings and spanner constructions to derive trade-offs and optimality claims.

Experimental results

Research questions

• RQ1 Can (1+ϵ)-distortion tree covers be constructed for doubling metrics with a number of trees sub-exponential in 1/ϵ and without Steiner points?
• RQ2 What is the optimal trade-off between distortion and number of trees in tree covers for general and doubling metrics?
• RQ3 Is there a super-constant separation in distortion between tree covers and Ramsey tree covers for small tree counts?
• RQ4 Can O(1)-tree covers achieve distortion 1+ϵ for planar or minor-free graphs, or is distortion Ω(log n) unavoidable?
• RQ5 What is the tight lower bound on distortion for Ramsey tree covers of size k in doubling metrics?

Key findings

• For any doubling metric with doubling constant λ, a (1+ϵ)-distortion tree cover exists with (1/ϵ)^O(log λ) trees, improving prior results by eliminating the need for Steiner points.
• The number of edges in the resulting spanner is n/ϵ^O(log λ), which is asymptotically optimal.
• For any α ≥1, a tree cover with distortion O(α) and O(λ^{1/α} · log λ · log α) trees is constructed, improving over [CGMZ05] and matching known lower bounds.
• A lower bound of Ω(n^{1/k}) is established for the distortion of any Ramsey (α,k)-tree cover in n-point doubling metrics, showing a strong separation from standard tree covers.
• The recursive N-cycle graph construction embeds [C_{2N}]^k_3 with constant distortion into a series-parallel graph of O(1) doubling dimension, enabling the lower bound to extend to planar and minor-free graphs.
• The paper shows that O(1)-tree covers for planar graphs can achieve constant distortion, but distortion 1+ϵ requires O((log n)/ϵ)^2 trees, indicating a trade-off.