[Paper Review] A light metric spanner

Lee-Ad Gottlieb|arXiv (Cornell University)|May 14, 2015

Computational Geometry and Mesh Generation39 references8 citations

TL;DR

This paper resolves a long-standing open problem by constructing a $(1+\varepsilon)$-stretch spanner for doubling metric spaces with lightness $(\mathrm{ddim}/\varepsilon)^{O(\mathrm{ddim})}$, using a hierarchical decomposition into sparse subgraphs and applying a light spanner construction on each. The result enables a significantly faster polynomial-time approximation scheme for the traveling salesman problem in doubling spaces.

ABSTRACT

It has long been known that $d$-dimensional Euclidean point sets admit $(1+ε)$-stretch spanners with lightness $W_E = ε^{-O(d)}$, that is total edge weight at most $W_E$ times the weight of the minimum spaning tree of the set [DHN93]. Whether or not a similar result holds for metric spaces with low doubling dimension has remained an important open problem, and has resisted numerous attempts at resolution. In this paper, we resolve the question in the affirmative, and show that doubling spaces admit $(1+ε)$-stretch spanners with lightness $W_D = (ddim/ε)^{O(ddim)}$. Important in its own right, our result also implies a much faster polynomial-time approximation scheme for the traveling salesman problemin doubling metric spaces, improving upon the bound presented in [BGK-12].

Motivation & Objective

• To resolve the open problem of whether doubling metric spaces admit light $(1+\varepsilon)$-stretch spanners, analogous to the known result in Euclidean spaces.
• To close the gap between Euclidean and doubling metric spaces in terms of spanner lightness, where Euclidean spaces achieve $\varepsilon^{-O(d)}$ lightness but doubling spaces previously only had $\Omega(\log n)$ bounds.
• To improve the runtime of polynomial-time approximation schemes for the traveling salesman problem in doubling metric spaces by leveraging light spanners.
• To develop a hierarchical decomposition technique that partitions doubling spaces into subgraphs with controlled sparsity and weight, enabling light spanner construction.

Proposed method

• Construct a full hierarchical decomposition of the metric space using net-respecting edges, leveraging the doubling dimension to control growth.
• Apply a segmentation procedure (Theorem 4.2) that partitions the graph into subgraphs $\mathcal{D}$, each with $c^{O(\mathrm{ddim})}f$-sparse minimum spanning trees.
• Ensure neighbor proximity by guaranteeing that every $c$-neighbor pair in the hierarchy is co-located in at least one subgraph $D_i$.
• Limit point occurrences to $c^{O(\mathrm{ddim})}\log\Delta$ sets per point to control total weight and construction time.
• Construct a $(1+\varepsilon/12)$-stretch spanner on each sparse subgraph using a light spanner construction (Theorem 3.4), then combine them into a global spanner.
• Use the hierarchical structure and sparsity to bound the total weight of the final spanner relative to the MST, achieving lightness $(\mathrm{ddim}/\varepsilon)^{O(\mathrm{ddim})}$.

Experimental results

Research questions

• RQ1 Can doubling metric spaces support $(1+\varepsilon)$-stretch spanners with lightness polynomial in $\mathrm{ddim}/\varepsilon$, matching the Euclidean case?
• RQ2 Does the absence of the leapfrog property in doubling spaces preclude light spanner constructions, or can alternative structural properties be exploited?
• RQ3 Can the light spanner construction be adapted to work efficiently in low-doubling-dimension spaces, given their more general structure than Euclidean spaces?
• RQ4 What is the impact of such a light spanner on the runtime of approximation algorithms for NP-hard problems like TSP in doubling metrics?

Key findings

• The paper establishes that doubling metric spaces admit $(1+\varepsilon)$-stretch spanners with lightness $W_D = (\mathrm{ddim}/\varepsilon)^{O(\mathrm{ddim})}$, resolving a long-standing open problem.
• The construction time is $(\mathrm{ddim}/\varepsilon)^{O(\mathrm{ddim})}n\log^2 n$, which is efficient for fixed $\mathrm{ddim}$ and $\varepsilon$.
• The light spanner construction enables a faster polynomial-time approximation scheme for the traveling salesman problem in doubling spaces, with runtime $2^{(\mathrm{ddim}/\varepsilon)^{O(\mathrm{ddim}^2)}}n + (\mathrm{ddim}/\varepsilon)^{O(\mathrm{ddim})}n\log^2 n$.
• The hierarchical segmentation technique ensures that each point appears in at most $c^{O(\mathrm{ddim})}\log\Delta$ subgraphs, controlling total weight and construction cost.
• The total weight of the spanner is bounded by $W_D \cdot w(\mathrm{MST}(G))$, with $W_D = (\mathrm{ddim}/\varepsilon)^{O(\mathrm{ddim})}$, proving lightness in the doubling dimension parameter.
• The result implies that the greedy hierarchical spanner alone may not yield a light spanner, but a refined decomposition and stitching approach does.