# Light Euclidean Steiner Spanners in the Plane

@inproceedings{Bhore2021LightES, title={Light Euclidean Steiner Spanners in the Plane}, author={Sujoy Kumar Bhore and Csaba D. T'oth}, booktitle={SoCG}, year={2021} }

Lightness is a fundamental parameter for Euclidean spanners; it is the ratio of the spanner weight to the weight of the minimum spanning tree of a finite set of points in $\mathbb{R}^d$. In a recent breakthrough, Le and Solomon (2019) established the precise dependencies on $\varepsilon>0$ and $d\in \mathbb{N}$ of the minimum lightness of an $(1+\varepsilon)$-spanner, and observed that additional Steiner points can substantially improve the lightness. Le and Solomon (2020) constructed Steiner…

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## 6 Citations

### A Unified Framework of Light Spanners II: Fine-Grained Optimality

- Computer Science, MathematicsArXiv
- 2021

This work presents a unified framework of light spanners in a variety of graph classes, and complements the upper bound with a highly nontrivial lower bound construction, for which any (1 + )-spanner must have lightness Ω( r + 1 2 ).

### Minimum Weight Euclidean $(1+\varepsilon)$-Spanners

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- 2022

The result generalizes to d -space for all d ∈ N : The minimum weight of a Euclidean (1 + ε )-spanner for n points in the unit cube d is O d ( ε (1 − d 2 ) /d n ( d − 1) /d ), and this bound is the best possible.

### Euclidean Steiner Spanners: Light and Sparse

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- 2022

This work improves several bounds on the lightness and sparsity of Euclidean Steiner (1 + ε )-spanners, and generalizes the notion of shallow light trees, which may be of independent interest, and use directional spanners and a modified window partitioning scheme to achieve a tight weight analysis.

### Online Spanners in Metric Spaces

- Computer Science, MathematicsArXiv
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Given a metric space M = ( X, δ ), a weighted graph G over X is a metric t -spanner of M if for every u, v ∈ X , δ ( u, v ) ≤ δ G ( u, v ) ≤ t · δ ( u, v ), where δ G is the shortest path metric in G…

### A Gap-ETH-Tight Approximation Scheme for Euclidean TSP

- Computer Science2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

This work revisits the classic task of finding the shortest tour of $n$ points in d-dimensional Euclidean space, and builds upon the celebrated quadtree-based methods initially proposed by Arora, but adds a new idea that is sparsity-sensitive patching that lets the granularity with which the tour depends on how sparse it is locally.

### Online Euclidean Spanners

- Computer Science, MathematicsESA
- 2021

It is proved that any online spanner algorithm for a sequence of n points in ℝ^d under the L₂ norm has competitive ratio Ω(f(n), where lim_{n → ∞} f(n) = ∞.

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