# Ramsey Spanning Trees and Their Applications

@article{Abraham2018RamseyST, title={Ramsey Spanning Trees and Their Applications}, author={Ittai Abraham and Shiri Chechik and Michael Elkin and Arnold Filtser and Ofer Neiman}, journal={ACM Transactions on Algorithms (TALG)}, year={2018}, volume={16}, pages={1 - 21} }

The metric Ramsey problem asks for the largest subset S of a metric space that can be embedded into an ultrametric (more generally into a Hilbert space) with a given distortion. Study of this problem was motivated as a non-linear version of Dvoretzky theorem. Mendel and Naor [29] devised the so-called Ramsey Partitions to address this problem, and showed the algorithmic applications of their techniques to approximate distance oracles and ranking problems. In this article, we study the natural…

## 15 Citations

### Advances in Metric Ramsey Theory and its Applications

- Mathematics, Computer ScienceArXiv
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This decomposition provides the first deterministic Bourgain-type embedding of finite metric spaces into Euclidean space, and an optimal multi-embedding into ultrametrics, thus improving its applications in approximation and online algorithms.

### Clan embeddings into trees, and low treewidth graphs

- Computer Science, MathematicsSTOC
- 2021

This paper designs Ramsey-type embedding and clan embedding analogs of the stochastic embedding for minor-free graphs of diameter prameterized by D, which were known to be stochastically embeddable into bounded treewidth graphs with expected additive distortion є D.

### Covering Metric Spaces by Few Trees

- Computer ScienceICALP
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Efficient algorithms to construct tree covers and Ramsey tree covers for general, planar and doubling metrics are devised and a large separation is shown between what can be achieved by tree covers vs. Ramsey tree cover.

### Deterministic Tree Embeddings with Copies for Algorithms Against Adaptive Adversaries

- Computer ScienceArXiv
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A new tree embedding is provided which addresses issues by deterministically embedding a graph into a single tree containing O(logn) copies of each vertex while preserving the connectivity structure of every subgraph and O(log n)-approximating the cost ofevery subgraph.

### Low Treewidth Embeddings of Planar and Minor-Free Metrics

- Computer Science, MathematicsArXiv
- 2022

A new embedding technique to improve the treewidth bound of Cohen-Addad et al. is devised and a deterministic embedding of planar graphs of diameter D into graphs oftreewidth O and additive distortion + (cid:15)D that can be constructed in nearly linear time is obtained.

### Hop-Constrained Metric Embeddings and their Applications

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

This paper improves the Ramsey-type embedding to obtain parameters $t=\beta=\frac{\tilde{O}(\log n)}{\epsilon}$, and generalize it to arbitrary distortion parameter $t$ (in the cost of reducing the size of $M$).

### Reliable Spanners: Locality-Sensitive Orderings Strike Back

- MathematicsArXiv
- 2021

This work constructs a locality-sensitive ordering for doubling metrics with a small number of orderings suitable for non-Euclidean metrics and constructs reliable spanners from the newly introduced locality- sensitive orderings via reliable 2-hop spanners for paths.

### Can't See the Forest for the Trees: Navigating Metric Spaces by Bounded Hop-Diameter Spanners

- Computer Science, MathematicsPODC
- 2022

Euclidean spanners can be viewed as means of compressing the pairwise distances of a d-dimensional Euclidean space into O(n) = O∈,d (n) spanner edges, so that the spanner distances preserve the original distances to within a factor of 1 + ε.

### Tree embeddings for hop-constrained network design

- Computer ScienceSTOC
- 2021

It is shown that hop-constrained distances can be approximated by distributions over ``partial tree metrics'' and built into a powerful and versatile algorithmic tool which, similarly to classic probabilistic tree embeddings, reduces hop- Constrained problems in general graphs to hop-unconstraining problems on trees.

### Locality-sensitive orderings and applications to reliable spanners

- Computer Science, MathematicsSTOC
- 2022

The theory of LSO’s in non-Euclidean metrics by introducing new types of LOs suitable for general and topologically structured metrics and building reliable spanners for trees and planar graphs with the optimal stretch of 2.

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