# Hop-Constrained Metric Embeddings and their Applications

@article{Filtser2021HopConstrainedME, title={Hop-Constrained Metric Embeddings and their Applications}, author={Arnold Filtser}, journal={2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)}, year={2021}, pages={492-503} }

In network design problems, such as compact routing, the goal is to route packets between nodes using the (approximated) shortest paths. A desirable property of these routes is a small number of hops, which makes them more reliable, and reduces the transmission costs. Following the overwhelming success of stochastic tree embeddings for algorithmic design, Haeupler, Hershkowitz, and Zuzic (STOC'21) studied hop-constrained Ramsey-type metric embeddings into trees. Specifically, embedding $f: G(V…

## 5 Citations

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

- Computer Science, Mathematics2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

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

### 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.

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

- Computer Science, Mathematics
- 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 ((log log n ) 2 ) and additive distortion + (cid:15)D that can be constructed in nearly linear time is obtained.

### 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.

### Light Spanners for High Dimensional Norms via Stochastic Decompositions

- Mathematics, Computer ScienceAlgorithmica
- 2022

This paper shows that any n-point subset of ℓp\documentclass[12pt]{minimal} \usepackage{amsmath} Ã‚¬wasysym} Â£1,000,000 Ã¡¬2,500,000 is a spanner which is both sparse and also light, i.e., its total weight is not much larger than that of the minimum spanning tree.

## References

SHOWING 1-10 OF 113 REFERENCES

### 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.

### Hopsets with Constant Hopbound, and Applications to Approximate Shortest Paths

- Computer Science, Mathematics2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
- 2016

The first construction of sparse hopsets with a constant number of hops is exhibited, and the applicability of the results for the fundamental problem of computing approximate shortest paths from s sources is demonstrated.

### Near-Additive Spanners and Near-Exact Hopsets, A Unified View

- Computer Science, MathematicsBull. EATCS
- 2020

An intriguing phenomenon is explored, Thorup-Zwick's later construction of near-additive spanners was shown to provide hopsets with analogous (to that of \cite{TZ06}) properties, and the basic proof techniques used for these results are sketched.

### Deterministic Constructions of Approximate Distance Oracles and Spanners

- Computer Science, MathematicsICALP
- 2005

The first deterministic linear time algorithm for constructing optimal spanners of weighted graphs is obtained by derandomizing the O(km) expected time algorithm of Baswana and Sen for constructing (2k–1)-spanners of size O(kn) of weighted undirected graphs without incurring any asymptotic loss in the running time or in the size of the spanners produced.

### Relaxed Voronoi: A Simple Framework for Terminal-Clustering Problems

- Computer Science, MathematicsSOSA
- 2019

The main contribution is to demonstrate that the Relaxed-Voronoi algorithm is applicable to restricted metrics, and actually leads to relatively simple algorithms and analyses.

### Ramsey Spanning Trees and Their Applications

- Mathematics, Computer ScienceSODA
- 2018

This article revisit the metric Ramsey problem and provide a new deterministic construction of a distance oracle, and proves that for every k, any n-point metric space has a subset S of size at least n1−1/k that embeds into an ultrametric with distortion 8k.

### Prioritized Metric Structures and Embedding

- Computer ScienceSTOC
- 2015

This paper shows that given a priority ranking of the graph vertices one can devise a metric data structure in which the stretch incurred by any pair containing a vertex xj will depend on the rank j of the vertex, and shows that other important parameters, such as the label size and (in some sense) the dimension, may depend only on j.

### When trees collide: an approximation algorithm for the generalized Steiner problem on networks

- Computer ScienceSTOC '91
- 1991

This work gives the first approximation algorithm for the generalized network Steiner problem, a problem in network design, and proves a combinatorial min-max approximate equality relating minimum-cost networks to maximum packings of certain kinds of cuts.

### Labelings vs. Embeddings: On Distributed Representations of Distances

- MathematicsSODA
- 2020

The worst-case bound for label size often ``translates'' to a prioritized one, but also a surprising exception to this rule when comparing the classical and prioritized settings.

### On Strong Diameter Padded Decompositions

- MathematicsAPPROX-RANDOM
- 2019

The state of the art for weak and strong decompositions for free graphs are matched, and aparse cover scheme is constructed to construct $(O(d),\tilde{O}(d))$-sparsecover scheme for such graphs.