# Hop-Constrained Metric Embeddings and their Applications

@article{Filtser2021HopConstrainedME, title={Hop-Constrained Metric Embeddings and their Applications}, author={Arnold Filtser}, journal={ArXiv}, year={2021}, volume={abs/2106.14969} }

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…

## 2 Citations

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

- Computer ScienceArXiv
- 2021

Spanners for metric spaces have been extensively studied, both in general metrics and in restricted classes, perhaps most notably in low-dimensional Euclidean spaces — due to their numerous…

On Strong Diameter Padded Decompositions

- Mathematics, Computer ScienceAPPROX-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.

## References

SHOWING 1-10 OF 102 REFERENCES

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.

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.

Prioritized Metric Structures and Embedding

- Computer Science, MathematicsSTOC
- 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.

On Light Spanners, Low-treewidth Embeddings and Efficient Traversing in Minor-free Graphs

- Computer Science, Mathematics2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
- 2020

The paper designs the first FPT approximation scheme for bounded-capacity vehicle routing on bounded-treewidth graphs (parameterized by the treewidth) and shows the two following structural results for minor-free metrics.

Efficient distributed approximation algorithms via probabilistic tree embeddings

- Computer ScienceDistributed Computing
- 2012

It is shown how to efficiently compute an (implicit) FRT embedding in a decentralized manner and how to use the embedding to obtain efficient expected O(log n)-approximate distributed algorithms for various problems, in particular the generalized Steiner forest problem, the minimum routing cost spanning tree problem, and the k-source shortest paths problem.

Hop-constrained oblivious routing

- Computer Science, MathematicsSTOC
- 2021

It is proved the existence of an oblivious routing scheme that is poly(logn)-competitive in terms of (congestion + dilation), thus resolving a well-known question in oblivious routing, as an analogue of the celebrated oblivious routing results of R'acke.

Integer programming formulations for the k-edge-connected 3-hop-constrained network design problem

- Mathematics, Computer ScienceNetworks
- 2016

New integer programming formulations for the k-edge-connected L-hop-constrained network design problem that are based on the transformation of the initial undirected graph into directed layered graphs are introduced.

Approximating Buy-at-Bulk and Shallow-Light k-Steiner Trees

- Mathematics, Computer ScienceAlgorithmica
- 2007

The results are recently used to give the first polylogarithmic approximation algorithm for the non-uniform multicommodity buy-at-bulk problem and an O(log 4n)-approximation algorithm for a relaxed version of Shallow-light k-Steiner tree.

k-node-disjoint hop-constrained survivable networks: polyhedral analysis and branch and cut

- Mathematics, Computer ScienceAnn. des Télécommunications
- 2018

An integer linear programming formulation for the k-node-disjoint hop-constrained network design problem is proposed, and new valid inequalities for the problem for L ∈{2,3,4}, and necessary and sufficient conditions for these inequalities to be facet defining are introduced.

Ramsey Spanning Trees and their Applications

- Computer Science, MathematicsSODA
- 2018

The natural extension of the metric Ramsey problem to graphs is studied, and the notion of Ramsey Spanning Trees is introduced, which provides the state-of-the-art deterministic construction of a distance oracle.