HopConstrained Metric Embeddings and their Applications
Abstract
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 hopconstrained Ramseytype metric embeddings into trees. Specifically, embedding $f:G(V,E)\rightarrow T$ has Ramsey hopdistortion $(t,M,\beta,h)$ (here $t,\beta,h\ge1$ and $M\subseteq V$) if $\forall u,v\in M$, $d_G^{(\beta\cdot h)}(u,v)\le d_T(u,v)\le t\cdot d_G^{(h)}(u,v)$. $t$ is called the distortion, $\beta$ is called the hopstretch, and $d_G^{(h)}(u,v)$ denotes the minimum weight of a $uv$ path with at most $h$ hops. Haeupler {\em et al.} constructed embedding where $M$ contains $1\epsilon$ fraction of the vertices and $\beta=t=O(\frac{\log^2 n}{\epsilon})$. They used their embedding to obtain multiple bicriteria approximation algorithms for hopconstrained network design problems. In this paper, we first improve the Ramseytype 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 $M$). This embedding immediately implies polynomial improvements for all the approximation algorithms from Haeupler {\em et al.}. Further, we construct hopconstrained clan embeddings (where each vertex has multiple copies), and use them to construct bicriteria approximation algorithms for the group Steiner tree problem, matching the state of the art of the non constrained version. Finally, we use our embedding results to construct hop constrained distance oracles, distance labeling, and most prominently, the first hop constrained compact routing scheme with provable guarantees.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.14969
 Bibcode:
 2021arXiv210614969F
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Discrete Mathematics