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 $f:G(V,E)\rightarrow T$ has Ramsey hop-distortion $(t,M,\beta,h)$ (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 hop-stretch, and $d_G^{(h)}(u,v)$ denotes the minimum weight of a $u-v$ 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 hop-constrained network design problems. In this paper, we first improve 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 $M$). This embedding immediately implies polynomial improvements for all the approximation algorithms from Haeupler {\em et al.}. Further, we construct hop-constrained 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.

翻译：在网络设计问题（如紧凑路由）中，目标是通过（近似）最短路径在节点间路由数据包。这些路径的一个理想特性是跳数较小，这既能提升可靠性，又能降低传输成本。基于随机树嵌入在算法设计中的巨大成功，Haeupler、Hershkowitz 和 Zuzic（STOC'21）研究了跳约束的Ramsey型度量嵌入到树中的问题。具体而言，嵌入 $f:G(V,E)\rightarrow T$ 具有Ramsey跳形变 $(t,M,\beta,h)$（其中 $t,\beta,h\ge1$ 且 $M\subseteq V$），需满足 $\forall u,v\in M$，$d_G^{(\beta\cdot h)}(u,v)\le d_T(u,v)\le t\cdot d_G^{(h)}(u,v)$。这里 $t$ 称为形变，$\beta$ 称为跳拉伸，$d_G^{(h)}(u,v)$ 表示至多 $h$ 跳的 $u-v$ 路径的最小权重。Haeupler 等人构造了满足 $M$ 包含 $1-\epsilon$ 比例顶点且 $\beta=t=O(\frac{\log^2 n}{\epsilon})$ 的嵌入，并利用该嵌入获得了多个跳约束网络设计问题的双准则近似算法。本文首先改进了Ramsey型嵌入，使参数达到 $t=\beta=\frac{\tilde{O}(\log n)}{\epsilon}$，并将其推广到任意形变参数 $t$（代价是缩小 $M$ 的规模）。该嵌入直接使得Haeupler等人的所有近似算法获得多项式改进。此外，我们构造了跳约束家族嵌入（其中每个顶点具有多个副本），并将其用于群组斯坦纳树问题的双准则近似算法，达到了与无约束版本相当的最高水平。最后，我们利用嵌入结果构造了跳约束距离预言、距离标记，并首次构造了具有可证明保证的跳约束紧凑路由方案。