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Hop-Constrained Metric Embeddings and their Applications

Arnold Filtser

TL;DR

This work advances hop-constrained metric embeddings by developing improved Ramsey-type embeddings into ultrametrics with distortion $t=β=̃O(k)$ and hop-stretch $β$-terms, while enabling broad measure-inclusion guarantees. It extends the framework with hop-constrained clan embeddings and subgraph-preserving embeddings, leading to bicriteria approximations that match or surpass non-hop-constrained state-of-the-art for group Steiner problems and related structures. The authors also design hop-constrained metric data structures—distance oracles, distance labeling, and a first hop-constrained compact routing scheme—with provable guarantees and polylogarithmic dependence on the hop parameter. Together, these results yield a unified toolkit for designing hop-aware routing and approximation algorithms in networks, with practical implications for reliability and delay-bounded communication. The work also outlines open questions on parameter optimization, expected distortion, and extensions to restricted graph classes.

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 hop-constrained Ramsey-type metric embeddings into trees. Specifically, embedding $f:G(V,E)\rightarrow T$ has Ramsey hop-distortion $(t,M,β,h)$ (here $t,β,h\ge1$ and $M\subseteq V$) if $\forall u,v\in M$, $d_G^{(β\cdot h)}(u,v)\le d_T(u,v)\le t\cdot d_G^{(h)}(u,v)$. $t$ is called the distortion, $β$ 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 $1-ε$ fraction of the vertices and $β=t=O(\frac{\log^2 n}ε)$. 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 $t=β=\frac{\tilde{O}(\log n)}ε$, 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.

Hop-Constrained Metric Embeddings and their Applications

TL;DR

This work advances hop-constrained metric embeddings by developing improved Ramsey-type embeddings into ultrametrics with distortion and hop-stretch -terms, while enabling broad measure-inclusion guarantees. It extends the framework with hop-constrained clan embeddings and subgraph-preserving embeddings, leading to bicriteria approximations that match or surpass non-hop-constrained state-of-the-art for group Steiner problems and related structures. The authors also design hop-constrained metric data structures—distance oracles, distance labeling, and a first hop-constrained compact routing scheme—with provable guarantees and polylogarithmic dependence on the hop parameter. Together, these results yield a unified toolkit for designing hop-aware routing and approximation algorithms in networks, with practical implications for reliability and delay-bounded communication. The work also outlines open questions on parameter optimization, expected distortion, and extensions to restricted graph classes.

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 hop-constrained Ramsey-type metric embeddings into trees. Specifically, embedding has Ramsey hop-distortion (here and ) if , . is called the distortion, is called the hop-stretch, and denotes the minimum weight of a path with at most hops. Haeupler {\em et al.} constructed embedding where contains fraction of the vertices and . 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 , and generalize it to arbitrary distortion parameter (in the cost of reducing the size of ). 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.

Paper Structure

This paper contains 45 sections, 48 theorems, 69 equations, 5 figures, 1 table, 8 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Metric embeddings
  4. Approximation algorithms
  5. Hop-constrained metric data structures
  6. Distance Oracles.
  7. Distance Labeling.
  8. Compact Routing Scheme.
  9. Related work
  10. Conclusion and open problems
  11. Preliminaries
  12. Metric Embeddings
  13. Bounded hop distances
  14. Paper overview
  15. Ramsey type embeddings
  16. ...and 30 more sections

Key Result

Theorem 1

Consider an $n$-vertex graph $G=(V,E,w)$ with polynomial aspect ratio, and parameters $k,{\color{red}h}\in [n]$. Then there is a distribution $\mathcal{D}$ over dominating ultrametrics, such that: In addition, for every $\epsilon\in (0,1)$, there is distribution $\mathcal{D}$ as above such that every $U\in\mathrm{supp}(\mathcal{D})$ has Ramsey hop-distortion $(O(\frac{\log n}{\epsilon}),M,{\color

Figures (5)

  • Figure 1: An illustration of an embedding of the unweighted $3\times6$ grid into an ultrametric with Ramsey hop-distortion $(t=3,M=\{a,b,c,p,q,r\},{\color{red}\beta=4},{\color{red}h=1})$. In the ultrametric, the distance between a pair of vertices equals to the label of the least common ancestor. The ultrametric defines an hierarchical partition, which corresponds to the blue and green dashed lines on the left. The embedding has hop stretch ${\color{red}\beta=4}$ as $\forall u,v\in V$, $d_{G}^{{\color{red}(4\cdot 1)}}(u,v)\le d_U(u,v)$. Note that $4$ is tight, as $d_{G}^{{\color{red}(4)}}(a,i)=4=d_U(a,i)<d_{G}^{{\color{red}(3)}}(a,i)=\infty$. The embedding has distortion $t=3$ as $\forall u\in M,v\in V$, $d_U(u,v)\le t\cdot d_{G}^{{\color{red}(1)}}(u,v)$. Note that $3$ is tight, as $d_U(a,b)=3=3\cdot d_G^{{\color{red}(1)}}(a,b)$. Interestingly, the exact same embedding also has Ramsey hop-distortion $(t=4,M=\{a,b,c,d,e,f,m,n,o,p,q,r\},{\color{red}\beta=4},{\color{red}h=1})$.
  • Figure 2: An illustration of a clan embedding of the unweighted $3\times6$ grid into an ultrametric with hop distortion $(t=4,{\color{red}\beta=5},{\color{red}h=1})$. The copies of each vertex $v$ are denoted by a subset of $v_1,v_2,v_3$. The vertex representing the copy chosen to be the chief has a red underline. The clan embedding has hop stretch $\beta=5$, indeed $\forall u,v\in V,~~d^{{\color{red}(5\cdot 1)}}_G(x,y)\le \min_{u'\in f(u),v'\in f(v)}d_U(u',v')$. Note that $5$ is tight, as $d^{{\color{red}(5)}}_G(a,l)=5=d_U(a_1,l_1)<d^{{\color{red}(4)}}_G(a,l)=\infty$. The clan embedding has distortion $t=4$, indeed $\forall u,v\in V,~~\min_{u'\in f(u)}d_U(u',\chi(v))\le 4\cdot d^{{\color{red}(1)}}_G(u,v)$. Note that $4$ is tight, as $\min_{j'\in f(j)}d_U(j',\chi(g))=d_U(j_1,g_2)=4=4\cdot d^{{\color{red}(1)}}_G(j,g)$. The clan embedding has hop-path-distortion $(4,{\color{red}h=2})$. Consider the $2$-respecting path $(e,h,k)$ in $G$ of weight $2$. Then the copies $e_2,h_2,k_1$ in $U$ corespond to a "path" of weight $8$. From the other hand the path $(e,h,k,n)$ is not $2$-respecting, and indeed there is no corresponding path in $U$ of finite weight.
  • Figure 3: The path $P=\{v_0,\dots,v_{30}\}$ is illustrated in black. Here $\mathcal{X}=\{0,2,4\}$ correspond to indices $i$ for which $Z_i=\overline{X}_1$, while $\mathcal{X}=\{1,3\}$ correspond to indices $i$ for which $Z_i=$$Y_1$.
  • Figure 4: The graph $G$ is illustrated on the left, while the path tree one-to-many embedding into the tree $T$ is illustrated on the right. Each edge $e\in T$ has an associated path $P^T_e$, for example $P^T_{e',d}=(e,d)$, and $P^T_{a',e}=(a,f,e)$. Each pair of vertices $u,v\in V(T)$, has an (not necessarily simple) induced path $P^T_{u,v}$, for example $P^T_{f,e'}=(f,b,c,d,e)$, and $P^T_{a',e'}=(a,f,e,b,c,d,e)$. The hop-bound ${\color{red}\beta}$ is the maximal number of hops in an inducted path. Here ${\color{red}\beta}=6$ (realized by $P_T^{a',e'}$).
  • Figure :

Theorems & Definitions (95)

  • Theorem 1: Hop-Constrained Ramsey Embedding
  • Theorem 2: Hop-Constrained Ramsey Embedding with small hop-stretch and arbitrary aspect-ratio
  • Theorem 3: Clan embedding into ultrametric
  • Theorem 4: Clan embedding into ultrametric with small hop-stretch and arbitrary aspect-ratio
  • Theorem 5: Subgraph Preserving Embedding
  • Theorem 6
  • Theorem 7
  • Theorem 8: Hop-constrained Distance Oracle
  • Theorem 9: Hop-constrained Distance Labeling
  • Theorem 10: Hop-constrained Compact Routing Scheme
  • ...and 85 more