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Low Sensitivity Hopsets

Authors Vikrant Ashvinkumar, Aaron Bernstein , Chengyuan Deng, Jie Gao , Nicole Wein

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ACM Subject Classification
  • Theory of computation → Shortest paths
  • Theory of computation → Sparsification and spanners
Keywords
  • Hopsets
  • Shortcuts
  • Sensitivity
  • Differential Privacy

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Abstract

Given a weighted graph G = (V,E,w), a (β, ε)-hopset H is an edge set such that for any s,t ∈ V, where s can reach t in G, there is a path from s to t in G ∪ H which uses at most β hops whose length is in the range [dist_G(s,t), (1+ε)dist_G(s,t)]. We break away from the traditional question that asks for a hopset H that achieves small |H| and small diameter β and instead study the sensitivity of H, a new quality measure. The sensitivity of a vertex (or edge) given a hopset H is, informally, the number of times a single hop in G ∪ H bypasses it; a bit more formally, assuming shortest paths in G are unique, it is the number of hopset edges (s,t) ∈ H such that the vertex (or edge) is contained in the unique st-path in G having length exactly dist_G(s,t). The sensitivity associated with H is then the maximum sensitivity over all vertices (or edges). The highlights of our results are:  
- A construction for (Õ(√n), 0)-hopsets on undirected graphs with O(log n) sensitivity, complemented with a lower bound showing that Õ(√n) is tight up to polylogarithmic factors for any construction with polylogarithmic sensitivity. 
- A construction for (n^o(1), ε)-hopsets on undirected graphs with n^o(1) sensitivity for any ε > 0 that is at least inverse polylogarithmic, complemented with a lower bound on the tradeoff between β, ε, and the sensitivity. 
- We define a notion of sensitivity for β-shortcut sets (which are the reachability analogues of hopsets) and give a construction for Õ(√n)-shortcut sets on directed graphs with O(log n) sensitivity, complemented with a lower bound showing that β = Ω̃(n^{1/3}) for any construction with polylogarithmic sensitivity. 
We believe hopset sensitivity is a natural measure in and of itself, and could potentially find use in a diverse range of contexts. More concretely, the notion of hopset sensitivity is also directly motivated by the Differentially Private All Sets Range Queries problem [Deng et al. WADS 23]. Our result for O(log n) sensitivity (Õ(√n), 0)-hopsets on undirected graphs immediately improves the current best-known upper bound on utility from Õ(n^{1/3}) to Õ(n^{1/4}) in the pure-DP setting, which is tight up to polylogarithmic factors.

Cite As Get BibTex

Vikrant Ashvinkumar, Aaron Bernstein, Chengyuan Deng, Jie Gao, and Nicole Wein. Low Sensitivity Hopsets. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 13:1-13:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.13

Author Details

Vikrant Ashvinkumar
  • Rutgers University, Piscataway, NJ, USA
Aaron Bernstein
  • New York University, NY, USA
Chengyuan Deng
  • Rutgers University, Piscataway, NJ, USA
Jie Gao
  • Rutgers University, Piscataway, NJ, USA
Nicole Wein
  • University of Michigan, Ann Arbor, MI, USA

Funding

  • Bernstein, Aaron: Funding through Sloan Fellowship, Google Research Award, and NSF Grant 1942010.
  • Gao, Jie: Ashvinkumar, Deng and Gao would like to acknowledge funding through NSF IIS-2229876, DMS-2220271, DMS-2311064, CCF-2208663, CCF-2118953.
  • Wein, Nicole: This work was done while the author was at DIMACS, Rutgers University, supported by a grant to DIMACS from the Simons Foundation (820931).

Acknowledgements

The authors would like to thank Greg Bodwin for helpful discussions about hopsets.

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