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Nearly Optimal Local Algorithms for Constructing Sparse Spanners of Clusterable Graphs

Authors Reut Levi , Moti Medina , Omer Tubul

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Author Details

Reut Levi
  • Efi Arazi School of Computer Science, Reichman University, Herzliya, Israel
Moti Medina
  • Faculty of Engineering, Bar-Ilan University, Ramat Gan, Israel
Omer Tubul
  • Faculty of Engineering, Bar-Ilan University, Ramat Gan, Israel

Funding

  • Levi, Reut: The author was supported by the Israel Science Foundation under Grant 1867/20.
  • Medina, Moti: The author was supported by the Israel Science Foundation under Grants 867/19 and 554/23.
  • Tubul, Omer: The author was supported by the Israel Science Foundation under Grants 867/19 and 554/23.

Cite AsGet BibTex

Reut Levi, Moti Medina, and Omer Tubul. Nearly Optimal Local Algorithms for Constructing Sparse Spanners of Clusterable Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 60:1-60:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.60

Abstract

In this paper, we study the problem of locally constructing a sparse spanning subgraph (LSSG), introduced by Levi, Ron, and Rubinfeld (ALGO'20). In this problem, the goal is to locally decide for each e ∈ E if it is in G' where G' is a connected subgraph of G (determined only by G and the randomness of the algorithm). We provide an LSSG that receives as a parameter a lower bound, ϕ, on the conductance of G whose query complexity is Õ(√n/ϕ²). This is almost optimal when ϕ is a constant since Ω(√n) queries are necessary even when G is an expander. Furthermore, this improves the state of the art of Õ(n^{2/3}) queries for ϕ = Ω(1/n^{1/12}). We then extend our result for (k, ϕ_in, ϕ_out)-clusterable graphs and provide an algorithm whose query complexity is Õ(√n + ϕ_out n) for constant k and ϕ_in. This bound is almost optimal when ϕ_out = O(1/√n).

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Locally Computable Algorithms
  • Sublinear algorithms
  • Spanning Subgraphs
  • Clusterbale Graphs

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