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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.

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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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