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Reconstructing Random Graphs from Distance Queries

Authors Michael Krivelevich , Maksim Zhukovskii

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  • Mathematics of computing → Random graphs
  • Mathematics of computing → Graph algorithms
Keywords
  • random graphs
  • graph reconstruction
  • distance queries
  • query complexity

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Abstract

We estimate the minimum number of distance queries that is sufficient to reconstruct the binomial random graph G(n,p) with constant diameter with high probability. We get a tight (up to a constant factor) answer for all p > n^{-1+o(1)} outside "threshold windows" around n^{-k/(k+1)+o(1)}, k ∈ ℤ_{> 0}: with high probability the query complexity equals Θ(n^{4-d}p^{2-d}), where d is the diameter of the random graph. This demonstrates the following non-monotone behaviour: the query complexity jumps down at moments when the diameter gets larger; yet, between these moments the query complexity grows. We also show that there exists a non-adaptive algorithm that reconstructs the random graph with O(n^{4-d}p^{2-d}ln n) distance queries with high probability, and this is best possible.

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Michael Krivelevich and Maksim Zhukovskii. Reconstructing Random Graphs from Distance Queries. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.30

Author Details

Michael Krivelevich
  • School of Mathematical Sciences, Tel Aviv University, Israel
Maksim Zhukovskii
  • School of Computer Science, The University of Sheffield, UK

Funding

  • Krivelevich, Michael: Research supported in part by NSF-BSF grant 2023688.

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