A Simple Algorithm for Graph Reconstruction

Authors Claire Mathieu, Hang Zhou

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Author Details

Claire Mathieu
  • CNRS, IRIF, Université de Paris, France
Hang Zhou
  • École Polytechnique, Institut Polytechnique de Paris, France


We want to thank the anonymous reviewers for their valuable comments.

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Claire Mathieu and Hang Zhou. A Simple Algorithm for Graph Reconstruction. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 68:1-68:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


How efficiently can we find an unknown graph using distance queries between its vertices? We assume that the unknown graph is connected, unweighted, and has bounded degree. The goal is to find every edge in the graph. This problem admits a reconstruction algorithm based on multi-phase Voronoi-cell decomposition and using Õ(n^{3/2}) distance queries [Kannan et al., 2018]. In our work, we analyze a simple reconstruction algorithm. We show that, on random Δ-regular graphs, our algorithm uses Õ(n) distance queries. As by-products, we can reconstruct those graphs using O(log² n) queries to an all-distances oracle or Õ(n) queries to a betweenness oracle, and we bound the metric dimension of those graphs by log² n. Our reconstruction algorithm has a very simple structure, and is highly parallelizable. On general graphs of bounded degree, our reconstruction algorithm has subquadratic query complexity.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Random network models
  • Networks → Network algorithms
  • reconstruction
  • network topology
  • random regular graphs
  • metric dimension


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