A simple algorithm for graph reconstruction

• Published in: Random structures & algorithms Vol. 63; no. 2; pp. 512 - 532
• Main Authors: Mathieu, Claire, Zhou, Hang
• Format: Journal Article
• Language: English
• Published: New York John Wiley & Sons, Inc 01.09.2023; Wiley Subscription Services, Inc
• Subjects: Algorithms; Apexes; Graph theory; Graphs; metric dimension; network topology; Queries; random regular graphs; Reconstruction
• ISSN: 1042-9832; 1098-2418; 1098-2418
• DOI: 10.1002/rsa.21143

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(n3/2)$$ \overset{\widetilde }{O}\left({n}^{3/2}\right) $$ distance queries. In our work, we analyze a simple reconstruction algorithm. We show that, on random Δ$$ \Delta $$‐regular graphs, our algorithm uses Õ(n)$$ \overset{\widetilde }{O}(n) $$ distance queries. As by‐products, with high probability, we can reconstruct those graphs using log2n$$ {\log}^2n $$ queries to an all‐distances oracle or Õ(n)$$ \overset{\widetilde }{O}(n) $$ queries to a betweenness oracle, and we bound the metric dimension of those graphs by log2n$$ {\log}^2n $$. 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.