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Quasi-Linear Distance Query Reconstruction for Graphs of Bounded Treelength

Authors Paul Bastide , Carla Groenland

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Author Details

Paul Bastide
  • LaBRI - Université de Bordeaux, France
  • TU Delft, The Netherlands
Carla Groenland
  • TU Delft, The Netherlands

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Paul Bastide and Carla Groenland. Quasi-Linear Distance Query Reconstruction for Graphs of Bounded Treelength. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 20:1-20:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.IPEC.2024.20

Abstract

In distance query reconstruction, we wish to reconstruct the edge set of a hidden graph by asking as few distance queries as possible to an oracle. Given two vertices u and v, the oracle returns the shortest path distance between u and v in the graph.
The length of a tree decomposition is the maximum distance between two vertices contained in the same bag. The treelength of a graph is defined as the minimum length of a tree decomposition of this graph. We present an algorithm to reconstruct an n-vertex connected graph G parameterized by maximum degree Δ and treelength k in O_{k,Δ}(n log² n) queries (in expectation). This is the first algorithm to achieve quasi-linear complexity for this class of graphs. The proof goes through a new lemma that could give independent insight on graphs of bounded treelength.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Distance Reconstruction
  • Randomized Algorithm
  • Treelength

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References

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