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A WSPD, Separator and Small Tree Cover for c-Packed Graphs

Authors Lindsey Deryckere , Joachim Gudmundsson , André van Renssen , Yuan Sha , Sampson Wong

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Well-separated pair decomposition
  • separator
  • tree cover
  • distance oracles
  • realistic graphs

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Abstract

The c-packedness property, proposed in 2010, is a geometric property that captures the spatial distribution of a set of edges. Despite the recent interest in c-packedness, its utility has so far been limited to Fréchet distance problems. An open problem is whether a wider variety of algorithmic and data structure problems can be solved efficiently under the c-packedness assumption, and more specifically, on c-packed graphs.
In this paper, we prove two fundamental properties of c-packed graphs: that there exists a linear-size well-separated pair decomposition under the graph metric, and there exists a constant size balanced separator. We then apply these fundamental properties to obtain a small tree cover for the metric space and distance oracles under the shortest path metric. In particular, we obtain a tree cover of constant size, an exact distance oracle of near-linear size and an approximate distance oracle of linear size.

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Lindsey Deryckere, Joachim Gudmundsson, André van Renssen, Yuan Sha, and Sampson Wong. A WSPD, Separator and Small Tree Cover for c-Packed Graphs. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.21

Author Details

Lindsey Deryckere
  • The University of Sydney, Australia
Joachim Gudmundsson
  • The University of Sydney, Australia
André van Renssen
  • The University of Sydney, Australia
Yuan Sha
  • The University of Sydney, Australia
Sampson Wong
  • The University of Copenhagen, Denmark

Funding

This research was partially funded by the Australian Government through the Australian Research Council (project number DP240101353).
  • Gudmundsson, Joachim: This research was partially funded by the Australian Government through the Australian Research Council (project number DP240101353).
  • van Renssen, André: This research was partially funded by the Australian Government through the Australian Research Council (project number DP240101353).
  • Wong, Sampson: This research was partially funded by the European Union through the Marie Skłodowska-Curie Actions Postdoctoral Fellowship (project number 101146276).

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