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Spanner for the 0/1/∞ Weighted Region Problem

Authors Joachim Gudmundsson , Zijin Huang , André van Renssen , Sampson Wong

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • weighted region problem
  • approximate shortest path
  • spanner

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Abstract

We consider the problem of computing an approximate weighted shortest path in a weighted planar subdivision, with weights assigned from the set {0, 1, ∞}. The subdivision includes zero-cost regions (0-regions) with weight 0 and obstacles with weight ∞, all embedded in a plane with weight 1. In a polygonal domain, where the 0-regions and obstacles are non-overlapping polygons (not necessarily convex) with in total N vertices, we present an algorithm that computes a (1 + ε)-approximate spanner of the input vertices in expected Õ(N/ε³) time, for 0 < ε < 1. Using our spanner, we can compute a (1 + ε)-approximate weighted shortest path between any two points (not necessarily vertices) in Õ(N/ε³) time. Furthermore, we prove that our results more generally apply to non-polygonal convex regions. Using this generalisation, one can approximate the weak partial Fréchet similarity [Buchin et al., 2009] between two polygonal curves in expected Õ(n²/ε²) time, where n is the total number of vertices of the input curves.

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Joachim Gudmundsson, Zijin Huang, André van Renssen, and Sampson Wong. Spanner for the 0/1/∞ Weighted Region Problem. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.33

Author Details

Joachim Gudmundsson
  • The University of Sydney, Australia
Zijin Huang
  • The University of Sydney, Australia
André van Renssen
  • The University of Sydney, Australia
Sampson Wong
  • University of Copenhagen, Denmark

Funding

  • 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).

References

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