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Computing the alpha complex using dual active set quadratic programming.

Erik Carlsson1, John Carlsson2

  • 1Department of Mathematics, UC Davis, 1 Shields Ave, Davis, CA, 95618, USA. ecarlsson@math.ucdavis.edu.

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This summary is machine-generated.

This study introduces a new method for computing the alpha complex, a key data structure in computational geometry. The novel approach avoids the computationally expensive Delaunay triangulation, making it efficient for complex datasets.

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Area of Science:

  • Computational Geometry
  • Topology
  • Data Structures

Background:

  • The alpha complex is a fundamental data structure encoding the topological type of a union of balls.
  • Existing algorithms rely on computing the Delaunay complex, which is computationally intractable in higher dimensions.
  • This limits the application of alpha complexes to low-dimensional problems.

Purpose of the Study:

  • To present a novel method for computing the alpha complex without computing the full Delaunay triangulation.
  • To overcome the dimensionality limitations of existing alpha complex algorithms.
  • To enable efficient computation of alpha complexes for complex datasets.

Main Methods:

  • The study applies Lagrangian duality and dual quadratic programming.
  • The algorithm focuses on ruling simplices out rather than ruling them in.
  • This avoids the need to compute the entire Delaunay triangulation.

Main Results:

  • A new algorithm for computing the alpha complex is presented.
  • The method is more efficient than existing approaches, particularly in higher dimensions.
  • It successfully computes the alpha complex without relying on the full Delaunay triangulation.

Conclusions:

  • The proposed method offers a more scalable and efficient approach to computing alpha complexes.
  • This advancement has implications for various applications in computational geometry and data analysis.
  • The technique overcomes previous dimensionality constraints, broadening the utility of alpha complexes.