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Expected Complexity of Barcode Reduction.

Barbara Giunti1, Guillaume Houry2, Michael Kerber3

  • 1Graz University of Technology and SUNY University at Albany, 1400 Washington Avenue, HD-125, Albany, USA.

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|November 28, 2025
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Summary
This summary is machine-generated.

This study analyzes the computational complexity of persistence barcodes for random filtrations. We developed a method to bound the expected complexity of matrix reduction, improving estimates for Čech, Vietoris-Rips, and Erdős-Rényi filtrations.

Keywords:
Average complexityBarcodeMatrix reductionPersistent homology

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Area of Science:

  • Computational Topology
  • Algorithmic Complexity
  • Data Analysis

Background:

  • Topological Data Analysis (TDA) utilizes filtrations to study shape.
  • Computing persistence barcodes from these filtrations involves matrix reduction.
  • The complexity of this reduction is a key bottleneck.

Purpose of the Study:

  • To analyze the algorithmic complexity of computing persistence barcodes.
  • To develop a general technique for bounding the expected complexity of boundary matrix reduction.
  • To obtain improved bounds for specific filtrations like Čech, Vietoris-Rips, and Erdős-Rényi.

Main Methods:

  • Developing a general technique to bound expected matrix reduction complexity.
  • Relating complexity to the density of the reduced boundary matrix.
  • Leveraging existing results on expected Betti numbers of topological complexes.
  • Analyzing Čech, Vietoris-Rips, and Erdős-Rényi filtrations.

Main Results:

  • Established upper bounds for the average fill-in of boundary matrices after reduction.
  • Derived bounds on the expected complexity of barcode computation for random filtrations.
  • Demonstrated that fill-in bounds for Čech and Vietoris-Rips are asymptotically tight (up to log factor).
  • Showed that computed bounds outperform worst-case estimates.

Conclusions:

  • The developed technique provides tighter bounds on barcode computation complexity.
  • The bounds are significantly better than worst-case scenarios for practical applications.
  • An Erdős-Rényi filtration realizing worst-case complexity was constructed.