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.
Journal of Applied and Computational Topology
|November 28, 2025
Summary
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.
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.
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