# Expected Complexity of Barcode Reduction

## TL;DR

This paper analyzes the computational complexity of calculating persistence barcodes in randomly generated filtrations, providing new bounds and insights into their efficiency.

## Contribution

The paper introduces a novel technique to bound the expected complexity of barcode computation based on matrix density and applies it to common filtrations.

## Key findings

- The fill-in bounds for Čech and Vietoris–Rips complexes are asymptotically tight up to a logarithmic factor.
- The proposed method yields better average-case complexity than worst-case estimates for barcode computation.
- An Erdős–Rényi filtration is shown to achieve worst-case fill-in and computation complexity.

## Abstract

We study the algorithmic complexity of computing the persistence barcode of a randomly generated filtration. We provide a general technique to bound the expected complexity of reducing the boundary matrix in terms of the density of its reduced form. We apply this technique finding upper bounds for the average fill-in (number of non-zero entries) of the boundary matrix on Čech, Vietoris–Rips and Erdős–Rényi filtrations after matrix reduction, thus obtaining bounds on the expected complexity of the barcode computation. Our method is based on previous results on the expected Betti numbers of the corresponding complexes. Our fill-in bounds for Čech and Vietoris–Rips complexes are asymptotically tight up to a logarithmic factor. In particular, both our fill-in and computation bounds are better than the worst-case estimates. We also provide an Erdős–Rényi filtration realizing the worst-case fill-in and computation.