# Barcodes of Towers and a Streaming Algorithm for Persistent Homology

## TL;DR

This paper introduces a streaming algorithm to efficiently compute persistent homology from towers of simplicial complexes, producing a filtration with the same barcode while optimizing space and computational resources.

## Contribution

It presents a novel streaming algorithm that constructs a filtration equivalent to a tower's barcode with minimal space complexity, combining coning strategies and matrix reduction techniques.

## Key findings

- Algorithm is asymptotically space-efficient
- Can compute barcode directly from towers in practice
- Space complexity independent of tower length

## Abstract

A tower is a sequence of simplicial complexes connected by simplicial maps. We show how to compute a filtration, a sequence of nested simplicial complexes, with the same persistent barcode as the tower. Our approach is based on the coning strategy by Dey et al. (SoCG 2014). We show that a variant of this approach yields a filtration that is asymptotically only marginally larger than the tower and can be efficiently computed by a streaming algorithm, both in theory and in practice. Furthermore, we show that our approach can be combined with a streaming algorithm to compute the barcode of the tower via matrix reduction. The space complexity of the algorithm does not depend on the length of the tower, but the maximal size of any subcomplex within the tower.