# Chunk Reduction for Multi-Parameter Persistent Homology

## TL;DR

This paper introduces a new algorithm for simplifying multi-parameter persistent homology computations by reducing input complex size while preserving homological features, improving efficiency and scalability.

## Contribution

It extends the chunk algorithm to multi-parameter settings, producing the smallest quasi-isomorphic chain complex and enabling parallelization.

## Key findings

- The algorithm produces minimal multi-filtered chain complexes.
- Experimental results show improved performance over existing methods.
- Parallelization scheme enhances scalability for large datasets.

## Abstract

The extension of persistent homology to multi-parameter setups is an algorithmic challenge. Since most computation tasks scale badly with the size of the input complex, an important pre-processing step consists of simplifying the input while maintaining the homological information. We present an algorithm that drastically reduces the size of an input. Our approach is an extension of the chunk algorithm for persistent homology (Bauer et al., Topological Methods in Data Analysis and Visualization III, 2014). We show that our construction produces the smallest multi-filtered chain complex among all the complexes quasi-isomorphic to the input, improving on the guarantees of previous work in the context of discrete Morse theory. Our algorithm also offers an immediate parallelization scheme in shared memory. Already its sequential version compares favorably with existing simplification schemes, as we show by experimental evaluation.