### Abstract

We introduce a theoretical and computational framework to use discrete Morse theory as an efficient preprocessing in order to compute zigzag persistent homology. From a zigzag filtration of complexes (K

_{i}), we introduce a zigzag Morse filtration whose complexes (A_{i}) are Morse reductions of the original complexes (K_{i}), and we prove that they both have same persistent homology. This zigzag Morse filtration generalizes the filtered Morse complex of Mischaikow and Nanda, defined for standard persistence. The maps in the zigzag Morse filtration are forward and backward inclusions, as is standard in zigzag persistence, as well as a new type of map inducing non trivial changes in the boundary operator of the Morse complex. We study in details this last map, and design algorithms to compute the update both at the complex level and at the homology matrix level when computing zigzag persistence. We deduce an algorithm to compute the zigzag persistence of a filtration that depends mostly on the number of critical cells of the complexes, and show experimentally that it performs better in practice.Original language | English |
---|---|

Title of host publication | Algorithms and Data Structures. WADS 2019 |

Place of Publication | Cham |

Publisher | Springer |

Pages | 538-552| |

ISBN (Print) | 978-3-030-24765-2 |

DOIs | |

Publication status | Published - 2019 |

Event | WADS 2019: 16th International Symposium - Edmonton, Canada Duration: 5 Aug 2019 → 7 Aug 2019 |

### Publication series

Name | Lecture Notes in Computer Science |
---|---|

Volume | 11646 |

### Conference

Conference | WADS 2019 |
---|---|

Country | Canada |

City | Edmonton |

Period | 5/08/19 → 7/08/19 |

## Fingerprint Dive into the research topics of 'Discrete Morse Theory for Computing Zigzag Persistence'. Together they form a unique fingerprint.

## Cite this

Maria, C., & Schreiber, H. (2019). Discrete Morse Theory for Computing Zigzag Persistence. In

*Algorithms and Data Structures. WADS 2019*(pp. 538-552|). (Lecture Notes in Computer Science ; Vol. 11646). Cham: Springer. https://doi.org/10.1007/978-3-030-24766-9_39