### Abstract

We introduce random discrete Morse theory as a computational scheme to measure the complexity of a triangulation. The idea is to try to quantify the frequency of discrete Morse matchings with few critical cells. Our measure will depend on the topology of the space, but also on how nicely the space is triangulated. The scheme we propose looks for optimal discrete Morse functions with an elementary random heuristic. Despite its naiveté, this approach turns out to be very successful even in the case of huge inputs. In our view, the existing libraries of examples in computational topology are "too easy" for testing algorithms based on discrete Morse theory. We propose a new library containing more complicated (and thus more meaningful) test examples.

Original language | English (US) |
---|---|

Pages (from-to) | 66-94 |

Number of pages | 29 |

Journal | Experimental Mathematics |

Volume | 23 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2 2014 |

Externally published | Yes |

### Fingerprint

### Keywords

- Collapsibility
- Knots in triangulations
- Library of triangulations
- Random discrete Morse theory
- Shellability

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Experimental Mathematics*,

*23*(1), 66-94. https://doi.org/10.1080/10586458.2013.865281

**Random discrete morse theory and a new library of triangulations.** / Benedetti, Bruno; Lutz, Frank H.

Research output: Contribution to journal › Article

*Experimental Mathematics*, vol. 23, no. 1, pp. 66-94. https://doi.org/10.1080/10586458.2013.865281

}

TY - JOUR

T1 - Random discrete morse theory and a new library of triangulations

AU - Benedetti, Bruno

AU - Lutz, Frank H.

PY - 2014/1/2

Y1 - 2014/1/2

N2 - We introduce random discrete Morse theory as a computational scheme to measure the complexity of a triangulation. The idea is to try to quantify the frequency of discrete Morse matchings with few critical cells. Our measure will depend on the topology of the space, but also on how nicely the space is triangulated. The scheme we propose looks for optimal discrete Morse functions with an elementary random heuristic. Despite its naiveté, this approach turns out to be very successful even in the case of huge inputs. In our view, the existing libraries of examples in computational topology are "too easy" for testing algorithms based on discrete Morse theory. We propose a new library containing more complicated (and thus more meaningful) test examples.

AB - We introduce random discrete Morse theory as a computational scheme to measure the complexity of a triangulation. The idea is to try to quantify the frequency of discrete Morse matchings with few critical cells. Our measure will depend on the topology of the space, but also on how nicely the space is triangulated. The scheme we propose looks for optimal discrete Morse functions with an elementary random heuristic. Despite its naiveté, this approach turns out to be very successful even in the case of huge inputs. In our view, the existing libraries of examples in computational topology are "too easy" for testing algorithms based on discrete Morse theory. We propose a new library containing more complicated (and thus more meaningful) test examples.

KW - Collapsibility

KW - Knots in triangulations

KW - Library of triangulations

KW - Random discrete Morse theory

KW - Shellability

UR - http://www.scopus.com/inward/record.url?scp=84896383697&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84896383697&partnerID=8YFLogxK

U2 - 10.1080/10586458.2013.865281

DO - 10.1080/10586458.2013.865281

M3 - Article

AN - SCOPUS:84896383697

VL - 23

SP - 66

EP - 94

JO - Experimental Mathematics

JF - Experimental Mathematics

SN - 1058-6458

IS - 1

ER -