# 252-1425-00L Geometry: Combinatorics and Algorithms

Semester | Autumn Semester 2022 |

Lecturers | B. Gärtner, E. Welzl, M. Hoffmann |

Periodicity | yearly recurring course |

Language of instruction | English |

### Courses

Number | Title | Hours | Lecturers | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

252-1425-00 V | Geometry: Combinatorics and Algorithms | 3 hrs |
| B. Gärtner, E. Welzl, M. Hoffmann | |||||||||||||||

252-1425-00 U | Geometry: Combinatorics and Algorithms | 2 hrs |
| B. Gärtner, E. Welzl, M. Hoffmann | |||||||||||||||

252-1425-00 A | Geometry: Combinatorics and Algorithms Project Work, no fixed presence required. | 2 hrs | B. Gärtner, E. Welzl, M. Hoffmann |

### Catalogue data

Abstract | Geometric structures are useful in many areas, and there is a need to understand their structural properties, and to work with them algorithmically. The lecture addresses theoretical foundations concerning geometric structures. Central objects of interest are triangulations. We study combinatorial (Does a certain object exist?) and algorithmic questions (Can we find a certain object efficiently?) |

Objective | The goal is to make students familiar with fundamental concepts, techniques and results in combinatorial and computational geometry, so as to enable them to model, analyze, and solve theoretical and practical problems in the area and in various application domains. In particular, we want to prepare students for conducting independent research, for instance, within the scope of a thesis project. |

Content | Planar and geometric graphs, embeddings and their representation (Whitney's Theorem, canonical orderings, DCEL), polygon triangulations and the art gallery theorem, convexity in R^d, planar convex hull algorithms (Jarvis Wrap, Graham Scan, Chan's Algorithm), point set triangulations, Delaunay triangulations (Lawson flips, lifting map, randomized incremental construction), Voronoi diagrams, the Crossing Lemma and incidence bounds, line arrangements (duality, Zone Theorem, ham-sandwich cuts), 3-SUM hardness, counting planar triangulations. |

Lecture notes | yes |

Literature | Mark de Berg, Marc van Kreveld, Mark Overmars, Otfried Cheong, Computational Geometry: Algorithms and Applications, Springer, 3rd ed., 2008. Satyan Devadoss, Joseph O'Rourke, Discrete and Computational Geometry, Princeton University Press, 2011. Stefan Felsner, Geometric Graphs and Arrangements: Some Chapters from Combinatorial Geometry, Teubner, 2004. Jiri Matousek, Lectures on Discrete Geometry, Springer, 2002. Takao Nishizeki, Md. Saidur Rahman, Planar Graph Drawing, World Scientific, 2004. |

Prerequisites / Notice | Prerequisites: The course assumes basic knowledge of discrete mathematics and algorithms, as supplied in the first semesters of Bachelor Studies at ETH. Outlook: In the following spring semester there is a seminar "Geometry: Combinatorics and Algorithms" that builds on this course. There are ample possibilities for Semester-, Bachelor- and Master Thesis projects in the area. |

### Performance assessment

Performance assessment information (valid until the course unit is held again) | |

Performance assessment as a semester course | |

ECTS credits | 8 credits |

Examiners | M. Hoffmann, B. Gärtner, E. Welzl |

Type | session examination |

Language of examination | English |

Repetition | The performance assessment is only offered in the session after the course unit. Repetition only possible after re-enrolling. |

Mode of examination | oral 30 minutes |

Additional information on mode of examination | 60% final oral exam: 30 minutes oral exam with 30 minutes preparation time (no material allowed) plus two graded homework (20% each). The two mandatory graded homework (compulsory continuous performance assessments) will be released throughout the semester, at specific dates that will be announced. Each graded homework will have a deadline two weeks after the release. The solutions must be typeset in LaTeX (or similar). |

This information can be updated until the beginning of the semester; information on the examination timetable is binding. |

### Learning materials

Main link | Course Homepage |

Only public learning materials are listed. |

### Groups

No information on groups available. |

### Restrictions

There are no additional restrictions for the registration. |