Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

Geometry: Combinatorics & Algorithms (252-1425-00L) HS23

Time & Place

Lectures: Monday 13:15-14:00, CAB G51 and Thursday 14:15-16:00, CAB G51.
The lecturers are:
Bernd Gärtner, OAT Z15,
Michael Hoffmann, OAT Z13.1,
Patrick Schnider, OAT Z13.1,
Emo Welzl, OAT Z13.2,

Exercise: Monday 14:15-15:45, CAB G51 (starting Sep 25, 2023) and Tuesday 14:15-15:45, CAB H52 (starting Sep 26, 2023).
The teaching assistants are:
Patryk Morawski,
Meghana M Reddy [contact assistant], OAT Z18,


Course Material

Lecture notes from last year

Date Content Exercises and links Lecture notes Lecturer

#1 21.09.2023 Information about the course, geometric graphs Chapter 1, Chapter 2.1-2.2, Slides MH

#2 25.09.2023 Graph planarity Exercise 1 Chapter 2.3-2.4, Slides BG

#3 28.09.2023 Unique embeddings Slides BG

#4 02.10.2023 Unique embeddings Exercise 2 Slides MH

#5 05.10.2023 Canonical Orderings, The Shift Algorithm Chapter 2.5, Slides MH

#6 09.10.2023 The Shift Algorithm Exercise 3 Slides MH

#7 12.10.2023 Crossings Homework 1 Chapter 3, Slides MH

#8 16.10.2023 Crossings, Polygons Exercise 4 Chapter 4, Slides MH

#9 19.10.2023 Polygons, Convexity Chapter 5, Slides PS

#10 23.10.2023 Convexity Exercise 5 Slides PS

#11 26.10.2023 Convexity algorithms Slides PS

#12 30.10.2023 Delaunay Triangulations Exercise 6 Chapter 6, Slides PS

#13 02.11.2023 Delaunay triangulation Slides PS

#14 06.11.2023 Incremental Construction Exercise 7 Chapter 7, PS

#15 13.11.2023 Voronoi Diagrams Exercise 8 Chapter 8, Slides PS

#16 16.11.2023 Voronoi Diagrams, Polytopes Homework 2 Chapter 9, Slides BG

#17 20.11.2023 Polytopes Exercise 9 BG

#18 23.11.2023 Polytopes BG

#19 27.11.2023 Polytopes, Line Arrangements Chapter 10, Slides BG

#20 30.11.2023 Line Arrangements MH

#21 04.12.2023 Line Arrangements MH

#22 07.12.2023 Simplicial Depth Chapter 11 EW

#23 11.12.2023 Simplicial Depth Exercise 10 EW

#24 14.12.2023 Simplicial Depth PS

#25 18.12.2023 Simplicial Depth Exercise 11 EW

Course Description

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?)

Our 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.

Covered topics include: 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, Crossing Lemma and incidence bounds, arrangements, ham-sandwich cuts, Davenport-Schinzel sequences, 3-SUM hardness, simplicial depth, and Gale Duality.

Procedures, Exercises, Exam

Every week we provide you with exercises. The students are split into small groups, and the members of each group work together. At the end of the session, for each exercise, a student from a group presents their solution to the rest of the students. In addition to the exercise sessions, we encourage you to solve the exercises in written form and to hand in your solutions to the teaching assistant. Your solutions are thoroughly commented, but they do not count towards your final grade. The motivation to work on the exercises stems from your interest in the topic (and possibly also the desire to succeed in the exam).

In addition, you receive two homework assignments during the semester. The homework is to be solved in written form and you have two weeks of time to return your solutions/reports, typeset in LaTeX. In contrast to the exercises, these assignments do count towards the final grade: Your two grades will account for 20% of your final grade each. Solving the homework in teams is not allowed. Besides one or two exercises, the homework may include a small research project, or you are asked to give a short talk about your last small research project. The format of this talk will be determined by the number of students who register for the course.

There is an oral exam of 30 minutes during the examination period. Your final grade consists to 60% of the grade for the exam and to 40% of the grade for the homework assignments.
You are expected to learn proofs discussed in the lecture, should be able to explain their basic ideas and reproduce more details on demand. You should also be able to give a short presentation on any topic treated throughout the course. One of the questions given to you during the exam is to solve one of the exercises posed throughout the semester. Roughly half an hour before the exam you get to know the exercise to be solved and one topic that you will be questioned about in particular, that is, you have 30 minutes preparation time. For this preparation, paper and pencil will be provided. You may not use any other material, like books or notes.

For PhD students, the same rules apply for obtaining credit points as for all other participants. Taking the exam and achieving an overall grade of at least 4.0 (computed as a weighted average of grades for homework and the oral final exam as detailed above) qualifies for receiving credits.

Complementary Courses & Semester/Master/Diploma Theses

This course is complemented by a seminar Geometry: Combinatorics & Algorithms in the following spring semester. After having completed the course, it is possible to do a semester, master or diploma thesis in the area. Students are also welcome at our graduate seminar.

Literature and Links

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