## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Geometry: Combinatorics & Algorithms (252-1425-00L) HS22

## Time & Place

Lectures: Monday 13:15-14:00, CAB G51 and Thursday 14:15-16:00, CAB G51.
The lecturers are:
 Bernd Gärtner, CAB G31.1, gaertner@inf.ethz.ch Michael Hoffmann, CAB G33.1, hoffmann@inf.ethz.ch Patrick Schnider, patrick.schnider@inf.ethz.ch Emo Welzl, CAB G39.2, emo@inf.ethz.ch
Exercise: Monday 14:15-15:45, CAB G51 (starting Sep 26, 2022) and Tuesday 14:15-15:45, CAB H52 (starting Sep 27, 2022).
The teaching assistants are:
 Wang Yanheng, yanhwang@student.ethz.ch Meghana M Reddy [contact assistant], CAB G39.3, meghana.mreddy@inf.ethz.ch

## Course Material

Lecture notes from last year
 Date Content Exercises and links Lecture notes Lecturer #1 22.09.2022 Information about the course, geometric graphs Chapter 1, Chapter 2.1-2.4, Recording, Slides MH #2 26.09.2022 Graph planarity Exercise 1 Slides, Recording MH #3 29.09.2022 Unique embeddings Slides, Recording MH #4 03.10.2022 Unique embeddings Exercise 2, Solution 3 Slides, Recording MH #5 06.10.2022 Polygons, Polygon triangulation, Art gallery problem Chapter 3, Recording MW #6 10.10.2022 Canonical Orderings Exercise 3 Chapter 2.5, Slides, Recording MH #7 13.10.2022 The Shift Algorithm Homework 1 Chapter 2.5 (cont.), Slides, Recording MH #8 17.10.2022 Crossings Exercise 4, Solution 4 Chapter 4, Slides, Recording MH #9 20.10.2022 Convexity Chapter 5, Recording BG #10 24.10.2022 Convexity Exercise 5 BG #11 27.10.2022 Convexity algorithms BG #12 31.10.2022 Convexity algorithms Exercise 6 BG #13 03.11.2022 Delaunay triangulation Chapter 6 BG #14 07.11.2022 Delaunay triangulation and Incremental Construction Exercise 7 BG #15 10.11.2022 Incremental Construction and Voronoi Diagrams Homework 2 Chapter 7, Chapter 8, Slides BG and PS #16 14.11.2022 Voronoi Diagrams, Polytopes Exercise 8 Chapter 9, Slides PS #17 17.11.2022 Polytopes, Higher dimensional triangulations Slides PS #18 21.11.2022 Line Arrangements Exercise 9 Chapter 10, Slides PS #19 24.11.2022 Line Arrangements PS #20 28.11.2022 Line Arrangements PS #21 01.12.2022 Line Arrangements PS #22 05.12.2022 DS-sequences, 3-SUM PS #23 08.12.2022 Simplicial Depth Chapter 11 EW #24 12.12.2022 Simplicial Depth Exercise 10 EW #25 15.12.2022 Simplicial Depth EW #26 19.12.2022 Simplicial Depth Exercise 11 EW #27 22.12.2022 Gale Duality 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, the Crossing Lemma and incidence bounds, line arrangements (duality, Zone Theorem, ham-sandwich cuts), 3-SUM hardness.

## 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).

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. In order to comply with new regulations recently issued by the department, merely attending the course and/or handing in exercises is no longer sufficient.

## 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.