# CS 519: Topics in Computational Geometry (Fall 2014)

Instructor: Amir Nayyeri (Office hours: Thr 4:00pm-5:00pm at KEC 3061)

Lectures: Tue, Thr 14:00-15:20 in WGND 132

The main focus of this course is metric embeddings, and I plan to cover these lecture notes by Jiri Matousek. I might have time to cover a couple of other related topics from computational geometry and topology.

Prerequisites: Graduate standing in computer science and an undergraduate course in algorithms.

Evaluation:

• Homework: There will be three homework assignments.
• Presentation in class: Each student should give one presentation in class.

Resources:

Homework:

Topics for student presectations (with suggested resources):

Lectures:

1. Tue 3/31: No class.

2. Thr 4/2: Metric space definition, exampls: Euclidean space, shortest paths metric of a graph, embeddings, isometry, approximate isometry (D-embeddings).
Reference: (1) Chapter 1 of the lecture notes.

3. Tue 4/7: Norm definittion, Frechet embedding, dimensions of isometric embeddngs, inclusion among classes.
Reference: (1) Chapter 1 of the lecture notes.

4. Thr 4/9: Embedding into random of trees, buy at bulk network design problem.
Reference: (1) Random tree embedding, section 3.3, Anupam Gupta. (2) Buy at bulk network design problem, sections 2, R. Ravi.

5. Tue 4/14: Problem solving session, we discuss a subset of the following problems from section one of the lecture notes: 1, 2, 4, 7, 9, 10, 14, 15.

6. Thr 4/16: Embedding into a distribution of trees, Bartal's theorem proof using CKR decomposition.
Reference: (1) Embedding into random trees, Anupam Gupta. (2) Embedding into random trees: random graph decomposition, sections 9.1, 9.3, Anupam Gupta.

7. Tue 4/21: Metric relaxation, solving min-cut and min multi-way cut using metric relaxation.
Reference: (1) Metric relaxation, section 3, R. Ravi. (2) Metric rounding for cut problems, Anupam Gupta.

8. Thr 4/23: Constructing embeddings, bounding the dimension for a given distortion.
Reference: (1) Section 4.1 of the lecture notes.

9. Tue 4/28: Problem solving session, we discuss the following problems: Problems 1 and 2 from these notes by Anastasios Sidiropoulos, and problems 3, 4, and 5, from these notes by Anupam Gupta.

10. Thr 4/31: No class.

11. Tue 5/5: Approximating the sparsest cut, an application of low-distortion embedding.
Reference: (1) Section 4.3 of the lecture notes.

12. Thr 5/7: Johnsen-Lindenstrauss lemma, a simple proof.
Reference: (1) Section 2 of the lecture notes. (2) Dimensionality reduction and random projection, Anupam Gupta.

13. Tue 5/12: Nearest neighbor data structure in high dimensions.
Reference: (1) Section 2.9. of the lecture notes. (2) Lecture notes by Malcolm Slaney and Michael Casey. (3) Nearest neighbor search on Wikipedia.

14. Thr 5/14: Locality sensetive hashing in Euclidean spaces.
Reference: (1) Section 2 of the lecture notes.

15. Tue 5/19: A review of all lectures.

16. Thr 5/21: Problem solving session, we discuss Homework 3.

17. Tue 5/26: Student presentation, Meng Meng and Hanzhong Xu, on spanners.

18. Thr 5/28: Student presentation, Fan Yang and Xiaotao Yang, on Tree covers.

19. Tue 6/2: Student presentation, Hung Le, Farzad Zafarani, and Baigong Zheng, on lower bounds on embedding.

20. Thr 6/4: Student presentation, Sanaz Golbabaei and Gabriel Hackebeil, on cutting surfaces to reduce topology.