CS 519: Topics in Computational Geometry (Spring 2016)

Instructor: Amir Nayyeri (Office hours: Tue 3:30pm-4:30pm at KEC 3061)

Lectures: Tue, Thr 14:00-15:20 in WGNR 285

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:

Resources:

Homework:

Lectures:

  1. Tue 3/29: Metric space definition, examples: Euclidean space, shortest paths metric of a graph, embeddings, isometry.
    Reference: (1) Chapter 1 of the lecture notes. (2) Introduction to metric embedding, Anupam Gupta.

  2. Thr 3/31: Frechet embedding, D-embedding/distoriton and examples.
    Reference: (1) Chapter 1 of the lecture notes. (2) Introduction to metric embedding, Anupam Gupta.

  3. Tue 4/5: No class.

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

  5. Tue 4/12: Probabilistic embedding into trees, and its appliation in approximation algorithms.
    Reference: (1) Random tree embedding, section 3.3, Anupam Gupta. (2) Embedding into random trees, University of Toronto.

  6. Thr 4/14: Probabilistic embedding into trees, and its appliation in approximation algorithms.
    Reference: (1) Random tree embedding, section 3.3, Anupam Gupta. (2) Embedding into random trees, University of Toronto.

  7. Tue 4/19: Probabilistic embedding into trees, a lower bound.
    Reference: (1) Chapter 3 of the lecture notes. (2) Notes on expanders, Luca Trevisan.

  8. Thr 4/21: Problem solving session, we discuss this problem set by Anastasios Sidiropoulos.

  9. Tue 4/26: Probablity recap, continuous random variables, Markov inequality, Chebychev inequality, Law of large numbers, Moment generating functions and Chernoff bounds.
    Reference: (1) This note, Vlad krokhmal.

  10. Thr 4/28: Johnsen-Lindenstrauss lemma, a simple proof.
    Reference: (1) Section 2 of the lecture notes. (2) Dimensionality reduction and random projection, Anupam Gupta.

  11. Tue 5/3: 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.

  12. Thr 5/5: 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.

  13. Tue 5/10: Problem solving session, we discuss Homework 3.

  14. Thr 5/12: No class.

  15. Tue 5/17: Randomized incremental algorithms, and the power of grids.
    Reference: (1) Smallest enclosing cycle lecture notes from Utrecht University. (2) Geometric approximation algorithms, Chapter 1, Sariel Har-peled.

  16. Thr 5/19: Polygon triangulation.
    Reference: (1) Slides by Subhash Suri.

  17. Tue 5/24: Point/line duality, and degeneracy test.
    Reference: (1) Lecture notes from ETH.