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:

• Lecture notes by Jiri Matousek.

Homework:

• Problems 1, 4, 9 from section one of the lecture notes.
  
• Problem 1 from these notes by Anastasios Sidiropoulos, and problem 3 from these notes by Anupam Gupta.
  
• Homework 3.

Topics for student presectations (with suggested resources):

• Tree covers, lecture notes by Ravi.
• Spanners, lecture notes by Ravi.
• Tree metrics and ultra-metrics, lecture notes by Ravi.
• Embeddings for outerplanar graphs, lecture notes by Anupam Gupta.
• Embeddings for series-parallel graphs, lecture notes by Anupam Gupta.
• Learning Mixtures of Gaussians, a paper by Sanjoy Dasgupta.

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.