Computational and Metric Geometry

Winter Quarter 2019

Instructor: Yury Makarychev Course: TTIC 31100 and CMSC 39010-1 Lectures: Monday & Wednesday, 1:30–2:50 pm, TTIC, room 530 Office Hours: Wednesday, 3:00pm–4:00pm or by appointment, TTIC, room 437 Canvas course site: https://canvas.uchicago.edu/courses/19809 Textbook: Computational Geometry by M. de Berg, O. Cheong, M. van Kreveld, M. Overmars. Requirements: There will be 3 or 4 homework assignments. There will be no exams. Description: The course covers fundamental concepts, algorithms and techniques in computational and metric geometry. Topics covered include: convex hulls, polygon triangulations, range searching, segment intersection, Voronoi diagrams, Delaunay triangulations, metric and normed spaces, low-distortion metric embeddings and their applications in approximation algorithms, padded decomposition of metric spaces, Johnson–Lindenstrauss transform and dimension reduction, approximate nearest neighbor search and locality-sensitive hashing.

W. Kandinsky: Mild Tension, 1923

Homework

• Problem Set 1
• Problem Set 2
• Problem Set 3

Lecture Notes and References

• Basic Properties of Metric and Normed Spaces
• Bourgain's Theorem
• Sparsest Cut Problem
• Partitioning Metric Spaces
• Survey on Geometry, Flows Graph Partitioning Algorithms by S. Arora, S. Rao and U. Vazirani (Communications of ACM, Oct 2008)
• Dimension Reduction

Tentative Schedule

1. January 7: Convexity
   convex sets, convex hulls, vertices, supporting lines, edges, different definitions and basic properties, Caratheodory's theorem
2. January 9: Convex Hulls and Line Segment Intersections
   Jarvis March, Andrew's algorithm (Chapter 1.2), sweep line algorithms, line segment intersection, Bentley–Ottmann algorithm (Chapter 2.1)
3. January 14: Orthogonal Range Searching
   binary search, kd-trees, range trees (Chapter 5)
4. January 16: Point Location
   trapezoidal maps, randomized algorithm (Chapter 6)
5. January 21: Martin Luther King Jr. Day TTIC is closed.
6. January 23: Voronoi Diagrams
   Voronoi diagrams, Fortune's algorithm (Chapter 7)
7. January 28: Delaunay Triangulations I
   triangulations, Delaunay and locally Delaunay triangulations: definitions, existence and equivalence (Chapter 9)
8. January 30: The lecture was cancelled due to severe weather.
9. February 4: Delaunay Triangulations II, Metric Spaces
   duality between Delaunay triangulations and Voronoi diagrams, angle optimality (Chapter 9); metric and normed spaces–basic definitions (see lecture notes, Section 1.1)
10. February 6: Normed Spaces, Low Distortion Metric Embeddings
    normed spaces, Lipschitz maps, distortion, embeddings into L_p and l_p (see lecture notes)
11. February 11: Bourgain's Theorem
    Bourgain's theorem
12. February 13: Sparsest Cut
    approximation algorithm for Sparsest Cut (see lecture notes)
13. February 18: Minimum Balanced Cut, Minimum Linear Arrangement, Sparsest Cut with Non-Uniform demands, Expanders
    polylog approximation algorithms for Balanced Cut and Minimum Linear Arrangement, expander graphs, integrality gap for Sparsest Cut, Sparsest Cut with non-uniform demands
14. February 20: Minimum Multiway Cut, Minimum Multicut
    approximation algorithms for Minimum Multiway Cut and Minimum Multicut (see lecture notes)
15. February 25: Minimum Multiway Cut, Padded Decomposition
    padded decomposition, HST (see lecture notes)
16. February 27: Padded Decomposition, Tree Metrics, Hierarchically Separated Trees (HST)
    padded decomposition, embedding into distributions of dominating trees, HST, applications (see lecture notes)
17. March 4: Tree Metrics (cont'd), Games, von Neumann's Minimax Theorem, Multiplicative Weight Update Method
    two player zero-sum games, von Neumann's minimax theorem, multiplicative weight update method
18. March 6: Räcke's Framework, Semidefinite Programming
    Räcke's framework, approximation algorithm for Minimum Bisection, positive semidefinite matrices, semidefinite programming, Goemans–Willaimson algorithm for Max Cut
19. March 11: Dimension Reduction, Nearest Neighbor Search
    dimension reduction, approximate nearest neighbor search, locality sensitive hashing
20. March 13: Locality Sensitive Hashing, p-Stable Random Variables
    locality sensitive hashing, p-stable random variables