Computational and Metric Geometry

Spring Quarter 2013

Instructor: Yury Makarychev
Lectures: Tuesday & Thursdays, 10:30-11:50, TTIC, room 530
Mailing list: geometry-class-spring-2013@ttic.edu . Follow this link https://groups.google.com/a/ttic.edu/group/geometry-class-spring-2013/subscribe to subscribe to the course mailing list.
Course: TTIC 31100 and CMSC 39010-1

There will be no lecture on Tuesday, June 4.

Description

The course covers fundamental concepts and algorithms in computational geometry. Topics covered include: convex hulls, polygon triangulations, range searching, segment intersection, Voronoi diagrams, Delaunay triangulations, metric and normed spaces, metric embeddings and applications, dimension reduction, locality–sensitive hashing.

Textbooks

The course textbooks are Computational Geometry by M. de Berg, O. Cheong, M. van Kreveld, M. Overmars, and Lectures on Discrete Geometry by J. Matoušek.

Requirements

There will be 3 or 4 homework assignments.

Homeworks

• Homework 1 (due on Tuesday, April 30)
• Homework 2 (due on Tuesday, May 14)
• Homework 3 (due on Thursday, May 30)

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 Arora, Rao and Vazirani (Communications of ACM, Oct 2008)
• Dimension Reduction

Lectures

1. April 2: Convexity
   convex sets, convex hulls, vertices, supporting lines, edges, different definitions and basic properties, Caratheodory's theorem
2. April 4: 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. April 9: Planar Graphs and Overlays
   graphs, graph drawings, plane and planar graphs, Euler's formula, data structure for plane graphs, computing overlays (Chapter 2)
4. April 11: Orthogonal Range Searching
   binary search, kd-trees, range trees (Chapter 5)
5. April 18: Point Location
   trapezoidal maps, randomized algorithm (Chapter 6)
6. April 23: Voronoi Diagrams
   Voronoi diagrams, Fortune's algorithm (Chapter 7)
7. April 25: Delaunay Triangulations I
   triangulations, Delaunay and locally Delaunay triangulations: definitions, existence and equivalence (Chapter 9)
8. April 30: 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)
9. May 2: Normed Spaces. Low Distortion Metric Embeddings.
   normed spaces, Lipschitz maps, distortion, embeddings into l_∞ and l_p (see lecture notes)
10. May 7: Bourgain's Theorem
    Bourgain's theorem
11. May 9: Sparsest Cut
    approximation algorithm for Sparsest Cut (see lecture notes)
12. May 14: 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
13. May 16: Minimum Multiway Cut, Minimum Multicut
    approximation algorithms for Minimum Multiway Cut and Minimum Multicut (see lecture notes)
14. May 21: Padded Decomposition, Tree Metrics, Hierarchically Separated Trees (HST)
    padded decomposition, HST (see lecture notes)
15. May 23: Padded Decomposition, Tree Metrics, Applications. Semi-definite Programming.
    padded decomposition, HST, applications (see lecture notes), semi-definite programming
16. May 28: Semidefinite Programming, Algorithm of Arora, Rao and Vazirani
    semi-definite programming, ARV (high-level overview), delta separated sets, matching covers
17. May 30: Dimension Reduction, Nearest Neighbor Search
    dimension reduction, approximate nearest neighbor search, locality sensitive hashing
18. June 6: Locality Sensitive Hashing, p–Stable Random Variables
    locality sensitive hashing, p–stable random variables