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.
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
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
- January 7: Convexity
convex sets, convex hulls, vertices, supporting lines, edges, different definitions and basic properties, Caratheodory's theorem
- 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)
- January 14: Orthogonal Range Searching
binary search, kd-trees, range trees (Chapter 5)
- January 16: Point Location
trapezoidal maps, randomized algorithm (Chapter 6)
- January 21: Martin Luther King Jr. Day. TTIC is closed.
- January 23: Voronoi Diagrams
Voronoi diagrams, Fortune's algorithm (Chapter 7)
- January 28: Delaunay Triangulations I
triangulations, Delaunay and locally Delaunay triangulations: definitions, existence and equivalence (Chapter 9)
- January 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)
- February 4: Normed Spaces. Low Distortion Metric Embeddings.
normed spaces, Lipschitz maps, distortion, embeddings into Lp and lp (see lecture notes)
- February 6: Bourgain's Theorem
- February 11: Sparsest Cut
approximation algorithm for Sparsest Cut (see lecture notes)
- February 13: 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
- February 18: Minimum Multiway Cut, Minimum Multicut
approximation algorithms for Minimum Multiway Cut and Minimum Multicut (see lecture notes)
- February 20: Padded Decomposition, Tree Metrics, Hierarchically Separated Trees (HST)
padded decomposition, HST (see lecture notes)
- February 25: Padded Decomposition, Tree Metrics, Applications. Semidefinite Programming.
padded decomposition, HST, applications (see lecture notes), semidefinite programming
- February 27: Semidefinite Programming
- March 4: Measure concentration
- March 6: Dimension Reduction, Nearest Neighbor Search
dimension reduction, approximate nearest neighbor search, locality sensitive hashing
- March 11: Locality Sensitive Hashing, p-Stable Random Variables
locality sensitive hashing, p-stable random variables
- March 13: TBD