Computational and Metric Geometry
Winter Quarter 2017
Instructor: Yury Makarychev
Course: TTIC 31100 and CMSC 39010-1
Lectures: Monday & Wednesday, 1:30–2:50 pm, TTIC, room 530
Mailing list: firstname.lastname@example.org . To subscribe to the course mailing list, send an email to email@example.com (with any subject and body). You should get a confirmation email in a few minutes; reply to this email or click on a link inside. If you cannot subscribe to the mailing list, please contact me.
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
- Problem Set 1 (due on Monday, February 6)
- Problem Set 2 (due on Wednesday, February 22)
- Problem Set 3 (due on Wednesday, March 8)
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 4: 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 11: Planar Graphs and Overlays
graphs, graph drawings, plane and planar graphs, Euler's formula, data structure for plane graphs, computing overlays (Chapter 2)
- January 16: Martin Luther King Jr. Day. TTIC is closed.
- January 18: SODA 2017. No lecture.
- January 23: Orthogonal Range Searching
binary search, kd-trees, range trees (Chapter 5)
- January 25: Point Location
trapezoidal maps, randomized algorithm (Chapter 6)
- January 30: Voronoi Diagrams
Voronoi diagrams, Fortune's algorithm (Chapter 7)
- February 1: Delaunay Triangulations I
triangulations, Delaunay and locally Delaunay triangulations: definitions, existence and equivalence (Chapter 9)
- February 6: 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 8: Normed Spaces. Low Distortion Metric Embeddings.
normed spaces, Lipschitz maps, distortion, embeddings into Lp and lp (see lecture notes)
- February 13: Bourgain's Theorem
- February 15: Sparsest Cut
approximation algorithm for Sparsest Cut (see lecture notes)
- February 20: 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 22: Minimum Multiway Cut, Minimum Multicut
approximation algorithms for Minimum Multiway Cut and Minimum Multicut (see lecture notes)
- February 27: Padded Decomposition, Tree Metrics, Hierarchically Separated Trees (HST)
padded decomposition, HST (see lecture notes)
- March 1: Padded Decomposition, Tree Metrics, Applications. Semi-definite Programming.
padded decomposition, HST, applications (see lecture notes), semi-definite programming
- March 6: Dimension Reduction, Nearest Neighbor Search
dimension reduction, approximate nearest neighbor search, locality sensitive hashing
- March 8: Locality Sensitive Hashing, p-Stable Random Variables
locality sensitive hashing, p-stable random variables