Computational and Metric Geometry
Winter Quarter 2019
Instructor: Yury Makarychev
Course: TTIC 31100 and CMSC 390101 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, lowdistortion metric embeddings and their applications in approximation algorithms, padded decomposition of metric spaces, Johnson–Lindenstrauss transform and dimension reduction, approximate nearest neighbor search and localitysensitive hashing.

W. Kandinsky: Mild Tension, 1923 
Homework
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
 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, kdtrees, 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: The lecture was cancelled due to severe weather.
 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)  February 6: Normed Spaces, Low Distortion Metric Embeddings
normed spaces, Lipschitz maps, distortion, embeddings into L_{p} and l_{p} (see lecture notes)  February 11: Bourgain's Theorem
Bourgain's theorem  February 13: Sparsest Cut
approximation algorithm for Sparsest Cut (see lecture notes)  February 18: Minimum Balanced Cut, Minimum Linear Arrangement, Sparsest Cut with NonUniform demands, Expanders
polylog approximation algorithms for Balanced Cut and Minimum Linear Arrangement, expander graphs, integrality gap for Sparsest Cut, Sparsest Cut with nonuniform demands  February 20: Minimum Multiway Cut, Minimum Multicut
approximation algorithms for Minimum Multiway Cut and Minimum Multicut (see lecture notes)  February 25: Minimum Multiway Cut, Padded Decomposition
padded decomposition, HST (see lecture notes)  February 27: Padded Decomposition, Tree Metrics, Hierarchically Separated Trees (HST)
padded decomposition, embedding into distributions of dominating trees, HST, applications (see lecture notes)  March 4: Tree Metrics (cont'd), Games, von Neumann's Minimax Theorem, Multiplicative Weight Update Method
two player zerosum games, von Neumann's minimax theorem, multiplicative weight update method  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  March 11: Dimension Reduction, Nearest Neighbor Search
dimension reduction, approximate nearest neighbor search, locality sensitive hashing  March 13: Locality Sensitive Hashing, pStable Random Variables
locality sensitive hashing, pstable random variables