Computational and Metric Geometry
Instructor: Yury Makarychev
Course: TTIC 31100 and CMSC 390101 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 
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
 Lecture 1: Convexity I
convex sets, convex hulls, different definitions and basic properties, extreme points, Caratheodory's theorem, Radon's theorem  Lecture 2: Convexity II
extreme points, KreinMilman theorem, Helly's theorem, separating hyperplanes  Lecture 3: 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)  Lecture 4: Orthogonal Range Searching
binary search, kdtrees, range trees (Chapter 5)  Lecture 5: Voronoi Diagrams
Voronoi diagrams, Fortune's algorithm (Chapter 7)  Lecture 6: Delaunay Triangulations I
triangulations, Delaunay and locally Delaunay triangulations: definitions, existence and equivalence (Chapter 9)  Lecture 7: 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)  Lecture 8: Normed Spaces, Low Distortion Metric Embeddings
normed spaces, Lipschitz maps, distortion, embeddings into L_{p} and l_{p} (see lecture notes)  Lecture 9: Bourgain's Theorem
Bourgain's theorem  Lecture 10: Sparsest Cut
approximation algorithm for Sparsest Cut (see lecture notes)  Lecture 11: 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  Lecture 12: Minimum Multiway Cut, Minimum Multicut
approximation algorithms for Minimum Multiway Cut and Minimum Multicut (see lecture notes)  Lecture 13: Minimum Multiway Cut, Padded Decomposition
padded decomposition, HST (see lecture notes)  Lecture 14: Padded Decomposition, Tree Metrics, Hierarchically Separated Trees (HST)
padded decomposition, embedding into distributions of dominating trees, HST, applications (see lecture notes)  Lecture 15: Games, von Neumann's Minimax Theorem, Multiplicative Weight Update Method
two player zerosum games, von Neumann's minimax theorem, multiplicative weight update method  Lecture 16: Räcke's Framework, Semidefinite Programming
Räcke's framework, approximation algorithm for Minimum Bisection, positive semidefinite matrices, semidefinite programming, Goemans–Willaimson algorithm for Max Cut  Lecture 17: Dimension Reduction, Nearest Neighbor Search
dimension reduction, approximate nearest neighbor search, locality sensitive hashing  Lecture 18: Locality Sensitive Hashing, pStable Random Variables
locality sensitive hashing, pstable random variables