Computational and Metric Geometry
Winter Quarter 2015
Instructor: Yury Makarychev
Course: TTIC 31100 and CMSC 390101 Lectures: Tuesday & Thursdays, 10:3011:50, TTIC, room 530 Mailing list: geometryclass2015@ttic.edu . Follow http://goo.gl/QZJMb1 to subscribe to the course mailing list. 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 
Homework
 Problem Set 1 (due on Tuesday, February 3)
 Problem Set 2 (due on Thursday, February 19)
 Problem Set 3 (due on Tuesday, March 17)
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
Tentative Schedule
 January 6: Convexity
convex sets, convex hulls, vertices, supporting lines, edges, different definitions and basic properties, Caratheodory's theorem  January 8: 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 13: Planar Graphs and Overlays
graphs, graph drawings, plane and planar graphs, Euler's formula, data structure for plane graphs, computing overlays (Chapter 2)  January 15: Orthogonal Range Searching
binary search, kdtrees (Chapter 5)  January 20:Orthogonal Range Searching (continued)
range trees (Chapter 5)  January 22: Point Location
trapezoidal maps, randomized algorithm (Chapter 6)  January 27: Voronoi Diagrams
Voronoi diagrams, Fortune's algorithm (Chapter 7)  January 29: Delaunay Triangulations I
triangulations, Delaunay and locally Delaunay triangulations: definitions, existence and equivalence (Chapter 9)  February 3: 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 5: Normed Spaces. Low Distortion Metric Embeddings.
normed spaces, Lipschitz maps, distortion, embeddings into L_{p} and l_{p} (see lecture notes)  February 10: Bourgain's Theorem
Bourgain's theorem  February 12: Sparsest Cut
approximation algorithm for Sparsest Cut (see lecture notes)  February 17: 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 19: Minimum Multiway Cut, Minimum Multicut
approximation algorithms for Minimum Multiway Cut and Minimum Multicut (see lecture notes)  February 24: Padded Decomposition, Tree Metrics, Hierarchically Separated Trees (HST)
padded decomposition, HST (see lecture notes)  February 26: Padded Decomposition, Tree Metrics, Applications. Semidefinite Programming.
padded decomposition, HST, applications (see lecture notes), semidefinite programming  March 3: Semidefinite Programming, Algorithm of Arora, Rao and Vazirani
semidefinite programming, ARV (highlevel overview), delta separated sets, matching covers  March 5: Dimension Reduction, Nearest Neighbor Search
dimension reduction, approximate nearest neighbor search, locality sensitive hashing  March 10: Locality Sensitive Hashing, p–Stable Random Variables
locality sensitive hashing, p–stable random variables