# Computational and Metric Geometry

## Spring Quarter 2013

**Instructor:**Yury Makarychev

**Lectures:**Tuesday & Thursdays, 10:30-11:50, TTIC, room 530

**Mailing list:**geometry-class-spring-2013@ttic.edu . Follow this link https://groups.google.com/a/ttic.edu/group/geometry-class-spring-2013/subscribe to subscribe to the course mailing list.

**Course:**TTIC 31100 and CMSC 39010-1

There will be no lecture on Tuesday, June 4.

## Description

The course covers fundamental concepts and algorithms in computational geometry. Topics covered include: convex hulls, polygon triangulations, range searching, segment intersection, Voronoi diagrams, Delaunay triangulations, metric and normed spaces, metric embeddings and applications, dimension reduction, locality–sensitive hashing.### Textbooks

The course textbooks are*Computational Geometry*by M. de Berg, O. Cheong, M. van Kreveld, M. Overmars, and

*Lectures on Discrete Geometry*by J. Matoušek.

### Requirements

There will be 3 or 4 homework assignments.## Homeworks

- Homework 1 (due on Tuesday, April 30)
- Homework 2 (due on Tuesday, May 14)
- Homework 3 (due on Thursday, May 30)

## 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

## Lectures

**April 2: Convexity***convex sets, convex hulls, vertices, supporting lines, edges, different definitions and basic properties, Caratheodory's theorem***April 4: 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)***April 9: Planar Graphs and Overlays**

*graphs, graph drawings, plane and planar graphs, Euler's formula, data structure for plane graphs, computing overlays (Chapter 2)***April 11: Orthogonal Range Searching***binary search, kd-trees, range trees (Chapter 5)***April 18: Point Location***trapezoidal maps, randomized algorithm (Chapter 6)***April 23: Voronoi Diagrams***Voronoi diagrams, Fortune's algorithm (Chapter 7)***April 25: Delaunay Triangulations I***triangulations, Delaunay and locally Delaunay triangulations: definitions, existence and equivalence (Chapter 9)***April 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)***May 2: Normed Spaces. Low Distortion Metric Embeddings.***normed spaces, Lipschitz maps, distortion, embeddings into l*_{∞}and l_{p}(see lecture notes)**May 7: Bourgain's Theorem***Bourgain's theorem***May 9: Sparsest Cut***approximation algorithm for Sparsest Cut (see lecture notes)***May 14: 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***May 16: Minimum Multiway Cut, Minimum Multicut***approximation algorithms for Minimum Multiway Cut and Minimum Multicut (see lecture notes)***May 21: Padded Decomposition, Tree Metrics, Hierarchically Separated Trees (HST)***padded decomposition, HST (see lecture notes)***May 23: Padded Decomposition, Tree Metrics, Applications. Semi-definite Programming.***padded decomposition, HST, applications (see lecture notes), semi-definite programming***May 28: Semidefinite Programming, Algorithm of Arora, Rao and Vazirani***semi-definite programming, ARV (high-level overview), delta separated sets, matching covers***May 30: Dimension Reduction, Nearest Neighbor Search***dimension reduction, approximate nearest neighbor search, locality sensitive hashing***June 6: Locality Sensitive Hashing,***p*–Stable Random Variables*locality sensitive hashing, p–stable random variables*