Geometric Algorithms
COMPCI 634 - Fall 2018
Instructor: Pankaj K Agarwal
TA: Aaron Lowe (Office Hours: Tuesdays 1pm-2pm D301)
Time: TTh 3:05-4:20pm
Location: LSRC D243

Overview

The field of Geometric Algorithms studies the design, analysis, and implementation of algorithms and data structures for geometric problems. These problems arise in a wide range of areas, including robotics, computer graphics, molecular biology, GIS, spatial databases, sensor networks, and machine learning. In addition to the tools developed in computer science, the study of geometric algorithms also requires ideas from various mathematical disciplines, including combinatorics, topology, and algebra.

The goal of this course is to provide an overview of the techniques developed in geometric algorithms as well as some of its application areas. The topics covered in the course will include:

  1. Geometric Fundamentals: Motivation, models of computation, geometric primitives, geometric transforms
  2. Convex hulls: Planar convex hulls, higher dimensional convex hulls, output-sensitive and dynamic algorithms
  3. Intersection detection: Segment intersection, line sweep, polyhedra intersection
  4. Geometric data structures: Segment and interval trees, point location, persistent data structure, range searching, nearest-neighbor searching
  5. Proximity problems: Voronoi diagram, Delaunay triangulation and their subgraphs, spanners, well separated pair decomposition
  6. Arrangements: Arrangements of lines and hyperplanes, sweep-line and incremental algorithms, lower envelopes, levels, and zones, applications
  7. Geometric sampling: Random sampling and e-nets, e-approximation and discrepancy, coresets
  8. Geometric optimization: Linear programming, geometric set cover, shape matching, clustering
  9. High-dimensional geometry: Bourgain's theorem, random-projection, low-distortion embeddings, locality sensitive hashing

 

Textbook

The main textbook for this course: M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars, Computational Geometry: Algorithms and Applications. Springer-Verlag, 3rd ed., 2008.

Grading

Assignments: 40% weight
Four assignments will be given during the semester, which each student has to complete individually without searching the material online.

Lecture Scribe: 10% weight
Each student will scribe one lecture.

Research Project: 50% weight
Intended to produce a work of publishable quality, the project should consist of a comprehensive survey on a topic plus new research work. Due on December 7, 2018.

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