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Geometric Algorithms

Compsci 634 – Spring 2024

Instructor:  Pankaj K. Agarwal
TA:  Keegan Yao
Time:  Tues, Thurs 3:05-4:20pm
Location:  LSRC D106

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. Geometric data structures: Segment and interval trees, point location, persistent data structure, orthogonal range searching, nearest-neighbor searching, geometric cuttings, simplex range searching
  3. Intersection detection: Segment intersection, line sweep, randomized incremental algorithm
  4. Convex hulls: Planar convex hulls, higher dimensional convex hulls, output-sensitive and dynamic algorithms
  5. Proximity problems:  Voronoi diagram, Delaunay triangulation and their subgraphs, well separated pair decomposition, locality sensitive hashing
  6. Arrangements: Arrangements of lines and hyperplanes, sweep-line and incremental algorithms, lower envelopes, levels, and zones, union of objects
  7. Geometric sampling: Random sampling and e-nets, e-approximation and discrepancy, coresets
  8. Geometric optimization: Linear programming, LP-type problems, shape matching, clustering, low-distortion embeddings

Textbooks

  1. M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars, Computational Geometry: Theory and Applications. Springer-Verlag, 3rd ed., 2008.
  2. Har-Peled. Geometric Approximation Algorithms. American Mathematical Society, 2011.

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 April 22, 2024.