CS 6050 Computational Geometry, Spring 2008

• Time and place: Mon Wed Fri 10:30am - 11:20am, Main 207
• Course website: http://www.cs.usu.edu/~mjiang/cs6050/spring2008/
• Professor: Dr. Minghui Jiang
  • Contact: mjiang at cc.usu.edu, 435-797-0347
  • Office hours: Mon Wed Fri 11:30am - 12:30pm, Main 402G
• Textbook: M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. Computational Geometry: Algorithms and Applications, Second Edition. Springer-Verlag. ISBN: 3-540-65620-0
• Course goals: The student will
  • Gain knowledge on a variety of computational and mathematical problems in discrete geometry and their applications.
  • Be able to utilize fundamental geometric data structures and algorithmic design techniques for the solution of new computational problems in discrete geometry.
  • Be able to implement basic geometric algorithms using standard programming languages.
  • Be prepared for theoretical research in discrete and computational geometry.
• Preparation: This is an advanced graduate-level course on discrete and computational geometry. Solid mathematical, algorithmic, and programming skills are required. The students are expected to explore the vast literatures of the field and work on current research problems under the guidance of the instructor. Prerequisite: CS5050.
• Grading:
  • Homework (40%):
    • Homework 1 (due at the beginning of class on Mon Jan 14): Read chapter 1. Browse through the whole book, pick your favorite chapter (the topic that you wish to be covered).
    • Homework 2 (due at the beginning of class on Fri Jan 18): What is the probability that the convex hull of k random points on the boundary of a circle encloses the circle center?
    • Homework 3 (due at the beginning of class on Wed Jan 30): Show that f(r,s)=(r+s-4 choose r-2)+1 is the solution to the recurrence f(r,s)=f(r-1,s)+f(r,s-1)-1 (Hint: induction).
    • Homework 4 (due at the beginning of class on Mon Feb 4): Prove: given a finite set of non-parallel lines on the plane not all through one point, there is a point intersected by exactly two lines.
    • Homework 5 (due at the beginning of class on Fri Feb 8): Derive the parameterized equation for an ellipse with specified origin and axes.
    • Homework 6 (due at the beginning of class on Fri Feb 15): Given a set of n points in the plane in general position, design an O(n^2) time algorithm that computes for each point the circular order of the other points.
    • Homework 7 (due at the beginning of class on Mon Mar 31): Build a rhombic dodecahedron and check whether Euler's formula works for this polyhedron.
  • Project (60%):
    • Project 1 (due Fri Feb 1): Convex hull fest.
    • Project 2 (due Mon Mar 3): Convex contour detection. Jonathan's peppers. Seth's mickey. Dr. Qi's data:
      
    • Project 3 (due Mon Mar 24): 2-Interval Pattern.
    • Final Project (proposal due Mon Mar 31): Topic of your choice.
• Course evaluation: Mon Mar 31.
• Resources:
  • Journals: IJCGA CGTA DCG
  • Conferences: SoCG CCCG
  • Papers:
    • Timothy M. Chan. Optimal output-sensitive convex hull algorithms in two and three dimensions. Discrete & Computational Geometry, 16:361-368, 1996.
    • Maxime Crochemore, Danny Hermelin, Gad Landau, Dror Rawitz, and Stephane Vialette. Approximating the 2-interval pattern problem. Theoretical Computer Science, to appear.
• Lectures (schedule subject to change):
  • Jan 7 9 (11): Introduction. Convex hull: implementation, Graham scan, merge-hull, quick-hull.
  • Jan 14 16 18: Convex hull: incremental hull, lower bound, reduction from sorting. Chan's algorithm: binary search, partitioning, doubling search.
  • Jan (21) 23 25: Erdos and Szekeres. Pigeonhole principle.
  • Jan 28 30 Feb 1: Ordinary line. Convex hull of segments.
  • Feb 4 6 8: Contour detection. Duality. Halfplane intersection and linear programming.
  • Feb 11 13 15: 2-interval pattern and RNA secondary structure.
  • Feb (18) 20 22: Approximation algorithms for weighted 2-interval pattern.
  • Feb 25 27 29: Local-improvement heuristics for weighted 2-interval pattern.
  • Mar 3 5 7: Covering a square by squares and by rectangles.
  • Mar (10 12 14): Spring Break.
  • Mar 17 19 21: Smallest enclosing circle: randomized incremental algorithm and backward analysis (section 4.7). Frechet simplification. Triangulation: art gallery theorem (section 3.1) and Euler's formula (section 9.1).
  • Mar 24 26 28: Delauney triangulation (sections 9.1, 9.2).
  • Mar 31 Apr 2 4: Delauney triangulation (sections 9.3, 9.4).
  • Apr 7 9 11: All things considered: Delauney triangulation, Voronoi diagram, convex hull, and half-space intersection (sections 7.1, 8.2, 8.5, 9.2, 11.1, 11.4, 11.5). Range searching (chapter 5).
  • Apr 14 16: Range searching (chapter 5).
  • Apr 18 21 23 25: Final project demonstration.
• Registration policy:
  • The last day to add this class is January 28.
  • The last day to drop this class without notation on your transcript is January 28.
  • Attending this class beyond January 28 without being officially registered will not be approved by the Dean's Office. Students must be officially registered for this course. No assignments or tests of any kind will be graded for students whose names do not appear on the class list.
• Code of conduct:
  • Every student should read and follow the department code of conduct.
  • Students are encouraged to discuss and exchange ideas on homework and projects, but each student must write up the solutions independently.
  • Students who are caught cheating immediately receive "Fail" grades.
• DRC statement: Students with physical, sensory, emotional or medical impairments may be eligible for reasonable accommodations in accordance with the Americans with Disabilities Act and Section 504 of the Rehabilitation Act of 1973. All accommodations are coordinated through the Disability Resource Center (DRC) in Room 101 of the University Inn, 797-2444 voice, 797-0740 TTY, or toll free at 1-800-259-2966. Please contact the DRC as early in the semester as possible. Alternate format materials (Braille, large print or digital) are available with advance notice.