CS 6050 Computational Geometry, Spring 2012

• Time and place: Tue Thu 12:00 - 1:15pm, Peterson Agriculture Science 338
• Course website: http://www.cs.usu.edu/~mjiang/cs6050/spring2012/
• Professor: Dr. Minghui Jiang
  • Contact: mjiang at cc.usu.edu, 435-797-0347
  • Office hours: Tue Thu 3:00 - 4:30pm, Main 402G
• Textbook: M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. Computational Geometry: Algorithms and Applications, 2nd or 3rd Edition, Springer-Verlag.
• Course objectives:
  • [IDEA 1 - Essential] Gaining factual knowledge (terminology, classification, methods, trends)
    • Gain knowledge on a variety of computational and mathematical problems in discrete geometry and their applications.
  • [IDEA 2 - Essential] Learning fundamental principles, generalizations, or theories
    • Be able to implement basic geometric algorithms using standard programming languages.
    • Be able to utilize fundamental geometric data structures and algorithmic design techniques for solving new computational problems in discrete geometry.
  • [IDEA 11 - Important] Learning to analyze and critically evaluate ideas, arguments, and points of view
    • Be able to recognize the conceptual connection between different computational problems.
    • 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 or consent of instructor.
• Grading:
  • (25%) Homework
    • Homework 1 (10 points; due 12 noon Tue Jan 17): Read Chapter 1 and handout. Solve Exercise 1.4 on triangle pqr.
    • Homework 2 (10 points; due 12 noon Thu Jan 26): Determine the probability that the convex hull of n random points on the boundary of a circle encloses the circle center.
    • Homework 3 (10 points; due 12 noon Tue Feb 7): Read Chapter 8 on arrangements and duality. Given a set P of n points in the plane in general position, design an O(n^2) time algorithm that computes, for each point p in P, the circular order of the other n-1 points sorted by polar angles with origin at p.
    • Homework 4 (5 points; due 12 noon Thu Feb 9): Build a Rhombic Dodecahedron.
    • Homework 5 (10 points; due 12 noon Tue Feb 14): Read Chapter 4 on linear programming. Given a convex polygon of n vertices in the plane, design an efficient algorithm that computes the largest disk contained in the polygon.
  • (75%) Project
    • Project 1 (Jan 31): Convex hull.
    • Project 2 (Mar 20): Triangulation.
    • Final Project (Apr 24 26):
      • Optimization problems on geometric intersection graphs: (Nare) circular-arc graph, (Will) circle graph, (Nate) trapezoid graph, (Daniel) box intersection graph, (Andrew) sphere intersection graph, (Goldee) dotted interval graph. (Mohammad) Convex hull for imprecise points. (Ahmet) Computational geometry on OpenCL.
      • (1) Search for research papers on your topic using Google Scholar and Interlibrary Loan. (2) Browse through research papers to survey known results. (3) Select one or two algorithms to study in depth. (4) Gather a list of open questions. (5) Either implement selected algorithms, or work on the open questions.
      • (due Apr 27 Friday 11:59pm): Submit a technical report in the format of a research article.
• Resources:
  • Journals: CGTA IJCGA DCG
  • Conferences: SOCG CCCG
  • Papers:
    • T.M. Chan. Optimal output-sensitive convex hull algorithms in two and three dimensions. Discrete & Computational Geometry, 16:361-368, 1996.
    • P. Erdős and G. Szekeres. A combinatorial problem in geometry. Compositio Mathematica, 2:463-470, 1935.
    • V. Chvatal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4:233-235, 1979.
    • G. Ramalingam and C. P. Rangan. A unified approach to domination problems on interval graphs. Information Processing Letters, 27:271-274, 1988.
    • J. Snoeyink. Maximum independent set for intervals by divide and conquer with pruning. Networks, 49:158-159, 2007.
    • M. C. Francis, D. Gonçalves, P. Ochem. The maximum clique problem in multiple interval graphs. arXiv:1201.0043, 2012.
    • R. Bar-Yehuda, M. M. Halldórsson, J. (S.) Naor, H. Shachnai, and I. Shapira. Scheduling split intervals. SIAM Journal on Computing, 36:1-15, 2006.
    • N. Alon. Piercing d-intervals. Discrete & Computational Geometry, 19:333-334, 1998.
    • D. R. Gaur, T. Ibaraki, and R. Krishnamurti. Constant ratio approximation algorithms for the rectangle stabbing problem and the rectilinear partitioning problem. Journal of Algorithms, 43:138-152, 2002.
• Lectures (schedule subject to change; no classes on [*] dates):
  • Jan 10 12: Introduction. Convex hull: implementation, gift wrapping (without sorting), upper/lower chains (with sorting), Graham scan (with sorting), merge-hull (with or without pre-sorting), quick-hull (without sorting), incremental (with sorting and search tree).
  • Jan 17 19: Chan's output-sensitive convex hull algorithm: grouping, binary search, doubling search. More on convex hull: convex layers, dynamic convex hull, lower bound. Binary search without upper bound. Tiling a hexagon by this thing. Testing ostrich eggs.
  • Jan 24 26: Pigeonhole principle. Erdos and Szekeres: monotone subsequence, convex subset.
  • Jan 31 Feb 2: Project 1 demonstration. Ordinary line. Duality (sections 8.2, 8.3). Halfplane intersection (section 4.2).
  • Feb 7 9: Randomized incremental algorithm and backward analysis: linear programming and smallest enclosing disk (sections 4.4, 4.7).
  • Feb 14 16: Double counting and handshake theorem. Euler's formula. Dual of planar graph. Pick's theorem. Dual of ordinary line revisited. Art gallery theorem (section 3.1).
  • Feb [21] 23: Delaunay triangulation (sections 9.1, 9.2).
  • Feb 28 Mar 1: Delaunay triangulation (sections 9.3, 9.4).
  • Mar 6 8: All things considered: Delaunay triangulation, Voronoi diagram, convex hull, and half-space intersection (sections 7.1, 8.2, 8.5, 9.2, 11.1, 11.4, 11.5).
  • Mar [13 15]: No classes for the week (spring break).
  • Mar 20 22: Project 2 demonstration. Overview of research on discrete and computational geometry.
  • Mar 27 29: Optimization problems on general graphs: O(n / log n)-approximations for clique ω, independent set α, coloring χ, clique partition ϑ; 2-approximation for vertex cover; H_n-approximation for dominating set γ and set cover. Duality and complement: χ versus ω; ϑ versus α.
  • Apr 3 5: Optimization problems on interval graphs: with interval representation O(n) after sorting by greedy for ω χ and α ϑ.
  • Apr 10 12: Optimization problems on interval graphs: without interval representation O(n + m) by dynamic programming for γ; with interval representation O(n) after sorting by dynamic programming for weighted α; O(n log k) without sorting by divide and conquer with pruning for α. Mycielskian. Optimization problems on multiple-interval graphs: double-counting and pigeon-hole principle and König's theorem for ω χ.
  • Apr 17 19: Optimization problems on multiple-interval graphs: linear programming and minimax theorem for α ϑ and τ.
  • Apr 24 26: Final project demonstration.
• Registration policy:
  • The last day to add this class is January 30, 2012.
  • The last day to drop this class without notation on your transcript is January 30, 2012.
  • Attending this class beyond January 30, 2012 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 and cheating policy.
  • 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 (the instructor is absolutely firm on this policy; he has previously given two "Fail" grades for this reason).
• 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.