CS 506, Computational Geometry, Spring 2026

Instructor

• Jared Saia
• Email: "last name" at cs.unm.edu.
• Office: FEC 3120, The best way to reach me for this class is generally via Piazza. I check it once a day, usually around noon.
• Office Hours: By appointment.
• Note: I will always be available in my office during office hours. At other times, if my door is open, feel free to come in. If the door is closed, I'm probably at work on a paper, grant or research problem. Please come by another time or make an appointment via email.

Class Info

Course Description

Text Book

Piazza Forum

• Piazza forum for this class

Syllabus

Prereq: CS 561, CS 362 or equivalent

Homework

• HW 1
• HW 2
• HW 3

Lecture Topics

• Convex Hull
• My Convex Hull Notes
• Jeff Erickson's Convex Hull Notes
• Convex Crepes!
• High Dimensional Convex Hull; See also Quickhull
• Duality
• My Halfplane Intersection and Duality Notes
• UFL lecture notes on Duality
• U. Maryland Notes, Pages 41-44 give a good connection between convex hulls, and upper/lower envelopes, Lecture 8 gives good connection between envelopes and linear programming. Lecture 16 gives good connections between convex hulls and Voronoi diagrams, and Delaunay triangulations
• Voronoi Diagrams, Delaunay Triangulations and More Dual Transformations
• My notes on Voronoi Diagrams, Delaunay Triangulations and More Dual Transformations
• Many of these problems have deep connections to each other via Projections and Duality Transformations
• Linear Programming
• My notes on Linear Programming and High-Dimensional Convex Hulls
• Linear Programming: Gupta's notes, Lecture 17
• Higher Dimensional Spaces and Johnson-Lindenstrauss
• My notes on High Dimensional Properties and the Johnson-Lindenstrauss Projection
• Arora Notes on Johnson-Lindenstrauss Projection
• You can embed an arbitrary metric into Euclidean space with O(log n) distortion (via Bourgain's theorem, see also here). Then, you can use Johnson-Lidenstrauss to project onto R^d where d = O(log n).
• Arora Notes on SVD (re low-rank approximation)
• Arora Notes on SVD (Part 2)
• SVD Low-rank Approximation
• My notes on the Singular Value Decomposition
• Arora Notes on SVD (re low-rank approximation)
• Arora Notes on SVD (Part 2)

Projects

• Project Deliverable: The class project should be no more than 10 pages not including bibliography and appendix; it can be 2 columns.
• More details on the class project and some project ideas are available here.
• MIT Algorithm's Projects This is a general description of how to find a good CS theory project. The specific project ideas in this class are, of course, different from our own class - if you'd like specific ideas, please talk to me.

Useful Links for this Class

• Asymptotic Notation (Review)
• Recurrences and Induction (Review)
• Proofs by Induction (via Jeff Erickson)
• Sanjeev Arora Notes on Convex Optimization
• My Last CS 506 Page
• Blum, Hopcroft and Kannon Foundations of Data Science
• David Mount's Computational Geometry Class with Notes and Homeworks The University of Maryland notes
• Convex Optimization Notes by Vishnoi Formal Treatment of Convex Optimization Algorithms
• MWU as Gradient Descent (Section 3):
• Convex Optimization by Boyd and Vandenberghe Particularly of interest: Section 2.3 "Operations that Preserve Convexity" ; Chapter 4 and onward discuss optimization algorithms (albeit informally, without proofs of convergence time; NB that many problems discussed (i.e. quadratic programming) are NP-Hard). See Vishnoi's notes above for a more formal treatment.