CS507A: Computational Geometry Lectures
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other students, send them to me and I'll include them on this page.
Lecture 1: 5/9/2
- Course description.
- What is Computational Geometry?
- Models of computation:
- The Cross
Product and applications.
- Convex Hull
CH(S) of a finite set S of points.
Lecture 2: 9/9/2
- properties of CH(S)+ algos :
- definition of a vertex of CH(S).
- A pt is a vertex iff it is not inside any triangle
=> O(n^4) algo.
- properties of an edge => O(n^3) algo.
- Jarvis march.
- Graham scan original
- Monotone Graham scan (linesweep, triangulation)
Lecture 3: 12/9/2
- Course projects.
- MergeHull, merging two convex polygons.
- Lower bounds:
- Adversary arguments.
- Algebraic decision trees, Dobkin-Lipton's theorem
- Read: Michiel Smid's lecture notes on lower bounds.
Lecture 4: 17/9/2 (covered by
Lecture 5: 19/9/2 (covered by
- Two ears theorem, method for finding two ears.
- Proof that triangulation is always possible (ear-cutting and
- Triangulate by finding diagonal.
- dual tree of a triangulation and 2 leaves = ears.
- O(n^3) triangulation, brute force.
- O(n^2) triangulation by finding ear in O(n), which needed:
- finding ear in O(n) by GSP method.
- Art gallery, n/3 guards by coloring.
- Decomposing a monotone polygon into monotone mountains.
- introduction of Weakly Externally Visible (WEV). You can use
3coins to find the CH of a WEV polygon.
- See also notes and links on Greg's
Lecture 6: 24/9/2
- Further motivation for triangulations.
- Triangulation of the exterior of a WEV polygon using the 3 coins
algorithm, in O(n) time.
- Weakly Internally Visible (WIV) polygons (visible from one of its
edges. O(n) 3 coins algorithm for triangulating those.
- Monotone polygons, O(n) time decomposition into WIV
- Linesweep to decompose a polygon into monotone pieces.
- Read: Textbook chapter 3
Lecture 7: 26/9/2
- Triangulation in O(nR) where R = # of reflex vertices, using
modified 3 coins.
- More on linesweep:
- Line segment intersection by linesweep.
- Degeneracies for line segment intersection.
- O(n) space for line segment intersection.
- Omega(n log(n)) lower bound for detecting a line segment
intersection, in the algebraic decision tree model.
- Planar graphs: Euler's formula
Lecture 8: 1/10/2
- Data structures for planar graphs, Doubly Connected Edge List
(DCEL), augmenting for fast walking. Space analysis.
- Point location data structures:
- By Triangle testing - O(n) query time and space.
- Ray shooting
- O(log(n)) query time and O(n) space for star-shaped polygons
- Dobkin-Lipton (slabs) - O(n^2) space and O(log(n)) query time
- Kirkpatrick's: Simplifying triangulations (analysis next time).
Lecture 9: 3/10/2
- Kirkpatrick's point location data structure:
- Construction by repeatedly removing independent vertices of
low degree, pointer structure.
- Performing searches.
- Proof that there exist at least n/24 vertices of degree
- Query time, depth of the search structure.
- Trapezoidal decompositions, search structure and incremental
Lecture 10: 8/10/2
- Randomized incremental trapezoidal decomposition algorithm,
review and analysis.
- Read: Book chapter 6.
Lecture 11: 10/10/2
- Triangulations of sets of points
- Delaunay triangulations:
- Maximizing the smallest angle
- Empty circle property
- Incremental construction
- Read: Book chapter 9.
Lecture 12: 15/10/2
Lecture 13: 17/10/2
- Midterm solutions.
- More on Delaunay triangulations.
Lecture 14: 22/10/2
- Analysis of randomized incremental Delaunay triangulation
- Delaunay triangulation by lifting points on a paraboloid and
computing the 3D convex hull.
Lecture 15: 24/10/2
- Voronoi diagrams.
- Read: Book chapter 7.
Lecture 16: 29/10/2
- Duality transforms:
- Motivation: intersection of halfplanes
- Point to line duality: (a,b) to y=ax+b
- From the incidence property, derived line to point: y=cx+d to
- vertical distance
- Dual vision of
- Vertical lines to points at infinity (homogeneous
- Collinear points
- Intersection of halfplanes to convex hull
- Brief mention of higher dimension duality transforms, point to
planes and Pluecker coordinates.
- Next time: Linear programming
Lecture 17: 31/10/2
- Linear Programming:
- Applicationn: Carving pumpkins.
- Definition, Simplex method (can be slow)
- Linear programming in R^1 in O(n)
- In R^2: Convex Hull -> O(n log n)
- In R^2: incremental -> O(n^2)
- In R^2: randomized incremental -> O(n)
- In R^d: randomized incremental -> O(d! n)
- Briefly: smallest enclosing circle
- Read: Textbook chapter 4
Lecture 18: 5/11/2
- Definitions: vertices, edges, faces
- Complexity: number of vertices, edges, faces
- Data structure: DCEL
- Construction: linesweep -> O(n^2 log n)
- Construction: incremental
- Zone theorem: complexity of a zone is O(n) => complexity
incremental construction is O(n^2)
- Read: Textbook chapter 8.
- Linear regression for n points:
- Least squares, not robust
- Thiel-Sen estimator: draw a line through each pair of points,
- find the median of their slopes
- The Thiel estimator is a line of that slope with at most n/2
points above and below it.
- Computation: list all O(n^2) lines to find median slope ->
O(n^2) time and space
- Dual: n lines. Find median vertex according to x coordinate.
Linesweep. O(n^2 log n) time and O(n) space.
- Counting vertices to the left of a vertical line by counting
inversions during sorting. O(n log n) time.
Webpage by David Eppstein on robust regression. Read the
"Median slope" paragraph for references to papers.
- Read: On this
Webpage by Jiri Matousek, you will find "Older lecture
notes on incremental geometric algorithms and some algorithms
for planar arrangements". Read chapter 10.2, p127, about
Lecture 19: 7/11/2
- Vertex selection
- Review of last time
- Ranking a vertex (how many are to its left), deciding if the
kth vertex is to the left or to the right of a vertical line
- Counting vertices inside a vertical slab
- Counting vertices -> generating a random vertex
- Randomized selection of the kth vertex in O(n log^2 n)
- Ham-sandwich cuts
- Bisectors of a set of points
- Dual of a bisector, median level
- Ham-sandwich cut = intersection of median levels in the
dual, existence, odd intersections
- vertical sidedness
- O(n log^2 n) randomized algorithm
Lecture 20: 12/11/2