Schedule (rough: expect impromptu changes)
| Date | Lecture Topic | |
| CONVEX HULLS ALGORITHMS [BKOS00, Chapter 1] | ||
| Aug 23 Th | Introduction, syllabus, course structure, etc. Computational problems, Algorithms, asymptotic complexity measures; Begin Convex Hulls; Convexity; Intuitive algorithms. Orientation test; Jarvis' march [J73 pdf ] Graham's scan [G72, pdf1 pdf2 A79] | |
| Aug 28 Tu | Convex hulls Details: Divide & conquer; Chan's alg.[C96], Membership of point in convex region; Dynamic hull maintanence. | |
| Aug 30 Th | Convex hulls Lower bounds and close relationship to sorting; Reductions; Extreme Point problem and its relation to Convex Hull; How geometrization of combinatorial problems helps in proving lower bounds. | |
| Sep 4 Tu | Geometrization of Element Distinctness and Extreme Point and
proving lower bounds for both | |
| Sep 6 Th | Relationships between Membership (of point in region),
Intersection (of 2 regions),
Subdivision (of region into subregions), Pointset Partition, Point location
(of point in subdivision); Membership in convex and starshaped polygons.
| |
| PLANE-SWEEP ALGORITHMS [BKOS00, Chapters 2 and 3 ] | ||
| Sep 11 Tu | Line segment intersections Plane-sweep [BO79]; membership in simple polygons. | |
| Sep 13 Th | TEST 1 | |
| Sep 18 Tu | Polygon Triangulation Triangulating monotone polygons [GJPT78] Partitioning simple polygons | |
| Sep 20 Th | Convex Partitioning Lower and upper bounds, A factor 4 approximation algorithm | |
| LINEAR PROGRAMMING [BKOS00, Chapter 4 ] | ||
| Sep 25 Tu | Manufacturing with Molds Necessary and Sufficient condition, Half-Plane Intersections | |
| Sep 27 Thu | Linear Programming Feasible Region, Optimal solution; Incremental and randomized algorithms | |
| ORTHOGONAL SEARCH [BKOS00, Chapters 5 and 10] | ||
| Oct 2 Tu | Geometric data structures; Range search Quad-tree; kd-tree [B75]; | |
| Oct 4 Thu | Improvements on range searching Range tree; fractional cascading [CG86] | |
| Oct 9 Tu | Inverse Range Search Segment tree [B77]; interval tree [E83]; priority search tree [M85] | |
| Oct 11 Thu | TEST 2 | |
| VORONOI DIAGRAMS & DELAUNAY TRIANGULATIONS [BKOS00, Chapters 7, 9, and 14] | ||
| Oct 16 Tu | Voronoi diagrams
Voronoi diagrams [AK00]; furthest point Voronoi diagram, Other distance metrics | |
| Oct 18 Thu | Voronoi diagrams
Fortune's plane sweep algorithm | |
| Oct 23 Tu | Delaunay triangulation Empty circles, local Delaunayhood [D34], edge-flip [L77], lifting, analysis, maxmin angles | |
| Oct 25 Th | Randomized incremental algorithm Incremental construction [GKS92]; backward analysis [S93] Relating Voronoi diagrams to Convex hull (unifying geometric problems and algorithms using duality and arrangements). | |
| Oct 30 Tu (possible that we move to ARRANGEMENTS already this week) | Point Location DAG structure for point location in triangulations [BKOS00, Chapters 6 and 9] Steiner triangulations Steiner triangulation [BE95]; quality measure; quad-trees [BE95]; | |
| Nov 1 Th | Delaunay refinement Circumcenter insertion [R95]; Sphere packing argument | |
| ARRANGEMENTS (possible that we use the previous week for this as well) [BKOS00, Chapter 8] | ||
| Nov 6 Tu | Zones Duality; line arrangements; complexity; incremental algorithm; zone theorem [ESS93] | |
| Nov 8 Th | Levels and discrepancy Geometric sampling | |
| Nov 13 Th | More on arrangements, duality, sampling | |
| Nov 15 Th | TEST 3 | |
| OTHER GEOMETRY APPLICATIONS [BKOS00, Chapters 13 and 15] | ||
| Nov 20 Tu | Geometric Approximation Algorithms TSP, Metric TSP, Euclidean TSP Polynomial Time Approximation Scheme (PTAS) | |
| Nov 22 Thu | THANKSGIVING | |
| Nov 27 Tu | Geometrization and its uses | |
| Nov 29 Th | Distance geometry, combinatorial rigidity | |
| Dec 4 Tu | Partial metric space embeddings and applications | |
| Dec 11 Tu | FINAL EXAM IN CLASS DURING EXAM WEEK |
BooksLecture Slides by Marc van Kreveld (one of the authors of the textbook) Surveys
[BKOS00] M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. Computational Geometry: Algorithms and Applications. Springer-Verlag, 3rd edition, 2008. [course textbook]
[E02] Lecture Notes on ConvexHulls by J. Erickson. [E07] Lecture Notes on Polygon Triangulations by J. Erickson. [BE95] M. Bern and D. Eppstein. Mesh generation and optimal triangulation. Computing in Euclidean Geometry (2nd ed.), D.-Z. Du and F. Hwang (eds.), World Scientific, 1995, 47-123. [E00] H. Edelsbrunner. Triangulations and meshes in computational geometry. Acta Numerica (2000), 133-213.
[A79] A. M. Andrew. Another efficient algorithm for convex hulls in two dimensions. Information Processing Letters, 9:216-219, 1979. [A left-to-right variant of Graham's scan] [AK00] F. Aurenhammer and R. Klein. Voronoi Diagrams. Handbook of Computational Geometry, Ed. J. Sack, J. Urrutia (eds.), 2000, 201-290. [BDH96] B. Barber, D. Dobkin, and H. Huhdanpaa. The Quickhull Algorithm for Convex Hulls. ACM Transactions on Mathematical Software Vol. 22, No. 4, December 1996, Pages 469¿483. [B75] J. L. Bentley. Multidimensional binary search trees used for associative searching. Commun. ACM, 18:509-517, 1975. [B77] J. L. Bentley. Solution to Klee's rectangle problems. Tech. Rep., Carnegie-Mellon Univ., Pittsburgh, 1975. [BO79] J. L. Bentley and T. A. Ottmann. Algorithms for reporting and counting geometric intersections. IEEE Transactions on Computers, C-28:643-647, 1979. [C96] T. Chan. Optimal output-sensitive convex hull algorithms in two and three dimensions. Discrete and Computational Geometry, 16:361-368, 1996. [CE92] B. Chazelle and H. Edelsbrunner. An optimal algorithm for intersecting line segments in the plane. Journal of the ACM 39:1-54, 1992. [CG86] B. Chazelle and L. J. Guibas. Fractional cascading. Algorithmica 1:133-162 and 163-191, 1986.