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

References

Books
[BKOS00] M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. Computational Geometry: Algorithms and Applications. Springer-Verlag, 2nd edition, 2000. [course textbook]
Surveys
[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.

Other Papers
[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.
[D34] B. N. Delaunay. Sur la Sphere vide. Izvestia Akademia Nauk SSSR, VII Seria, Otdelenie Matematicheskii i Estestvennyka Nauk 7:793-800, 1934.
[E83] H. Edelsbrunner. A new approach to rectangle intersections. International Journal Computational Mathematics 13:209-219 and 221-229, 1983.
[GJPT78] M. R. Garey, D. S. Johnson, F. P. Preparata, and R. E. Tarjan. Triangulating a simple polygon. Information Processing Letters, 7:175-179, 1978.
[G72] R. L. Graham. An efficient algorithm for determining the convex hull of a finite planar set. Information Processing Letters, 1:132-133, 1972.
[GKS92] L. J. Guibas, D. E. Knuth, M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica 7:381-413, 1992.
[J73] R. A. Jarvis. On the identification of the convex hull of a finite set of points in the plane. Information Processing Letters, 2:18-21, 1973.
[KS86] D. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM J. Computing, 12:1:287-299, 1986.
[L77] C. L. Lawson. Software for C1 surface interpolation. Mathematical Software III, J. Rice ed., Academic Press, New York, 1977, 161-194.
[M85] E. M. McCreight. Priority search trees. SIAM Journal on Computing, 14:257-276, 1985.

Links
[TOPP] The Open Problems Project by Erik D. Demaine - Joseph S. B. Mitchell - Joseph O'Rourke
[OP] Open Problems by Jeff Erickson
[OPDCG] Open Problems on Discrete and Computational Geometry by Jorge Urrutia
[SP] Sample Computational Geometry Projects from McGill University

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