Reading

Books and Surveys

  1. F. Aurenhammer and R. Klein. Voronoi Diagrams, CRC Press.
  2. B. Chazelle. The Discrepancy Method: Randomness and Complexity Cambridge Univ.Press, 2001. [C]
  3. M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag, 3rd ed., 2008. (course textbook) [BCKO]
  4. H. Edelsbrunner. Geometry and Topology for Mesh Generation. Cambridge Univ. Press, England, 2001. [Ed]
  5. J. E. Goodman and J. O'Rourke (eds.), Handbook of Discrete and Computational Geometry, 2nd edition, CRC Press LLC, Boca Raton, FL, 2004. [GO]
  6. Sariel Har-peled. Geometric Approximation Algorithms (Mathematical Surveys and Monographs). American Mathematical Society (2011). [HP]
  7. J. Matousek, Lectures on Discrete Geometry, Springer, 2002. [Ma]
  8. J. O'Rourke, Computational Geometry in C, 2nd edition, Cambridge University Press, New York, 1998. [O]
  9. J. Pach and P. K. Agarwal, Combinatorial geometry, Wiley-Interscience, New York, 1995. [PA]
  10. F. P. Preparata and M. I. Shamos. Computational Geometry: An Introduction. Springer-Verlag, New York, 1985. [PS]
  11. J.-R. Sack and J. Urrutia (eds.), Handbook of Computational Geometry. North-Holland, 2000. [SU]

top

Papers

Geometric Fundamentals

  1. A. Björner, L. Lovász, and A.C. Yao. Linear decision trees: Volume estimates and topological bounds. In Proc. 24th Annu. ACM Sympos. Theory Comput., pp. 170-177, 1993.
  2. B. Chazelle, L. J. Guibas, D.T. Lee. The Power of Geometric Duality. BIT 25 (1985), 76-90. [CGL]
  3. B. Chazelle and B. Rosenberg. Computing partial sums in multidimensional arrays. In Proc. 5th Annu. ACM Sympos. Comput. Geom., pp. 131-139, 1989.
  4. F. P. Preparata and M.I. Shamos. Computational Geometry: An Introduction. Springer-Verlag, New York, 1985.
  5. A. C. Yao. Decision tree complexity and Betti numbers. In Proc. 25th Annu. ACM Sympos. Theory Comput., 1994.
  6. R. Seidel, M. Sharir. Top-Down Analysis of Path Compression. In SIAM J. Comp., 2005.

top

Geometric Data Structures I

  1. P. K. Agarwal. Range searching, CRC Handbook of Discrete and Computational Geometry, 2nd ed., (J. Goodman and J. O'Rourke, eds.), CRC Press, New York, 2004, pp.809-837.
  2. P. K. Agarwal and J. Erickson. Geometric range searching and its relatives. Advances in Discrete and Computational Geometry, Contemp. Math. 223, Amer. Math. Soc. Press, 1999, pp. 1-56.
  3. B. Chazelle and L. J. Guibas. Fractional cascading: I. A data structuring technique. Algorithmica, 1:133-162, 1986.
  4. B. Chazelle and L. J. Guibas. Fractional cascading: II. Applications. Algorithmica, 1:163-191, 1986.
  5. H. Edelsbrunner, L. J. Guibas, and J. Stolfi. Optimal point location in a monotone subdivision. SIAM J. Comput., 15:317-340, 1986.
  6. E. M. McCreight. Priority search trees. SIAM Journal on Computing, 14(2):257-276, 1985.
  7. K. Mehlhorn. Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry. Springer-Verlag, 1984
  8. K. Mehlhorn and S. Näher. Dynamic fractional cascading. Algorithmica, 5:215-241, 1990
  9. F. P. Preparata and R. Tamassia. Efficient point location in a convex spatial cell-complex. SIAM J. Comput., 21:267-280, 1992.
  10. N. Sarnak and R. E. Tarjan. Planar point location using persistent search trees. Commun. ACM , 29:669-679, 1986. Link.[ST]
  11. J. Snoeyink. Point location. CRC Handbook of Discrete and Computational Geometry, 2nd ed., (J. Goodman and J. O’Rourke, eds.), CRC Press, New York, 2004, pp.809-837. pp. 767-786.

top

Intersection Detection

  1. I. J. Balaban. An optimal algorithm for finding segment intersections. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pp. 211-219, 1995.
  2. J. L. Bentley and T. A. Ottmann. Algorithms for reporting and counting geometric intersections. IEEE Trans. Comput., C-28:643-647, 1979.
  3. B. Chazelle and H. Edelsbrunner. An optimal algorithm for intersecting line segments in the plane. J. ACM, 39:1-54, 1992.
  4. K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry, II. Discrete Comput. Geom., 4:387-421, 1989.
  5. D. P. Dobkin, and D. G. Kirkpatrick. A linear algorithm for determining the separation of convex polyhedra. J. Algorithms, 6:381-392, 1985.
  6. D. Dobkin, J. Hershberger, D. Kirkpatrick, and S. Suri. Computing the intersection-depth of polyhedra. Algorithmica, 9:518-533, 1993.

top

Convex Hull

  1. T. M. Y. Chan. Output-sensitive results on convex hulls, extreme points, and related problems. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pp. 10-19, 1995.
  2. K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry, II. Discrete Comput. Geom., 4:387-421, 1989.
  3. R. Seidel. Convex Hull Computations. Ch. 19 in Handbook of Discrete and Computational Geometry (J. E. Goodman and J. O'Rourke, eds.). CRC Press, Boca Raton, FL, pp. 361-375, 1997.
  4. R. Seidel. Small-Dimensional Linear Programming and Convex Hulls Made Easy, DCG 6 (1991) 423-434. [Sei]
  5. R. Seidel. The upper bound theorem for polytopes: an easy proof of its asymptotic version. Comput. Geom. Theory Appl., 5:115-116, 1995.
  6. G. M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, 1994.

top

Proximity Problems

  1. F. Aurenhammer and R. Klein. Voronoi diagrams. In Handbook of Computational Geometry (J.R. Sack and J. Urrutia, Eds.). North Holland, 1998.
  2. P. B. Callahan and S.R. Kosaraju. A decomposition of multidimensional point sets with applications to k-nearest neighbors and potential fields. JACM, 42: 67-90, 1995.
  3. S. J. Fortune. A sweepline algorithm for Voronoi diagrams. Algorithmica, 2:153-174, 1987.
  4. L. J. Guibas, D. E. Knuth, and M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica, 7:381-413, 1992.

top

Arrangements

  1. P. K. Agarwal and M. Sharir. Arrangements and their applications. In Handbook of Computational Geometry (J. Sack and J. Urrutia, eds.). North-Holland, New York, pp. 49-119, 2000.
  2. T. K. Dey. Improved Bounds for Planar k -Sets and Related Problems. Discrete & Computational Geometry 19(3): 373-382 (1998).
  3. J. Matousek. Lower envelopes. Ch 7 in Lectures on Discrete Geometry. Springer-Verlag, New York, 2002.
  4. J. Matousek. Number of faces in arrangements. Ch 6 in Lectures on Discrete Geometry. Springer-Verlag, New York, 2002.
  5. J. Pach and P.K. Agarwal. Combinatorial Geometry. Wiley Interscience, 1995 (Chapters 10, 11)
  6. F. M. Sharir and P. K. Agarwal. Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York, 1995.

top

Triangulation

  1. M. Bern. Triangulations and mesh generation. In Handbook of Discrete and Computational Geometry, 2nd ed. (J. E. Goodman and J. O'Rourke, eds.). CRC Press, New York, pp. 563-582, 2004.
  2. L. P. Chew. Constrained Delaunay triangulations. Proceedings of the third annual symposium on Computational geometry, p.215-222, June 08-10, 1987, Waterloo, Ontario, Canada.
  3. H. Edelsbrunner. Triangle Meshes. Ch. 2 in Geometry and Topology for Mesh Generation. Cambridge University Press, 2001.
  4. W. Mulzer and G. Rote. Minimum Weight Triangulation is NP-hard. Proceedings of the 22nd Annual ACM Symposium on Computational Geometry (SoCG), Sedona, USA, 2006, pp. 1-10.
  5. J. Remy and A. Steger. A quasi-polynomial time approximation scheme for minimum weight triangulation. Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, May 21-23, 2006, Seattle, WA, USA.
  6. J. Ruppert. A Delaunay Refinement Algorithm for Quality 2-Dimensional Mesh Generation. J. Algorithms 18(3): 548-585 (1995).
  7. R. Seidel. A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons. Comput. Geom. Theory Appl. 1:51-64, 1991.
  8. J. R. Shewchuk. Delaunay refinement algorithms for triangular mesh generation. Computational Geometry: Theory and Applications 22(1-3):21-74, 2002.

top

Geometric Sampling

  1. P. K. Agarwal, S. Har-Peled, and K. R. Varadarajan, Geometric approximation via coresets, in Combinatorial and Computational Geometry (J. Goodman, ed.), Cambridge University Press, New York, 2005. [AHV] Link
  2. B. Chazelle. The Discrepancy Method: Randomness and Complexity. Cambridge University Press, 2000.
  3. P. Indyk. Streaming algorithms for geometric problems. In Proc. 24th Intl. Conf. Foundat. Software Tech. Theoret. Comput. Sci., pages 32-34, 2004.
  4. J. Matousek. Ch. 10 in Lectures on Discrete Geometry. Springer-Verlag New York, 2002.
  5. J. Matousek. Lectures on Discrete Geometry. Springer-Verlag, New York, 2002.
  6. S. Muthukrishnan. Data streams: algorithms and applications, Proceedings of the fourteenth annual ACM-SIAM Symposium on Discrete Algorithms, January 12-14, 2003, Baltimore, Maryland.
  7. J. Pach and P. K. Agarwal. Combinatorial Geometry, Wiley Interscience, 1995 (Chapters 15, 16).

top

Geometric Data Structures II

  1. P. K. Agarwal. Range searching, CRC Handbook of Discrete and Computational Geometry, 2nd ed., (J. Goodman and J. O'Rourke, eds.), CRC Press, New York, 2004, pp.809-837.
  2. P. K. Agarwal and J. Erickson. Geometric range searching and its relatives. Advances in Discrete and Computational Geometry, Contemp. Math. 223, Amer. Math. Soc. Press, 1998, pp. 1-56.
  3. M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. Ch. 16 in Computational Geometry: Algorithms and Applications. Springer-Verlag, 2nd ed., 2000.
  4. J. Matousek. Efficient partition trees. Discrete Comput. Geom., 8:315-334, 1992.

top

Geometric Optimization

    Linear Programming

  1. P. K. Agarwal and M. Sharir. Efficient algorithms for geometric optimization. ACM Comput. Surv. 30:412-458, 1998.
  2. S. Arora. Approximation schemes for NP-hard geometric optimization problems: A survey, Math Programming, 97, 2003.
  3. G. B. Dantzig. Linear Programming, (Historical Account).
  4. R. Motwani and P. Raghavan. Ch 9 in Randomized Algorithms. Cambridge University Press, 1995.
  5. K. L. Clarkson. Las Vegas Algorithms for Linear and Integer Programming when the Dimension is Small. J. ACM 42(2): 488-499 (1995) [Cla]

    6. (Geometric Set Cover - to be added)

    Shape Matching

  6. H. Alt and L. Guibas. Discrete geometric shapes: Matching, interpolation, and approximation, in Handbook of Computational Geometry, (J.-R. Sack and J.Urrutia, eds.), Elsevier, 1999, 121-153.
  7. P. Besl and N. McKay. A method for registration of 3-D shapes. IEEE Trans. on Pattern Analysis and Machine Intelligence, 14 (1992).
  8. R. Kolodny, P. Koehl, and M. Levitt. Comprehensive evaluation of protein structure alignment methods: scoring by geometric measures. J. Mol. Biol., 346, 1173-1188 (2005).
  9. H. Pottmann, Q. Huang, Y. Yang, and S. Hu. Geometry and convergence analysis of algorithms for registration of 3D Shapes. International Journal of Computer Vision, 67(3): 277 - 296, 2006
  10. R. C. Veltkamp, Shape matching: Similarity measures and algorithms. Tech Rept, Utrecht University, 2001.
  11. H. Wolfson and I. Rigoutsos. Geometric hashing: An overview. IEEE Computational Science and Engineering, 4 (1997), 10-21.

top

Dimension Reduction

  1. A. Gionis, P. Indyk, R. Motwani. Similarity Search in High Dimensions via Hashing. VLDB 1999: 518-529
  2. A. Andoni, P. Indyk. Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. Commun. ACM 51(1): 117-122 (2008)
  3. J. Matousek. Lectures on Discrete Geometry, Springer, 2002 (Chapter 15).

top