Fall 2017

Prerequisite:
Algorithms

Announcements:

**Instructor**:

Matya Katz (
matya@cs.bgu.ac.il )

Office hours: Sunday 12:00-14:00, Alon
building (37), room 212, Tel: (08) 6461628

**Teaching Assistant:**

Omrit
Filtser

**Class Time:**

Wednesday 12-14 (building 34, room 3)

Thursday 10-12 (building 90, room 134)

**Course Description: **

This
is an introductory course to computational geometry and its applications. We
will present data structures, algorithms and general techniques for solving
fundamental geometric problems, such as convex hull computation, line segment
intersection, orthogonal range searching, construction of Voronoi
diagram and Delaunay triangulation, polygon triangulation, and linear
programming. We will also present several geometric (optimization) algorithms
(both exact and approximate) for problems in robotics, computer graphics, GIS
(geographic information systems), communication networks, facility location,
and VLSI systems design.

The **main** textbook of the course is

[dBCvKO] *Computational Geometry: Algorithms and Applications (3rd edition),*

M. de Berg, O. Cheong, M. van Kreveld and M. Overmars,
Springer-Verlag, 2008.

**Additional** textbooks

[DO] * Discrete and Computational
Geometry*, S. Devadoss
and J. O’Rourke, Princeton University Press, 2011.

[E]
* Algorithms in Combinatorial Geometry, *H. Edelsbrunner,
Springer-Verlag, 1987.

[M]
* Computational Geometry: An Introduction Through Randomized Algorithms,
*K. Mulmuley, Prentice Hall, 1994.

[O]
*Computational Geometry in C (2nd
edition), *J. O'Rourke, Cambridge University Press, 1998.

[PS]
* Computational Geometry: An Introduction (2nd edition), *F. Preparata and M. Shamos,
Springer-Verlag, 1988.

The final grade will be determined by 3-5 homework
assignments (4% each) and a final exam.

Some of the exercises in the HW assignments are taken
from [dBCvKO].

Assignment no. 1 (due Nov. 29,
2017)

Assignment no. 2 (due Dec. 18,
2017)

Assignment no. 3 (due Jan. 10,
2018)

Assignment no. 4 (due Jan. 18,
2018)

Some old exams: exam 2005 A;
exam 2005 B;
exam 2007 A;
exam 2007 B;
exam 2009 A;
exam 2009 B

The following list of topics is tentative.

The *convex hull* of
a set of points in the plane (applications: computing the diameter and width of
a point set).

An *output sensitive*
algorithm for computing the intersection points of a set of line segments; the *plane
sweep* technique.

A representation for *planar
maps* (based on doubly-connected edge lists).

Computing the *overlay*
of two planar maps; Boolean operations on two polygons (union, intersection,
and difference).

The art gallery theorem;
introduction to *polygon triangulation*.

An O(n
log n) polygon triangulation algorithm (partitioning a polygon into y-monotone
pieces; triangulating a y-monotone polygon).

Orthogonal *range
searching*.

Computing the intersection
of n half planes in O(n log n) time.

*Linear programming* -
introduction; A randomized incremental algorithm for linear programming in the
plane.

Planar *point location*,
vertical decomposition / trapezoidal map, a randomized incremental algorithm.

Nearest site queries,
nearest site *Voronoi** diagram*.

Triangulation of a set of
points in the plane; the *Delaunay triangulation*.

*Arrangement* of
lines; *duality*; computing the *discrepancy* of a set of points in
the unit square.

*Segment trees*;
computing the area of a set of n axis-parallel rectangles in O(n
log n) time.

Simplex *range searching.*

*Hidden surface removal*:
problem definition, image space / object space, the z-buffer algorithm, depth
order, the painter's algorithm. Output sensitive hidden surface removal
algorithm for horizontal fat triangles.

Introduction to *geometric
optimization* through *facility location* optimization and *wireless
networks*.

Measures of similarity
between curves.

Below you will find, after each class, a brief summary of
the topics covered in class.

This should not be taken as a complete description of the course's content.

**25.10.17**

Introduction

**26.10.17**

The *convex hull* of a set of
points in the plane (gift wrapping, quickhull, an O(n log n)-time
incremental algorithm and a divide and conquer algorithm)

**1.11.17**

Rabin’s memorial

**2.11.17**

Sweeping; an output sensitive algorithm for line segment intersection (Omrit Filtser)

**8.11.17**

Sweeping –
extensions. Computing the diameter and width of a point set.

**9.11.17**

The doubly-connected edge list representation for planar maps

**15.11.17**

The art gallery theorem, introduction to guarding problems, introduction to polygon triangulation

**16.11.17**

Partitioning a polygon into y-monotone pieces

**22.11.17**

Triangulating a y-monotone polygon,
introduction to orthogonal range searching

**23.11.17**

Kd-Trees

**29.11.17**

Orthogonal
range trees, segment trees

**30.11.17**

Computing
the area of the union of axis-parallel rectangles

**6.12.17**

Linear
programming – an intro, computing the intersection of half-planes

**7.12.17**

A
randomized incremental algorithm for 2d linear programming

**13.12.17**

Planar point location, vertical decomposition /
trapezoidal map, a randomized incremental algorithm

**14.12.17**

Voronoi diagram

**20.12.17**

Voronoi diagram extensions and generalizations

**21.12.17**

Triangulation
of a planar point set, Delaunay triangulation

**27.12.17**

Delaunay
triangulation continued, EMST, GG, RNG

**28.12.17**

Duality and
arrangement of lines in the plane

**3.1.18**

Euclidean
TSP, Christofides algorithm

**4.1.18**

Piercing
rectangles and covering points by rectangles

**10.1.18**

Searching
in an n x n sorted matrix in O( time (Frederickson and Johnson)

**11.1.18**

**17.1.18**

**18.1.18**

**Last
update: January 4, 2017**